A Topological Approach to Indices of Geometric Operators on Manifolds with Fibered Boundaries
 171 Downloads
Abstract
In this paper, we investigate topological aspects of indices of twisted geometric operators on manifolds equipped with fibered boundaries. We define Kgroups relative to the pushforward for boundary fibration, and show that indices of twisted geometric operators, defined by complete \(\Phi \) or edge metrics, can be regarded as the index pairing over these Kgroups. We also prove various properties of these indices using groupoid deformation techniques. Using these properties, we give an application to the localization problem of signature operators for singular fiber bundles.
1 Introduction
In this paper, we consider pairs of the form \((M, \pi : \partial M \rightarrow Y)\), where M is a compact manifold with boundary \(\partial M\) which is a closed manifold, and \(\pi : \partial M \rightarrow Y\) is a smooth submersion, equivalently a fiber bundle structure, to a closed manifold Y. We call such pairs manifolds with fibered boundaries. We investigate topological aspects of indices of geometric operators, namely \(spin^c\)Dirac operators, signature operators and their twisted versions, on such manifolds. There are two purposes of this paper. The first one is to formulate the index pairing on such manifolds. We define Kgroups relative to the pushforward for boundary fibration, and show that indices of twisted geometric operators, defined by complete metrics of the form (1.2), can be regarded as the index pairing over these Kgroups. The second one is to prove properties of these indices using groupoid deformation techniques. Using these properties, we give an application to the localization problem of signature operators for singular fiber bundles.
Singular spaces arise in various areas in mathematics. In particular, stratified pseudomanifolds include many important examples of singular spaces, such as manifolds with corners and algebraic varieties. Manifolds with fibered corners arise as resolutions of stratified pseudomanifolds [ALMP12], and the simplest case, stratified manifolds of depth 1, corresponds to manifolds with fibered boundaries. There are some classes of metrics which is suited to encode the singularities of such spaces, including (complete) \(\Phi \)metrics and edge metrics. To study pseudodifferential operators with respect to such metrics, the corresponding pseudodifferential calculi, called \(\Phi \)calculus and ecalculus, were introduced by [MM98] and [Maz91]. Since then, analysis of elliptic operators in these calculi, in particular Fredholm theory and spectral theory of geometric operators, has been developed by many authors and there have been many applications to geometry of singular spaces, for example see [ALMP12, DLR15] and [LMP06].
Most of those works are analytic in nature, in the sense that they analyze individual operators under these calculi. On the other hand, it is natural to expect more topological description of Fredholm indices of these operators, as in the case of closed manifolds. One of related works in this direction is [MR06], in which they formulate the index theorem for fully elliptic operators, as an equality of analytic and topological indices defined on abelian groups of stable homotopy classes of full symbols \(K_{\Phi cu}(\phi )\), which corresponds to \(K_1(\Sigma ^{\mathring{M}}(G_\Phi ))\) in our paper. We go in this direction further, and show that, once we fix a class \(\star \) of geometric operators we are concerned with (for example \(\star \) can be \(spin^c\) or sign), the indices of twisted operators can be formulated in terms of the pairing on more primary Kgroups, \(K_*(\mathcal {A}^\star _\pi )\), “Kgroups relative to the \(\star \) pushforward for boundary fibration”. This paper is considered as a step to understand elliptic theory on singular spaces from a more topological, or Ktheoretical viewpoint.
The first main purpose of this paper is to generalize this index pairing to the case of compact manifolds with fibered boundaries. It is stated in terms of Ktheory for some \(C^*\)algebras. The \(C^*\)algebras depend on the “geometric structure” we choose to deal with, so here we explain the case of \(spin^c\)structures.
From now on, we assume n, the dimension of M, is even. A pair of the form \((E, \tilde{D}_\pi ^E)\), where E is a complex vector bundle over M satisfying \(\pi _![E] = 0\in K^0(Y)\), and the operator \(\tilde{D}_\pi ^E\) is an invertible perturbation of the fiberwise \(spin^c\)Dirac operators by lower order odd selfadjoint operators, gives a class \([(E, [\tilde{D}_\pi ^E])] \in K_1(\mathcal {A}_\pi )\) (Lemma 5.7; the bracket in \([\tilde{D}_\pi ^E]\) means that we actually only have to consider the homotopy equivalence class of invertible perturbations). Furthermore, a pair \((P'_M, P'_Y)\) of (equivalence classes of) \(spin^c\)structures on M and Y which is compatible with the one on \(\pi \) at the boundary, gives a class in \(K^1(\mathcal {A}_\pi )\). This is the element \([(P'_M, P'_Y)] \otimes _{\Sigma ^{\mathring{M}}(G_\Phi )} \partial ^{\mathring{M}}(G_\Phi ) \in KK^1(\mathcal {A}_\pi , \mathbb {C})\) appearing in Theorem 5.24.
In this situation, for each i the pair \((\pi ^{1}(V_i), \pi :\pi ^{1}(\partial V_i)\rightarrow \partial V_i )\) is a compact manifold with fibered boundary, and \(\pi \) has the structure group G. The idea is to fix an invertible perturbation of universal family of signature operators defined on the classifying space, and pullback the perturbation to define the \(\Phi \)indices of signature operators for each \(V_i\). We verify this idea and construct functions \(\sigma \) with the desired properties in the main theorem of Sect. 6, Theorem 6.2. In Sect. 7, we give a particular example of this localization problem, where the typical fiber is the two dimensional oriented closed manifold with genus \(g\ge 2\), and the group G is the hyperelliptic diffeomorphism group. This is similar to the situation considered in [End00] for the case where M is four dimensional, but we consider a more general situation where the dimension of M can be higher.
This paper is organized as follows. In Sect. 2, we give preliminaries on representable Ktheory, Lie groupoids and \(\Phi \), ecalculi. In Sects. 3 and 4, we define the indices of twisted geometric operators in \(\Phi \) and edge metrics, and prove various properties, using the groupoid deformation technique. Section 3 is about the case without the invertible perturbations, and Sect. 4 is for the case with invertible perturbations. Although the results in Sect. 3 are covered by those in Sect. 4, we separate this primitive case, because the author believes it makes it easier to understand what is going on. We note that the properties proved in these sections are not used in Sect. 5, so the readers who are only interested in the index pairing need only to check the definitions of indices given in Definitions 4.16 and 4.24, and proceed to Sect. 5. In Sect. 5, we give the formulation of the indices as index pairings over the Kgroups relative to the boundary pushforward. In Sect. 6, we give the application of the those indices to the localization problem of signature for singular fiber bundles, and in Sect. 7, we apply this to the case of singular hyperelliptic fiber bundles.
2 Preliminaries
2.1 Representable Ktheory
In this subsection, we recall the definitions for representable Ktheory in [AS]. We only work with complex coefficients.
Let H be a separable infinite dimensional Hilbert space. Let \(\hat{H} := H \oplus H\) be the \(\mathbb {Z}_2\)graded separable infinite dimensional Hilbert space. Let B(H) and K(H) denote the spaces of bounded operators and compact operators on H, respectively. For two topological spaces X and Y, let [X, Y] denote the set of homotopy classes of continuous maps from X to Y.
Definition 2.1
 (0)Let \(\mathrm {Fred}^{(0)}(\hat{H})\) denote the space of selfadjoint odd bounded Fredholm operators \(\tilde{A}\) on \(\hat{H}\) such that \(\tilde{A}^2  I \in K(\hat{H})\), with the topology coming from its embeddingHere we denoted by \(B(\hat{H})_{\mathrm {c.o.}}\) the space of bounded operators equipped with compact open topology, and by \(K(\hat{H})_{\mathrm {norm}}\) the space of compact operators equipped with norm topology.$$\begin{aligned} \mathrm {Fred}^{(0)}(\hat{H}) \rightarrow B(\hat{H})_{\mathrm {c.o.}} \times K(\hat{H})_{\mathrm {norm}} \ , \ \tilde{A} \mapsto (\tilde{A}, \tilde{A}^21). \end{aligned}$$
 (1)Let \(\mathrm {Fred}^{(1)}(H)\) denote the space of selfadjoint bounded Fredholm operators A on H such that \(A^2I \in K(H)\), with the topology coming from its embedding$$\begin{aligned} \mathrm {Fred}^{(1)}({H}) \rightarrow B({H})_{\mathrm {c.o.}} \times K({H})_{\mathrm {norm}} \ , \ {A} \mapsto ({A}, {A}^21). \end{aligned}$$
Fact 2.2
Fact 2.3
([AS04, Proposition A2.1]). The space of unitary operators on H equipped with compact open topology, denoted by \(U(H)_{\mathrm {c.o.}}\), is contractible.
Corollary 2.5
The spaces \(GL^{(0)}(\hat{H})\), \(U^{(0)}(\hat{H})\), \(GL^{(1)}({H})\) and \(U^{(1)}({H}) \) are contractible.
Proof
By Fact 2.3, the spaces \(U^{(0)}(\hat{H})\) and \(U^{(1)}({H}) \) are contractible. The map \((A, t) \rightarrow AA^{t}\) for \(t \in [0, 1]\) gives a retraction from \(GL^{(0)}(\hat{H})\) to \(U^{(0)}(\hat{H})\) and from \(GL^{(1)}({H})\) to \(U^{(1)}({H}) \), respectively. So we get the result. \(\quad \square \)
The definition of Hilbert bundles, which is suitable for our purposes, is as follows.
Definition 2.6
(Hilbert bundles). Let X be a space. A separable infinite dimensional Hilbert bundle\(\mathcal {H} \rightarrow X\) is a fiber bundle whose typical fibers are separable infinite dimensional Hilbert space H, with structure group \(U(H)_{\mathrm {c.o.}}\).
A \(\mathbb {Z}_2\)graded separable infinite dimensional Hilbert bundle\(\hat{\mathcal {H}} \rightarrow X\) is a fiber bundle whose typical fibers are \(\mathbb {Z}_2\)graded separable infinite dimensional Hilbert space H, with structure group \(U(\hat{H})_{\mathrm {c.o.}}\).
By [AS04, Proposition 3.1], the action of \(U(\hat{H})_{\mathrm {c.o.}}\) on \(\mathrm {Fred}^{(0)}(\hat{H})\) is continuous. Thus given a \(\mathbb {Z}_2\)graded Hilbert bundle \(\hat{\mathcal {H}} \rightarrow X\), we also get the associated \(\mathrm {Fred}^{(0)}(\hat{H})\)bundle \(\mathrm {Fred}^{(0)}(\hat{\mathcal {H}})\rightarrow X\). The analogous construction applies to the ungraded case.
By Fact 2.3, we have the following.
Corollary 2.7
Any separable infinite dimensional Hilbert bundle is trivial, and any choices of trivialization are homotopic.
2.2 \(\mathbb {C}l_1\)invertible perturbations

Let X be a topological space.

Let \(\hat{\mathcal {H}} = \{\hat{\mathcal {H}}_x\}_{x \in X} \rightarrow X\) be a \(\mathbb {Z}_2\)graded separable Hilbert bundle (see Definition 2.6).

Let \(\gamma \) be the involution on \(\hat{\mathcal {H}}\) defining the \(\mathbb {Z}_2\)grading.

Let \(\mathrm {Fred}^{(0)}(\hat{\mathcal {H}}) = \{\mathrm {Fred}^{(0)}(\hat{\mathcal {H}}_x)\}_{x \in X} \rightarrow X\) be the \(\mathrm {Fred}^{(0)}(\hat{H})\)fiber bundle associated to \(\hat{\mathcal {H}}\).

Assume we are given an element \(F \in \Gamma (X; \mathrm {Fred}^{(0)}(\hat{\mathcal {H}}) )\).
Definition 2.8

\(\tilde{\mathbb {F}}\in \Gamma (X\times [0,1]; \mathrm {Fred}^{(0)}(\hat{\mathcal {H}}) )\)

\(\tilde{\mathbb {F}}_{X \times \{0\}} = F\).

\(\tilde{\mathbb {F}}_{X \times \{1\}} \) is a family of invertible operators.
We introduce a natural homotopy equivalence relation on \(\tilde{\mathcal {I}}(F)\),
Definition 2.9

\(\tilde{\mathbb {F}}'' \in \Gamma (X\times [0,1]\times [0,1]; \mathrm {Fred}^{(0)}(\hat{\mathcal {H}}))\).

\(\tilde{\mathbb {F}}''_{X \times [0, 1] \times \{0\}} = \tilde{\mathbb {F}}\) and \(\tilde{\mathbb {F}}''_{X \times [0,1] \times \{1\}} = \tilde{F}'\).

\(\tilde{\mathbb {F}}''_{X \times [0,1] \times \{u\}} \in \tilde{\mathcal {I}}(F)\) for all \(u \in [0,1]\).
The following lemma follows directly from Fact 2.2.
Lemma 2.10
The element F admits a \(\mathbb {C}l_1\)invertible perturbation if and only if \([F] = 0 \in K^0(X)\).
Lemma 2.11
Suppose F satisfies \([F] = 0 \in K^0(X)\). Then \(\mathcal {I}(F)\) has a natural structure of affine space over \(K^{1}(X) (:= [X, \Omega \mathrm {Fred}^{(0)}(\hat{H})])\).
Proof
Assume we are given two elements in \(\mathcal {I}(F)\). By Corollary 2.7, we choose a trivialization of the Hilbert bundle \(\hat{\mathcal {H}} \simeq \hat{H}\times X\), which is unique up to homotopy. Take any representative of these elements and denote them by \(\tilde{\mathbb {F}}^0, \tilde{\mathbb {F}}^1 \in \tilde{\mathcal {I}}(F)\), respectively. We explain the definition of the difference class \([\tilde{\mathbb {F}}^1  \tilde{\mathbb {F}}^0] \in K^{1}(X)\).
Conversely, if we are given an element \([\tilde{\mathbb {F}}^0] \in \mathcal {I}(F)\) and an element \([\mathcal {F}] \in K^{1}(X)\), it is easy to construct the unique element \([\tilde{\mathbb {F}}^1] \in \mathcal {I}(F)\) such that \([\tilde{\mathbb {F}}^1  \tilde{\mathbb {F}}^0] = [\mathcal {F}] \). Also it is easy to see that this defines an affine structure of \(\mathcal {I}(F)\) over \(K^{1}(X)\). \(\square \)

Let \(\pi : M \rightarrow X\) be a smooth fiber bundle with closed fibers, equipped with a smooth fiberwise riemannian metric \(g_\pi \).

Let \(E \rightarrow M\) be a smooth hermitian \(\mathbb {Z}_2\)graded vector bundle.

Let \(D= \{D_x\}_{x \in X}\), \(D_x : C^\infty (\pi ^{1}(x) ; E_{\pi ^{1}(x)}) \rightarrow C^\infty (\pi ^{1}(x) ; E_{\pi ^{1}(x)})\) be a smooth family of odd formally selfadjoint elliptic operators of positive order.

Let us denote \(\hat{\mathcal {H}} = \{\hat{\mathcal {H}}_x = L^2(\pi ^{1}(x) ; E_{\pi ^{1}(x)})\}_{x \in X}\) with the natural Hilbert bundle structure over X. The operator D also denotes the closed extension to \(D : \Gamma (X; \hat{\mathcal {H}}) \rightarrow \Gamma (X; \hat{\mathcal {H}})\).
Definition 2.12

\(\tilde{D}=\{\tilde{D}_x\}_{x \in X}\) is a smooth family of invertible odd formally selfadjoint operators.

\(\tilde{D}  D = A = \{A_x\}_{x \in X}\), where \(A_x\) is a pseudodifferential operator of order 0 for each \(x \in X\).
Fact 2.15
([MP97]). The family D admits a \(\mathbb {C}l_1\)smooth invertible perturbation if and only if \([D] = 0 \in K^0(X)\).
If \([D] = 0 \in K^0(X)\), then \(\mathcal {I}_{\mathrm {sm}}(D)\) has a natural structure of an affine space over \(K^{1}(X)\), described as follows.
Remark 2.16
Actually, in [MP97] they define \(\mathbb {C}l_1\)invertible perturbations as perturbations by fiberwise smoothing operators. Our class of smooth \(\mathbb {C}l_1\)invertible perturbations in Definition 2.12 is larger because we allow the perturbations to be zeroth order operators. But divided by the homotopy equivalences, they are canonically isomorphic.
Since the above affine structure corresponds to the affine structure on \(\mathcal {I}(\psi (D))\) under the canonical map (2.13), we have the following corollary.
Corollary 2.17
With an abuse of notation we write \(\mathcal {I}(D):= \mathcal {I}(\psi (D))\) for a positive order elliptic family D.
2.3 Groupoids
2.3.1 Basic definitions.
We recall basic definitions on groupoids and pseudodifferential calculus on them. The material is taken from [DL10].
Definition 2.18

An injective map \(u : G^{(0)} \rightarrow G\), called the unit map. We often identify \(G^{(0)}\) with its image \(u(G^{(0)}) \subset G\). \(G^{(0)}\) is called the space of units.

Two surjective maps \(r, s : G \rightarrow G^{(0)}\), satisfying \(r\circ u = s \circ u = id_{G^{(0)}}\). These are called range and source map, respectively.

An involution \(i : G \rightarrow G , \gamma \mapsto \gamma ^{1}\), called the inverse map. It satisfies \(s \circ i = r\).

A map \(m : G^{(2)} \rightarrow G, (\gamma _1, \gamma _2) \mapsto \gamma _1 \cdot \gamma _2\), called product, where \(G^{(2)} = \{(\gamma _1, \gamma _2) \in G \times G s(\gamma _1) = r(\gamma _2)\}\). Moreover for \((\gamma _1, \gamma _2) \in G^{(2)}\), we have \(r(\gamma _1 \cdot \gamma _2) = r(\gamma _1)\) and \(s(\gamma _1 \cdot \gamma _2) = s(\gamma _2)\).
 The product is associative: for any \(\gamma _1\), \(\gamma _2\), \(\gamma _3\) in G such that \(s(\gamma _1) = r(\gamma _2)\) and \(s(\gamma _2) = r(\gamma _3)\), the following equality holds.$$\begin{aligned} (\gamma _1 \cdot \gamma _2) \cdot \gamma _3 = \gamma _1 \cdot (\gamma _2 \cdot \gamma _3). \end{aligned}$$

For any \(\gamma \) in G, we have \(r(\gamma ) \cdot \gamma = \gamma \cdot s(\gamma )= \gamma \) and \(\gamma \cdot \gamma ^{1} = r(\gamma )\).
Definition 2.19
(Lie groupoids). We call \(G \rightrightarrows G^{(0)}\) a Lie groupoid when G and \(G^{(0)}\) are secondcountable smooth manifolds with \(G^{(0)}\) Hausdorff, and all the structural homomorphisms are smooth and s is a submersion (for definitions of submersions between manifolds with corners, we refer to [LN01, Definition 1]).
Note that by requiring s to be a submersion, for each \(x \in G^{(0)}\), the sfiber \(G_x\) is a smooth manifold without boundary or corners.

The involution given by \(f^*(\gamma )= \overline{f(\gamma ^{1})}\).

The convolution product given by \(f*g(\gamma )=\int _{G_{s(\gamma )}} f(\gamma \eta ^{1})g(\eta )\).
Definition 2.20
Remark 2.21
In general, there are many possible \(C^*\)completion of \(C^\infty _c(G; \Omega ^{\frac{1}{2}})\) which are not necessarily isomorphic to \(C^*(G)\). For example the full \(C^*\)algebra of G is the completion of \(C^\infty _c(G; \Omega ^{\frac{1}{2}})\) with respect to all continuous representations. All the groupoids we actually use in this paper are amenable, so the full and reduced \(C^*\)algebras coincide. We use reduced \(C^*\)algebras in this paper, in order to make the argument in Sect. 2.3.3 work.
Definition 2.22

The bracket \([{\cdot },{\cdot }]_{\mathfrak {A}}\) is \(\mathbb {R}\)bilinear, antisymmetric and satisfies the Jacobi identity.

\([X, fY]_{\mathfrak {A}} = f[X ,Y]_{\mathfrak {A}} + p(X)(f)Y\) for all \(X, Y \in C^\infty (M; \mathfrak {A}) \) and \(f \in C^\infty (M)\).

\(p([X, Y]_{\mathfrak {A}}) = [p(X) , p(Y)]\) for all \(X, Y \in C^\infty (M; \mathfrak {A})\).
Given a Lie groupoid G, we associate a Lie algebroid as follows. The vector bundle is given by \(\ker (ds)_{G^{(0)}} = \cup _{x \in G^{(0)}}TG_x \rightarrow G^{(0)}\). This has the structure of a Lie algebroid over \(G^{(0)}\) with the anchor map dr. We denote this Lie algebroid by \(\mathfrak {A}G\) and call it the Lie algebroid of G.
For a Lie groupoid G, a submanifold \(V \subset G^{(0)}\) is said to be transverse to G if for each \(x \in V\), the composition \(p_x \circ \sharp _x : \mathfrak {A}_xG \rightarrow (N_V^M)_x = T_xM / T_x V\) is surjective.
Definition 2.23
 The operator D restricts to a continuous family \(\{D_x\}_{x \in G^{(0)}}\) of linear operators \(D_x : C^\infty _c(G_x; r^{*}E \otimes \Omega ^{\frac{1}{2}}) \rightarrow C^\infty (G_x; r^{*}F \otimes \Omega ^{\frac{1}{2}})\) such that$$\begin{aligned} Df(\gamma ) = D_{s(\gamma )}f_{s(\gamma )}(\gamma ) \ \forall f \in C^\infty _c(G; r^{*}E \otimes \Omega ^{\frac{1}{2}}) . \end{aligned}$$
 The following equivariance property holds:where \(U_\gamma \) is the map induced by the right multiplication by \(\gamma \).$$\begin{aligned} U_\gamma D_{s(\gamma )}= D_{r(\gamma )}U_\gamma , \end{aligned}$$

Its Schwartz kernel \(k_D\) is a distribution on G that is smooth outside \(G^{(0)}\).
 For every distinguished chart \(\psi : U \subset G \rightarrow \Omega \times s(U) \subset \mathbb {R}^{np} \times \mathbb {R}^p\) of G, the operator \((\psi ^{1})^*D\psi ^* : C^\infty _c(\Omega \times s(U); (r\circ \psi ^{1})^*E) \rightarrow C^\infty _c(\Omega \times s(U); (r\circ \psi ^{1})^*F)\) is a smooth family parametrized by s(U) of pseudodifferential operators of order m on \(\Omega \).
One can show that the space \(\Psi ^*_c(G; E)\) of compactly supported pseudodifferential Goperators on E is an involutive algebra.
Now we give important examples of Lie groupoids which are building blocks of groupoids appearing in this paper. For more examples including the ones below, see [DL10, Example 6.2 and Example 6.4].
Example 2.24
(Vector bundle groupoids). If we are given a smooth vector bundle \(\pi : E \rightarrow X\), we get a Lie groupoid \(E \rightrightarrows X\) by setting \(s = r = \pi \) and multiplication induced from the addition on \(E_x\) for each x. Choosing any smooth family of fiberwise riemannian metric on E, the \(C^*\)algebra \(C^*(E)\) is the fiberwise convolution algebra of E, and we have \(C^*(E) \simeq C_0(E^*)\) by the fiberwise Fourier transform. An Epseudodifferential operator \(D_E\) is equivalent to a family of pseudodifferential operators \(\{D_x\}_{x \in X}\) parametrized by X, and each \(D_x\) is an operator on the space \(E_x\) which is translation invariant.
Example 2.25
(Groupoids associated to fiber bundles). If we are given a smooth fiber bundle \(\pi : M \rightarrow X\), we get a Lie groupoid \(M \times _\pi M = \{(m, n) \in M \times M \  \ \pi (m) = \pi (n)\} \rightrightarrows M\). Here \(s(m, n) = n\), \(r(m ,n) = m\) and \((m, n)\cdot (n, l) = (m, l)\). Choosing any smooth family of fiberwise riemannian metric for \(\pi \), the \(C^*\)algebra \(C^*(M \times _\pi M)\) is isomorphic to \(\mathcal {K}(L^2_X(M))\), where \(L^2_X(M)\) is the Hilbert \(C_0(X)\)module given by the completion of \(C_c^\infty (M)\) by the canonical \(C_0(X)\)valued inner product, and the symbol \(\mathcal {K}\) denotes the \(C^*\)algebra of compact operators in the sense of a Hilbert module. We have the canonical Morita equivalence (for the notion of Morita equivalence, see [DL10, Section 1.2]) between \(C^*(M \times _\pi M)\) and \(C_0(X)\). An \(M\times _\pi M\)pseudodifferential operator \(D_\pi \) is equivalent to a family of pseudodifferential operators \(\{D_x\}_{x \in X}\) parametrized by X, and each \(D_x\) is an operator on \(\pi ^{1}(x)\).
2.3.2 Geometric operators.
Here we define geometric operators, such as spin Dirac operators and signature operators, on a given Lie groupoid G. For detailed discussion and other examples, we refer to [LN01].
In this paper, we often deal with \(\mathbb {Z}_2\)graded vector bundles and algebras. If we are given two \(\mathbb {Z}_2\)graded vector bundles V and W, or algebras A and B, we always consider their graded tensor product \(V \hat{\otimes } W\) and \(A \hat{\otimes } B\), following the conventions in [LM89, Section 1.1].
First we define spin Dirac operators. In order to do this, we first define our convention on spin and \(spin^c\) structures on vector bundles. Denote \(\widetilde{GL^+_k(\mathbb {R})} \rightarrow GL^+_k(\mathbb {R})\) the unique nontrivial covering of \(GL^+_k(\mathbb {R})\) for \(k \ge 2\). For \(k=1\), denote \(\widetilde{GL^+_k(\mathbb {R})} := GL^+_k(\mathbb {R}) \times \mathbb {Z}_2 \rightarrow GL^+_k(\mathbb {R})\) the projection to the first factor.
Definition 2.26
 A prespin structure on E consists of the following data \((o, P')\).

An orientation o on the vector bundle \(E \rightarrow X\).

A principal \(\widetilde{GL^+_k(\mathbb {R})}\)bundle \(P' \rightarrow X\) equipped with a bundle map \(P' \rightarrow P_{GL^+}(E)\) which is equivariant with respect to the canonical homomorphism \(\widetilde{GL^+_k(\mathbb {R})} \rightarrow GL^+_k(\mathbb {R})\). Here we denoted \(P_{GL^+}(E)\) the oriented frame bundle of E defined by o.

 A spin structure on E consists of the following data (o, g, P).

An orientation o and a riemannian metric g on the vector bundle \(E\rightarrow X\).

A principal \(Spin_k\)bundle \(P \rightarrow X\) equipped with a bundle map \(P \rightarrow P_{SO}(E)\) which is equivariant with respect to the canonical homomorphism \(Spin_k \rightarrow SO_k\).

Note that a prespin structure on E together with any riemannian metric g on E defines a spin structure on E uniquely. A prespin structure on E can also be regarded as a homotopy class of spin structures on E. See [LM89, pp. 131–132].
Definition 2.27
 A pre\(spin^c\) structure on E consists of the following data \((o, P')\).

An orientation o on the vector bundle \(E \rightarrow X\).

A principal \(\widetilde{GL^+_k(\mathbb {R})} \times _{\mathbb {Z}_2} \mathbb {C}^*\)bundle \(P' \rightarrow X\) equipped with a bundle map \(P' \rightarrow P_{GL^+}(E)\) which is equivariant with respect to the canonical homomorphism \(\widetilde{GL^+_k(\mathbb {R})} \times _{\mathbb {Z}_2} \mathbb {C}^* \rightarrow GL^+_k(\mathbb {R})\).

 A \(spin^c\) structure on E consists of the following data (o, g, P).

An orientation o and a riemannian metric g on the vector bundle \(E\rightarrow X\).

A principal \(Spin^c_k\)bundle \(P \rightarrow X\) equipped with a bundle map \(P \rightarrow P_{SO}(E)\) which is equivariant with respect to the canonical homomorphism \(Spin^c_k \rightarrow SO_k\).

 A differential \(spin^c\) structure on E consists of the following data \((o, g, P, \nabla ^L)\).

A \(spin^c\) structure (o, g, P) on E.

A unitary connection \(\nabla ^L\) on the determinant line bundle L.

Definition 2.28
A spin (prespin, \(spin^c\), pre\(spin^c\), differential \(spin^c\)) structure on a Lie groupoid G is a spin (prespin,\(spin^c\), pre\(spin^c\), differential \(spin^c\) ) structure on its Lie algebroid \(\mathfrak {A}G \rightarrow G^{(0)}\) (regarded as a vector bundle).
Suppose that we are given a spin structure on G. Let \(S \rightarrow G^{(0)}\) be the associated complex spinor bundle. The LeviCivita connection on \(r^*\mathfrak {A}G\) lifts uniquely to a connection \(\nabla ^S : C^\infty (G; r^*S) \rightarrow C^\infty (G;r^*S\otimes r^*\mathfrak {A}^*G )\) and it has a right invariance property as above. Let us denote \(c : \mathrm {Cliff}(\mathfrak {A}G) \rightarrow \mathrm {End}(S)\) the Clifford action on the spinor bundle.
Definition 2.30
Equivalently, the definition of \(D^S\) can also be described as follows. Given a spin structure on G, for each \(x \in G^{(0)}\) the spin structure on \(G_x\) is associated. If we denote \(D^S_x\) the spin Dirac operator for each \(G_x\), the family \(D^S = \{D^S_x\}_{x \in G^{(0)}}\) forms a right invariant family, and coincides with the definition given above.
This construction generalizes to Clifford modules and Dirac operators on a Lie groupoid G, defined as follows.
Definition 2.31
 A continuous linear map \(\nabla ^W : C^\infty (G; r^*W) \rightarrow C^\infty (G; r^*W \otimes r^*\mathfrak {A}^*G)\) is called an admissible connection iffor all \(\xi \in C^\infty (G; r^*W)\) and \(X,Y \in C^\infty (G; r^*\mathfrak {A}G)\).$$\begin{aligned} \nabla ^W_X (c(Y) \xi ) = c(\nabla ^{LC}_XY)\xi + c(Y) \nabla _X^W(\xi ), \end{aligned}$$

A right invariant admissible connection \(\nabla ^W\) is called a Clifford connection on W.
 For a \(\mathrm {Cliff}(\mathfrak {A}G)\)module bundle W equipped with a Clifford connection \(\nabla ^W\), the Dirac operator \(D^W \in \mathrm {Diff}^1(G; W)\) is defined byusing a local orthonormal frame \(\{e_\alpha \}_\alpha \) for \(\mathfrak {A}G\).$$\begin{aligned} D^W := \sum _\alpha c(e_\alpha )\nabla ^W_{r^*e_\alpha }, \end{aligned}$$
Example 2.32
(\(Spin^c\)Dirac operators). Let G be a Lie groupoid equipped with a differential \(spin^c\)structure. The \(spin^c\)structure on \(\mathfrak {A}G\) gives the spinor bundle \(S(\mathfrak {A}G) \rightarrow G^{(0)}\) with a \(\mathrm {Cliff}(\mathfrak {A}G)\)module structure. Moreover, as in the classical case, the unitary connection \(\nabla ^L\) on the determinant line bundle, together with the fiberwise LeviCivita connection \(\nabla ^{LC}\) for G as in (2.29), determines a Clifford connection \(\nabla ^S\) on the complex spinor bundle of \(\mathfrak {A}G\), denoted by \(S(\mathfrak {A}G)\). We call the associated Dirac operator \(D^S \in \mathrm {Diff}^1(G; S(\mathfrak {A}G))\) the \(spin^c\)Dirac operator.
Example 2.33
(Twisted \(spin^c\) Dirac operators). Let G be a Lie groupoid equipped with a differential \(spin^c\) structure. Let \(E \rightarrow G^{(0)}\) be a \(\mathbb {Z}_2\)graded hermitian vector bundle with unitary connection \(\nabla ^E\) which preserves the grading. If we denote by \(c : \mathrm {Cliff}(\mathfrak {A}G) \rightarrow \mathrm {End}(S(\mathfrak {A}G))\) the Clifford action on the spinor bundle, \(c \hat{\otimes } 1 : \mathrm {Cliff}(\mathfrak {A}G) \rightarrow \mathrm {End}(S(\mathfrak {A}G)\hat{\otimes }E)\) gives a Clifford module structure on \(S(\mathfrak {A}G)\hat{\otimes }E\).
Example 2.34
(The signature operator). Let G be a Lie groupoid equipped with a metric on \(\mathfrak {A}G\). As in the classical case, the complexified exterior algebra bundle \(\wedge _{\mathbb {C}} \mathfrak {A}^*G \rightarrow G^{(0)}\) has the \(\mathrm {Cliff}(\mathfrak {A}G)\)module structure. The fiberwise LeviCivita connection as in (2.29) induces a Clifford connection on \(\wedge _{\mathbb {C}} \mathfrak {A}^*G \rightarrow G^{(0)}\). We call the associated Dirac operator \(D^{\mathrm {sign}} \in \mathrm {Diff}^1(G; \wedge _{\mathbb {C}} \mathfrak {A}^*G )\) the signature operator on G. Of course this is the family consisting of the signature operator on \(G_x\) for each \(x\in G^{(0)}\). If the rank of \(\mathfrak {A}G\) is even (let us denote it by n), the exterior algebra bundle \(\wedge _{\mathbb {C}}\mathfrak {A}^*G\) is \(\mathbb {Z}_2\)graded by the Hodge star operator. We only consider this grading on complexified exterior algebra bundles of evenrank real vector bundles in this paper. Under this grading, the signature operators are odd.
2.3.3 Ellipticity and index classes.
From now on we assume that \(G^{(0)}\) is compact.
A Gpseudodifferential operator D is called elliptic if \(\sigma (D)\) is invertible. If \(D \in \Psi _c^m(G; E, F)\) is elliptic, as in the classical situations, it has a parametrix \(Q \in \Psi _c^{m}(G; F, E)\) such that \(DQ  \mathrm {Id} \in \Psi ^{\infty }_c(G; F)\) and \(QD  \mathrm {Id} \in \Psi ^{\infty }_c(G; E)\).
2.3.4 Deformation goupoids and blowup groupoids.
Here we recall the two constructions of groupoids; deformation to the normal cone and blowup. For details we refer to [DS17].

the inclusion \(Y \times \mathbb {R}^* \rightarrow DNC(Y, X)\).

the map \(\Theta : \Omega ' :=\{((x,\xi ), \lambda ) \in N_X^Y \times \mathbb {R} \  \ (x, \lambda \xi ) \in U' \} \rightarrow DNC(Y, X)\) defined by \(\Theta ((x, \xi ), 0) = ((x, \xi ), 0) \) and \(\Theta ((x, \xi ), \lambda )=(\theta (x, \lambda \xi ), \lambda ) \in Y\times \mathbb {R}^*\) if \(\lambda \ne 0\).
There exists a canonical action of the group \(\mathbb {R}^*\) on the manifold DNC(Y, X), called the gauge action. This is defined by, for an element \(\lambda \in \mathbb {R}^*\), \(\lambda .(w, t) = (w, \lambda t)\) and \(\lambda . ((x, \xi ), 0) = ((x, \lambda ^{1}\xi ), 0)\) (with \(t \in \mathbb {R}^*\), \(w \in Y\), \(x \in X\) and \(\xi \in ((N_X^Y)_x)\)). This action is free and locally proper on the open subset \(DNC(Y, X) \setminus X \times \mathbb {R}\).
The functoriality of Blup is described as follows. Let \(f : (Y, X) \rightarrow (Y', X') \) be a smooth map between the pair as above. Let \(U_f := DNC(Y, X)\setminus DNC(f)^{1}(X' \times \mathbb {R})\). Denote \(Blup_f(Y, X) := U_f / \mathbb {R}^*\). Then we obtain a smooth map \(Blup(f) : Blup_f(Y, X) \rightarrow Blup(Y', X')\). Similarly we obtain a smooth map \(SBlup(f) : SBlup_f(Y, X) \rightarrow SBlup(Y', X')\).
Let us explain the case where Y is a manifold with corners and X meets \(\partial Y\). X is called an interior psubmanifold of Y if it is a smooth submanifold which meets all the boundary faces of Y transversally, and covered by coordinate neighborhoods \(\{U, (v, w)\}\) in Y such that v is a tuple of boundary defining functions on Y and \(U \cap X = \{w_i = 0 \  \ i = 1, \ldots , \mathrm {codim} X\}\). If X is an interior psubmanifold of Y, we consider the inward normal bundle \((N_X^{Y})^+\) and we can define \(DNC_+(Y, X) = (N_X^Y)^+ \times \{0\}\sqcup Y \times \mathbb {R}^*_+\). This manifold admits the gauge action by \(\mathbb {R}^*_+\). We define SBlup(Y, X) by the same formula as above.
2.4 b, \(\Phi \), ecalculus and corresponding groupoids

Let \((M, \partial M)\) be a compact manifold with closed boundaries. Here closed means that \(\partial M\) is a compact manifold without boundary.

Let \(\partial M = \sqcup _{i = 1}^m H_i\) be the decomposition into connected components.

Let \(\pi _i : H_i \rightarrow Y_i\) be a smooth oriented fiber bundle structure with closed fibers. The typical fibers are allowed to vary from one component to another. We also denote \(Y = \sqcup _{i} Y_i\) and \(\pi : \sqcup _{i} \pi _i\).

Let \(x \in C^\infty (M)\) be a boundary defining function. Here a boundary defining function is a smooth function x on M such that \(x^{1}(0) = \partial M\), \(x > 0\) on \(\mathring{M}\) and \(dx(p) \ne 0\) for all \(p \in \partial M\).
Denote by \(\Omega ^b\), \(\Omega ^\Phi \) and \(\Omega ^e\) the bundle of smooth densities on the vector bundle \(T^bM\), \(T^\Phi M\) and \(T^eM\), respectively, and we call them b, \(\Phi \), edensity bundles, respectively.
We define the space of b, \(\Phi \), and epseudodifferential operators. Let \(\mathrm {Diff}^*_b(M)\) denote the filtered algebra generated by \(\mathcal {V}_b(M)\) and \(C^\infty (M)\). An element in this algebra is called a bdifferential operator. The space of bpseudodifferential operators contains this algebra. We define the algebra \(\mathrm {Diff}^*_\Phi (M)\) and \(\mathrm {Diff}^*_e(M)\) in the analogous way, and the analogous result holds. This space of pseudodifferential operators can be described in two ways, microlocal approach and groupoid approach. The microlocal approach originates from Melrose [Mel93] for the bcase, and the \(\Phi \)case was given by Mazzeo and Melrose [MM98] and the ecase was given by Mazzeo [Maz91]. In this paper, we use the groupoid approach, which is more suited with Kthoretic approach using \(C^*\)algebras, as explained below. For relations between these two approaches, see [PZ19, Section 6.6].
2.4.1 The groupoid approach.

The bgroupoids.
We start with the pair groupoid \(M \times M \rightrightarrows M\). Note that this does not satisfy the definition of Lie groupoid given in Definition 2.19, since s is not a submersion; however it is easy to see that the spherical blowup construction is also valid in this case. Consider the subgroupoid \(\sqcup _i (H_i \times H_i) \rightrightarrows \partial M\) of \(M\times M\). Then bgroupoid of M is defined by\(\mathring{M}\) and \(H_i\), \(1 \le i \le m\) are saturated subsets of \(G_b\), and we have$$\begin{aligned} G_b := SBlup_{r, s}(M \times M, \sqcup _i (H_i \times H_i) ) \rightrightarrows M. \end{aligned}$$$$\begin{aligned} G_b = \mathring{M} \times \mathring{M} \sqcup \sqcup _i (H_i \times H_i \times \mathbb {R}) \rightrightarrows M. \end{aligned}$$ 
The \(\Phi \)groupoids.
Consider the subgroupoid \(\partial M \times _\pi \partial M = \sqcup _{i} (H_i \times _{\pi _i} H_i) \rightrightarrows \partial M\) of \(G_b\). Then \(\Phi \)groupoid of M is defined byLet us look at the singular part. The inward normal bundle groupoid of \(H_i \times _{\pi _i} H_i\) in \(G_b\) is$$\begin{aligned} G_\Phi := SBlup_{r, s}(G_b, \partial M \times _\pi \partial M ) \rightrightarrows M. \end{aligned}$$And the gauge action by \(\lambda \in \mathbb {R}_+^*\) is given by \((x, v, y, a, b) \mapsto (x, \lambda v, y, a, \lambda b)\). Thus dividing by this gauge action, we get an isomorphism$$\begin{aligned} H_i \times _{\pi _i} TY_i \times _{\pi _i} H_i \times \mathbb {R} \times \mathbb {R}_+&\rightrightarrows H_i \times \mathbb {R}_+ \\ s(x, v, y, a, b)&= (y, b) \\ r(x, v, y, a, b)&= (x, b) \\ m((x, v, y, a, b), (y, w, z, a', b))&= (x, v+w, z, a+a', b). \end{aligned}$$(this can be seen by restricting to \(b = 1\)). In other words we have$$\begin{aligned} G_\Phi _{H_i} \simeq H_i \times _{\pi _i} H_i \times _{\pi _i} TY_i\times \mathbb {R} \end{aligned}$$$$\begin{aligned} G_\Phi = \mathring{M} \times \mathring{M} \sqcup \partial M \times _\pi \partial M \times _\pi TY \times \mathbb {R} \rightrightarrows M. \end{aligned}$$ 
The egroupoids.
Consider the groupoid \(M \times M \rightrightarrows M\) and its subgroupoid \(\partial M \times _\pi \partial M =\sqcup _i (H_i \times _{\pi _i} H_i) \rightrightarrows \partial M\). Then egroupoid of M is defined byLet us look at the singular part. The inward normal bundle groupoid of \( H_i \times _{\pi _i} H_i\) in \(M \times M\) is$$\begin{aligned} G_e = SBlup_{r, s}(M \times M, \partial M \times _\pi \partial M ) \rightrightarrows M. \end{aligned}$$And the gauge action by \(\lambda \in \mathbb {R}_+^*\) is given by \((x, v, y, a, b) \mapsto (x, \lambda v, y, \lambda a, \lambda b)\). So dividing by this action, we get$$\begin{aligned} H_i \times _{\pi _i} TY_i \times _{\pi _i} H_i \times ( \mathbb {R}_+)^2&\rightrightarrows H_i \times \mathbb {R}_+ \\ s(x, v, y, a, b)&= (y, b) \\ r(x, v, y, a, b)&= (x, a) \\ m((x, v, y, a, b), (y, w, z, b, c))&= (x, v+w, z, a, c). \end{aligned}$$where \(\mathbb {R}_+^*\) acts on \(TY_i\) by multiplication.$$\begin{aligned} G_e_{H_i} \simeq H_i \times _{\pi _i} H_i \times _{\pi _i} (TY_i \rtimes \mathbb {R}_+^*) \end{aligned}$$
3 Indices of Geometric Operators on Manifolds with Fibered Boundaries: The Case Without Perturbations
3.1 The definition of indices
In Sects. 3.1 and 3.2, for simplicity we only consider spin Dirac operators, without any twists or perturbations. For our conventions on spin structures and prespin structures on vector bundles, see Definition 2.26.
For a given even dimensional compact manifold with fibered boundary \((M^{\mathrm {ev}}, \pi : \partial M \rightarrow Y)\) equipped with prespin structures on TM and TY as well as a riemannian metric on the vertical tangent bundle of the boundary fibration, \(T^V\partial M\), for which the fiberwise spin Dirac operator forms an invertible family, we associate its index in \(\mathbb {Z}\). This index can be realized using either \(\Phi \)metrics or emetrics. In the next section, we show that they actually coincide. Also we show some properties of this index, using groupoid deformation techniques. For simplicity, we only work in the case where Y is odd dimensional. The case where Y is even dimensional can be treated similarly.
Remark 3.1
For a manifold with fibered boundary \((M, \pi : \partial M \rightarrow Y)\), assume that we are given prespin structures on TM and TY. The prespin structure on TM induces a prespin structure on \(T\partial M\). Choose any splitting \(T\partial M = T^V\partial M \oplus \pi ^*TY\). We introduce the pullback prespin structure on \(\pi ^*TY\). Then a prespin structure on \(T^V\partial M\) is induced, and it does not depend on the choice of the splitting of \(T\partial M\). We always consider this choice of prespin structure on \(T^V\partial M\). In particular, when we are given prespin structures on TM and TY as well as a riemannian metric on \(T^V\partial M\), a spin structure on \(T^V\partial M\) is canonically induced and the fiberwise spin Dirac operator \(D_\pi \) is defined.
First, we show that for a fixed spin structure on \(T^\Phi M\) or \(T^e M\) which has a product decomposition at the boundary, we get the Fredholmness from the invertibility of the fiberwise Dirac operators.
Let \((M^{\mathrm {ev}}, \pi : \partial M \rightarrow Y^{\mathrm {odd}})\) be a compact manifold with fibered boundary, equipped with prespin structures on TM and TY, as well as a riemannian metric \(g_\pi \) on \(T^V\partial M\).
Proposition 3.2
In the above settings, assume that the family \(D_\pi \) is invertible. Then both \(D_\Phi \) and \(D_e\) are Fredholm, as operators on \(\mathring{M}\) with metric induced from \(g_\Phi \) and \(g_e\), respectively.
Proof
 (1)
Prespin structures \(P'_M\) and \(P'_Y\) on TM and TY, respectively, are fixed.
 (2)
A riemannian metric \(g_\pi \) on \(T^V\partial M\) is fixed. Assume that the associated fiberwise spin Dirac operator \(D_\pi \) is invertible.
 (3)A smooth riemannian metric \(g_\Phi \) for \(\mathfrak {A}G_\Phi \simeq T^\Phi M \rightarrow M\), whose restriction to \(\mathfrak {A}G_\Phi _{\partial M} = T^V\partial M \oplus \pi ^*TY \oplus \mathbb {R}\) can be written aswhere \(g_{TY \oplus \mathbb {R}}\) is some riemannian metric on the vector bundle \(TY \oplus \mathbb {R} \rightarrow Y\).$$\begin{aligned} g_\Phi _{\partial M} = g_\pi \oplus \pi ^*g_{TY \oplus \mathbb {R}} , \end{aligned}$$
 (4)A smooth riemannian metric \(g_e\) for \(\mathfrak {A}G_e \simeq T^e M \rightarrow M\), whose restriction to \(\mathfrak {A}G_e _{\partial M} = T^V\partial M \oplus \pi ^*TY \oplus \mathbb {R}\) can be written aswhere \(g_{TY \oplus \mathbb {R}}\) is some riemannian metric on the vector bundle \(TY \oplus \mathbb {R} \rightarrow Y\).$$\begin{aligned} g_e_{\partial M} = g_\pi \oplus \pi ^*g_{TY \oplus \mathbb {R}} , \end{aligned}$$
 (5)
Let us denote the spin Dirac operators associated to \(g_\Phi \) and \(g_e\) by \(D_\Phi \) and \(D_e\), respectively.
Proposition 3.6
(Stability). Under the above situations, \(\mathrm {Ind}_{\mathring{M}}(D_\Phi )\) and \(\mathrm {Ind}_{\mathring{M}}(D_e)\) only depend on the data (1) and (2) above. It does not depend on the choice of \(g_\Phi \) and \(g_e\) which satisfy the conditions (3) and (4) above.
Proof
This can be proved by a simple homotopy argument. We prove in the \(\Phi \)case. The ecase is similar. Let \(g_\Phi ^0\) and \(g_\Phi ^1\) be two choices of smooth metrics on \(T^\Phi M\) which satisfies the condition (3) (for the same fiberwise metric \(g_\pi \)). Let us denote \(D_\Phi ^0\) and \(D_\Phi ^1\) the spin Dirac operator with respect to these metrics. Letting \(g_\Phi ^t = tg_\Phi ^0 + (1t)g_\Phi ^1\) for \(t \in [0,1]\), we get a smooth path of riemannian metrics \(T^\Phi M\) connecting \(g_\Phi ^0\) and \(g_\Phi ^1\). Note that for all \(t\in [0, 1]\), \(g_\Phi ^t\) satisfies the condition (3).
By Proposition 3.6, in order to define the indices of spin Dirac operator \(D_\Phi \) and \(D_e\), we only have to specify the data (1) and (2) listed before the Proposition 3.6. So we define the index of the triple \((P'_M, P'_Y, g_\pi )\) by the above number.
Definition 3.7
3.2 Properties
First, we show that two indices \(\mathrm {Ind}_\Phi (P'_M, P'_Y, g_\pi )\) and \(\mathrm {Ind}_e(P'_M, P'_Y, g_\pi ) \) actually coincide.
Proposition 3.8
Proof

\(\mathfrak {A}_{M \times \{t\}} = \mathfrak {A}G_e\) for all \(t \in (0, 1]\).

\(\mathfrak {A}_{M \times \{0\}} = \mathfrak {A}G_\Phi \).
 The metric \(g_{\mathfrak {A}}\) on \(\mathfrak {A}\), defined as (see (3.9))gives a smooth metric on \(\mathfrak {A}\).$$\begin{aligned} g_{\mathfrak {A}} := {\left\{ \begin{array}{ll} g_e(t) &{} \text{ on } M \times (0, 1]_t \\ g_\Phi &{} \text{ on } M \times \{0\}, \end{array}\right. } \end{aligned}$$
Next we show the gluing formula.
Proposition 3.10

\(M^0\) and \(M^1\) are manifolds with fibered boundaries as above, equipped with prespin structures \(P'_{M^i}\) and \(P'_{Y^i}\) on \(TM^i\) and \(TY^i\), respectively, and a riemannian metric \(g_{\pi ^i}\) on \(T^V\partial {M^i}\), for \(i = 0, 1\).

Assume that on some components of \(\partial M^0\) and \(\partial M^1\), we are given isomorphisms of the data \((\pi ^i, P'_{M^i}, P'_{Y^i}, g_{\pi ^i})\) restricted there.

\((M, \partial M, \pi ', Y')\) : the manifold with fibered boundary obtained by the above isomorphism of some boundary components. This manifold is equipped with the prespin structures \(P'_M\) and \(P'_{Y'}\) on TM and \(TY'\), respectively, and a riemannian metric \(g_{\pi '}\) on \(T^V\partial M\) induced by the ones on \(M^i\).

Assume that on each boundary components of \(M^0\) and \(M^1\), the fiberwise spin Dirac operators are invertible.
Proof
We use a similar argument to the one in Proposition 3.8. For simplicity we consider the case where the boundary of each \(M^0\) and \(M^1\) consists of one component, and the isomormorphism is given between \(\partial M^0\) and \(\partial M^1\). In particular, the resulting manifold M is a closed manifold in this case. The general case can be shown in an analogous way. We denote the image of \(\partial M^0 \simeq \partial M^1\) in M by \(H \subset M\), which is a closed hypersurface. Also we denote \(\pi : H \rightarrow Y\) the fiber bundle structure induced from the ones on \(\partial M^0 \simeq \partial M^1\) and the given fiberwise metric as \(g_\pi \).

\(\mathfrak {A}_{M \times \{t\}} = \mathfrak {A}(M \times M)=TM\) for all \(t \in (0, 1]\).

\(\mathfrak {A}_{M \times \{0\}} = \mathfrak {A}G_e^0 \cup _{H} \mathfrak {A} G_e^1\). Here we denoted the egroupoid of \(M^i\) by \(G_e^i \) for \(i = 0, 1\).
 The metric \(g_{\mathfrak {A}}\) on \(\mathfrak {A}\), defined asgives a smooth metric on \(\mathfrak {A}\). Here x is a defining function for \(H \subset M\) and t is the [0, 1]coordinate in \(M \times [0, 1]\). The metric \(g_Y\) can be any metric on Y.$$\begin{aligned} g_{\mathfrak {A}} := \frac{dx^2}{x^2 + t^2}\oplus \frac{\pi ^*g_Y}{x^2+t^2} \oplus g_\pi \end{aligned}$$
Proposition 3.12
Proof
Next we show the vanishing formula for the case where the spin fiber bundle structure (preserving the boundary) extends to the whole manifold, and the fiberwise operators are invertible for the whole family.
Proposition 3.13

Let \((M^{\mathrm {ev}}, \partial M, \pi : \partial M \rightarrow Y^{\mathrm {odd}})\) be a compact manifold with fibered boundary, equipped with prespin structures \(P'_M\) and \(P'_Y\) on TM and TY, respectively, and a riemannian metric \(g_\pi \) on \(T^V\partial M\), for which the fiberwise spin Dirac operator \(D_\pi \) is an invertible family.
 There exist data \((\pi ', X, P'_X, g_{\pi '})\) such that

X is a compact manifold with boundary \(\partial X\), with a fixed diffeormorphism \(\partial X \simeq Y\). We identify \(\partial X\) with Y.

\(\pi ' : (M, \partial M) \rightarrow (X, \partial X)\) is a fiber bundle structure which preserves the boundary, and \(\pi '_{\partial M} = \pi \). Note that the typical fibers of \(\pi \) and \(\pi '\) are the same.

\(P'_X\) is a prespin structure on TX which satisfies \(P'_X _Y = P'_Y\).

\(g_{\pi '}\) is a riemannian metric on \(T^VM\) (the fiberwise tangent bundle of the fiber bundle \(\pi '\)) satisfying \(g_{\pi '} _{Y} = g_\pi \). We denote \(D_{\pi '}\) the family of fiberwise spin Dirac operators for \(\pi '\).

Proof
The first equality follows from Proposition 3.8. Consider the subgroupoid \(M \times _{\pi '} M \subset G_e\) and define \(\mathcal {G} := DNC(G_e, M \times _{\pi '}M)_{M \times [0, 1]}\). Denote the closed saturated subset \(M_1 := M \times \{0\} \cup \partial M \times [0, 1] \subset M \times [0, 1]\) for this groupoid. Note that we have \(\mathcal {G}_{\partial M \times [0, 1]} = \partial M \times _\pi \partial M \times _\pi DNC(TY \rtimes \mathbb {R}^*_+, Y)_{Y \times [0, 1]}\). We can also see that the restriction \(\mathcal {G}_{M \times \{0\}}\) is of the form \(M \times _{\pi '} M \times _{\pi '} E_X\), where \(E_X \rightarrow X\) is a vector bundle over X. In particular, there exists canonical direct sum decomposition of \(\mathfrak {A}\mathcal {G}_{M_1}\) such that one component is \(T^VM_1\). Choose any riemannian metric \(g_\mathfrak {A}\) on \(\mathfrak {A}\mathcal {G}\) such that, on \(M_1\), the two direct sum components are orthogonal, and \(T^VM_1\)component is equal to \(g_{\pi '}\cup g_\pi \times [0, 1]\). We consider the spin structure on \(\mathfrak {A}\mathcal {G}\) defined by the given data and metric \(g_\mathfrak {A}\) chosen above, and consider the spin Dirac operator \(\mathcal {D} \in \mathrm {Diff}^1(\mathcal {G}; S(\mathfrak {A}\mathcal {G}))\).
3.3 The cases of twisted \(spin^c\) and signature operators
The above argument easily generalizes to the cases of twisted \(spin^c\)Dirac operators and twisted signature operators, as follows. Let \((M^{\mathrm {ev}}, \pi : \partial M \rightarrow Y^{\mathrm {odd}})\) be a compact manifold with fibered boundary, and \(E \rightarrow M\) be a \(\mathbb {Z}_2\)graded complex vector bundle.
3.3.1 Twisted \(spin^c\) Dirac operators.
 (D1)
Pre\(spin^c\) structures \(P'_M\) and \(P'_Y\) on TM and TY, respectively.
 (D2)
A differential \(spin^c\) structure \(P_\pi \) on \(T^V\partial M\), which is compatible with the pre\(spin^c\)structure induced from \(P'_M\) and \(P'_Y\) (see Remark 3.1).
 (D3)A hermitian structure on \(E_{\partial M}\) as well as a smooth family of fiberwise unitary connection for the boundary fibration, i.e., a continuous mapgiven by a family of unitary connections \(\{\nabla _y^E\}_{y \in Y}\) on the vector bundle \(E_{\pi ^{1}(y)} \rightarrow \pi ^{1}(y)\) for each \(y\in Y\).$$\begin{aligned} \nabla ^E_\pi : C^\infty (\partial M; E_{\partial M}) \rightarrow C^\infty (\partial M; E_{\partial M} \otimes (T^{V}\partial M)^*) \end{aligned}$$
 (D4)
Denote the fiberwise twisted \(spin^c\)Dirac operators for \(\pi \) by \(D^E_\pi = \{D^E_{\pi ^{1}(y)}\}_{y \in Y}\). Here \(D^E_{\pi ^{1}(y)}\) acts on \( C^\infty (\pi ^{1}(y); S(\pi ^{1}(y)) \hat{\otimes } E)\). We assume that \(D^E_\pi \) forms an invertible family.
 (d1)A differential \(spin^c\) structure on \(\mathfrak {A}G_\Phi \) (\(\mathfrak {A}G_e\)) such that

it is compatible with the pre\(spin^c\) structures in (D1).

it has a product structure with respect to the decomposition \(\mathfrak {A}G_\Phi _{\partial M} = T^V\partial M \oplus \pi ^*TY \oplus \mathbb {R}\) (\(\mathfrak {A}G_e_{\partial M} = T^V\partial M \oplus \pi ^*TY \oplus \mathbb {R}\)) at the boundary.

the \(T^V\partial M\)component coincides with the one in (D2).

 (d2)
A hermitian structure on E which restricts to the one given in (D3), and a unitary connection \(\nabla ^E\) which restricts to \(\nabla ^E_\pi \) in (D3).
Definition 3.15
We can show the equality \(\mathrm {Ind}_\Phi (P'_M,P'_Y, P_\pi , E, \nabla ^E_\pi ) = \mathrm {Ind}_e(P'_M, P'_Y, P_\pi , E, \nabla ^E_\pi )\) as in Proposition 3.8. The gluing formula as in Proposition 3.10 and the vanishing property as in Proposition 3.13 hold analogously.
3.3.2 Twisted signature operators.
 (D1)
A riemanian metric \(g_\pi \) on \(T^V\partial M\).
 (D2)A hermitian structure on \(E_{\partial M}\) as well as a smooth family of fiberwise unitary connection for the boundary fibration, i.e., a continuous mapgiven by a smooth family of unitary connections \(\{\nabla _y^E\}_{y \in Y}\) on \(E_{\pi ^{1}(y)} \rightarrow \pi ^{1}(y)\) for each \(y\in Y\).$$\begin{aligned} \nabla ^E_\pi : C^\infty (\partial M; E_{\partial M}) \rightarrow C^\infty (\partial M; E_{\partial M} \otimes (T^{V}\partial M)^*) \end{aligned}$$
 (D3)
Denote the fiberwise twisted signature operators for \(\pi \) by \(D^{\mathrm {sign}, E}_\pi = \{D^{\mathrm {sign}, E}_{\pi ^{1}(y)}\}_{y \in Y}\). Here \(D^{\mathrm {sign}, E}_{\pi ^{1}(y)}\) acts on \( C^\infty (\pi ^{1}(y); \wedge _{\mathbb {C}} T^*(\pi ^{1}(y)) \hat{\otimes } E)\). We assume that \(D^{\mathrm {sign}, E}_\pi \) forms an invertible family.
 (d1)
 A smooth riemannian metric \(g_\Phi \) for \(\mathfrak {A}G_\Phi \simeq T^\Phi M \rightarrow M\), whose restriction to \(\mathfrak {A}G_\Phi _{\partial M} = T^V\partial M \oplus \pi ^*TY \oplus \mathbb {R}\) can be written aswhere \(g_{TY \oplus \mathbb {R}}\) is some riemannian metric on \(TY \oplus \mathbb {R} \rightarrow Y\).$$\begin{aligned} g_\Phi _{\partial M} = g_\pi \oplus \pi ^*g_{TY \oplus \mathbb {R}} , \end{aligned}$$
 (d1)\('\)
 A smooth riemannian metric \(g_e\) for \(\mathfrak {A}G_e \simeq T^e M \rightarrow M\), whose restriction to \(\mathfrak {A}G_e _{\partial M} = T^V\partial M \oplus \pi ^*TY \oplus \mathbb {R}\) can be written aswhere \(g_{TY \oplus \mathbb {R}}\) is some riemannian metric on \(TY \oplus \mathbb {R} \rightarrow Y\).$$\begin{aligned} g_e_{\partial M} = g_\pi \oplus \pi ^*g_{TY \oplus \mathbb {R}} , \end{aligned}$$
 (d2)

A hermitian structure on E which restricts to the one given in (D3), and a unitary connection \(\nabla ^E\) which restricts to \(\nabla ^E_\pi \) in (D3).
Definition 3.16
We can show the equality \(\mathrm {Sign}_\Phi ( M, g_\pi , E, \nabla ^E_\pi ) = \mathrm {Sign}_e(M, g_\pi , E, \nabla ^E_\pi )\) as in Proposition 3.8. The gluing formula as in Proposition 3.10 and the vanishing property as in Proposition 3.13 holds analogously.
4 Indices of Geometric Operators on Manifolds with Fibered Boundaries: The Case with Fiberwise Invertible Perturbations

on the interior \(\mathring{M}\), \(\tilde{D}\) differs from \(D_\Phi \) (\(D_e\)) by an operator of order 0.

the boundary operator of \(\tilde{D}\) is given by \(\tilde{D}_\pi \hat{\otimes } 1 + 1 \hat{\otimes }D_{TY \times \mathbb {R}}\) (\(\tilde{D}_\pi \hat{\otimes } 1 + 1 \hat{\otimes }D_{TY \rtimes \mathbb {R}}\)).
4.1 The general situation

Let M be a compact manifold possibly with boundaries and corners.

Let \(G \rightrightarrows M\) be a Lie groupoid.

Let \(V \subset M\) be a closed saturated subset for G.

Let \((\sigma _M, \tilde{F}_V) \in C(\mathfrak {S}^*G_M) \oplus _V \overline{\Psi _c^0(G_V)}\) be an invertible element.

Let \((\sigma _M, F_V) \in \Sigma ^{M \setminus V}(G)\) be an element such that \(\sigma _M \in C(\mathfrak {S}^*G_M)\) is invertible.
 Let \(F_{V \times [0, 1]} = \{F_{V \times \{t\}}\}_{t \in [0, 1]}\) be a continuous path of operators \(F_{V \times \{t\}} \in \overline{\Psi _c^0(G_V)}\) parametrized by \(t \in [0, 1]\) such thatWe call such a path “an invertible perturbation for \(F_V\)”.

\(F_{V\times \{0\}} = F_V\).

\(F_{V\times \{t\}}\) is elliptic for all \(t \in [0, 1]\).

\(F_{V \times \{1\}}\) is invertible.

Remark 4.1

An element \(F_V \in \overline{\Psi ^0_c(G_V)}\) for which \(\sigma (F_V) \in C(\mathfrak {S}^*G_V)\) is invertible, and

An invertible element \(\tilde{F}_V \in \overline{\Psi ^0_c(G_V)}\) which satisfies \(\tilde{F}_V  F_V \in C^*(G_V)\).
Definition 4.3
Remark 4.5
As in Sect. 2.2, we denote by \(\tilde{\mathcal {I}}(F_V)\) the set of invertible perturbations for the operator \(F_V\). This set has the obvious homotopy relation, and we denote \(\mathcal {I}(F_V)\) the set of homotopy classes of elements in \( \tilde{\mathcal {I}}(F_V)\). We can show that \(\mathcal {I}(F_V)\) is nonempty if and only if \(\mathrm {Ind}(F_V) = 0 \in K_0(C^*(G_V))\). The above definition of the difference class induces the affine space structure on \(\mathcal {I}(F_V)\) modeled on \(K_1(C^*(G_V))\).
Remark 4.6
Proposition 4.7
Proof

\(M_t := M \cup _{V \times \{0\}} V \times [0, t]\) and \(\mathring{M}_t := M \cup _{V \times \{0\}} V \times [0, t)\),

\(G_t := G \cup _{V \times \{0\}} G_V \times [0, t] \rightrightarrows M_t\) and \(\mathring{G}_t := G_t_{\mathring{M}_t}\)

The inclusion which gives a KKequivalence \( i_t : C^*(G_{M\setminus V}) \rightarrow C^*(\mathring{G}_t) \).
If we have two invertible perturbations \(F^i_{V \times [0, 1]}\) (\(i = 0, 1\)) which define the same class in \(\mathcal {I}(F_V)\), the difference class \([F^1_{V \times [0, 1]}  F^0_{V \times [0, 1]}]\) vanishes, so we have \(\mathrm {Ind}_{M \setminus V}(\sigma _M, F_{V \times [0, 1]}^1) = \mathrm {Ind}_{M \setminus V}(\sigma _M, F_{V \times [0, 1]}^0 )\). \(\quad \square \)
Remark 4.9
4.2 The connecting elements of \(G_\Phi \) and \(G_e\)
Lemma 4.10
 \((\Phi )\)

Under the Morita equivalence between \(G_\Phi _{\partial M} \) and \( TY \times \mathbb {R}\), the element \(\partial ^{\mathring{M}}(G_\Phi ) \in KK^1(C^*(G_\Phi _{\partial M}) , C^*(G_\Phi _{\mathring{M}})) \simeq KK(C^*(TY), \mathbb {C})\) identifies with the element \([\sigma _Y]\).
 (e)

Under the Morita equivalence between \(G_\Phi _{\partial M}\) and \( TY \rtimes \mathbb {R}_+^*\) and the \(KK^1\)equivalence between \(C^*(TY \rtimes \mathbb {R}_+^*)\) and \( C^*(TY) \) given by the ConnesThom isomorphism, the element \(\partial ^{\mathring{M}}(G_e) \in KK^1(C^*(G_e_{\partial M}) , C^*(G_e_{\mathring{M}})) \simeq KK(C^*(TY), \mathbb {C})\) identifies with the element \([\sigma _Y]\).
Proof
4.3 The definitions and relative formulas for the \(\Phi \) and eindices
We apply this general construction to our settings.
4.3.1 Twisted \(spin^c\)Dirac operators.
Here we explain the case for twisted \(spin^c\)Dirac operator. First we give a fundamental remark on the space of \(\mathbb {C}l_1\)invertible perturbations of geometric operators.
Remark 4.14
Let X be a closed manifold equipped with a pre\(spin^c\) structure, and \(E \rightarrow X\) be a \(\mathbb {Z}_2\)graded complex vector bundle. In order to define the twisted \(spin^c\)Dirac operator \(D^E\), we have to specify a differential \(spin^c\)structure, a hermitian metric on E and a unitary connection on E. However, since the space of these choices is contractible, the sets of homotopy classes of \(\mathbb {C}l_1\)invertible perturbations, \(\mathcal {I}(D^E)\), for two different choices are canonically isomorphic.
 (D1)
Pre\(spin^c\) structures \(P'_M\) and \(P'_Y\) on TM and TY, respectively. These induce a pre\(spin^c\) structure on \(T^V\partial M\), denoted by \(P'_\pi \).
 (D2)
A homotopy class of \(\mathbb {C}l_1\)invertible perturbation \(Q_\pi \in \mathcal {I}(P'_\pi , E)\).
 (d1)A differential \(spin^c\) structure on \(\mathfrak {A}G_\Phi \) (\(\mathfrak {A}G_e\)) such that

it is compatible with the pre\(spin^c\) structures in (D1).

it has a product structure with respect to the decomposition \(\mathfrak {A}G_\Phi _{\partial M} = T^V\partial M \oplus \pi ^*TY \oplus \mathbb {R}\) (\(\mathfrak {A}G_e_{\partial M} = T^V\partial M \oplus \pi ^*TY \oplus \mathbb {R}\)) at the boundary.

 (d2)
A hermitian structure on E and a unitary connection \(\nabla ^E\). Denote the fiberwise twisted \(spin^c\)Dirac opeartor \(D_\pi ^E\).
 (d3)
A family of operators \(\tilde{D}^E_\pi \in \tilde{\mathcal {I}}_{\mathrm {sm}}(D^E_\pi )\) which is a representative of the class \(Q_{\pi } \in \mathcal {I}(P'_\pi , E)\) in (D2).
In the ecase, \(D_e_{\partial M}\) also has the product form as in (3.5), so we define an invertible operator \(\tilde{D}^{S\hat{\otimes }E}_{e, \partial M}\) in an analogous way.
Definition 4.16
Proposition 4.18
(The gluing formula).

Let \((M^0, \pi ^0 : \partial M^0 \rightarrow Y^0, E^0 \rightarrow M^0)\) and \((M^1, \pi ^1 : \partial M^1 \rightarrow Y^1, E^1 \rightarrow M^1)\) be manifolds with fibered boundaries equipped with complex vector bundles.

Assume we are given data \((P'_{M^i}, P'_{Y^i}, Q_{\pi ^i})\) satisfying the conditions (D1) and (D2) above for each \(i = 0, 1\).

Assume that on some components of \(\partial M^0\) and \(\partial M^1\), we are given isomorphisms of the data \((\pi ^i, P'_{M^i}, P'_{Y^i}, E^i, Q_{\pi ^i})\) restricted there.

Let us denote \((M, \pi ' : \partial M \rightarrow Y')\) the manifold with fibered boundary obtained by identifying isomorphic boundary components. This manifold is equipped with data \(( P'_{M}, P'_{Y'}, E, Q_{\pi '})\) induced from those on \(M^0\) and \(M^1\).
Proposition 4.19

Let \((M^{\mathrm {ev}}, \partial M, \pi : \partial M \rightarrow Y^{\mathrm {odd}})\) be a compact manifold with fibered boundary, equipped with a complex vector bundle \(E \rightarrow M\).

Let \((P'_M, P'_{Y}, Q_{\pi })\) be data satisfying the conditions in (D1) and (D2).
 Assume that there exists data \((\pi ', X, P'_X, Q_{\pi '})\) such that

A compact manifold X with boundary \(\partial X\), with a fixed diffeormorphism \(\partial X \simeq Y\). We identify \(\partial X\) with Y.

A fiber bundle structure \(\pi ' : (M, \partial M) \rightarrow (X, \partial X)\) which preserves the boundary, and \(\pi '_{\partial M} = \pi \). Note that the typical fibers of \(\pi \) and \(\pi '\) are the same.

A pre\(spin^c\) structure \(P'_X\) on TX which restricts to \(P'_Y\).

Assume that the induced pre\(spin^c\)structure induced on \(T^VM\) restricts to \(P'_\pi \) at the boundary.

An element \(Q_{\pi '}\) in \(\mathcal {I}(P'_{\pi '}, E)\) which satisfies \(Q_{\pi '}_{\partial M} = Q_\pi \).

Next we show the relative formula for such indices. Recall that, for a family \(D_\pi \) of \(\mathbb {Z}_2\)graded selfadjoint operators parametrized by Y, if we are given two elements \(Q_\pi ^0\) and \(Q_\pi ^1\) in \(\mathcal {I}(D_\pi )\), their difference class \([Q_\pi ^1  Q_\pi ^0]\) is defined in \(K^{1}(Y)\).
Proposition 4.20
Proof
The first equality follows from (4.17). Choose any additional data (d1), (d2) and (d3) to define the operator \(D^E_e\). For each \(i = 0, 1\), choose any representative \(\tilde{D}^{E, i}_\pi \in \tilde{\mathcal {I}}_{\mathrm {sm}}(P'_\pi , E)\) for the class \(Q_\pi ^i \in \mathcal {I}(P'_\pi , E)\). By the general relative formula, Proposition 4.7, it is enough to show that the difference class of the invertible perturbations \(D^{E,i}_{\partial M}:=\tilde{D}^{E, i}_\pi \hat{\otimes } 1 + 1 \hat{\otimes }D_{TY \rtimes \mathbb {R}^*_+}\) for \(i = 0, 1\), defined in \(K_1(C^*(G_e_{\partial M})) (\simeq K_1(C^*(TY \rtimes \mathbb {R}^*_+)) \simeq K_0(Y))\), maps to \(\langle [Q_\pi ^1Q_\pi ^0], [D_Y]\rangle \) under the boundary map \(\partial ^{\mathring{M}} (G_e): K_1(C^*(G_e_{\partial M})) \rightarrow K_0(C^*(G_e_{\mathring{M}}))\).

\([\tilde{D}^{E, 1}_{\pi } \tilde{D}^{E, 0}_{\pi }] \in K_0(C^*((\partial M \times _\pi \partial M) \times (0, 1)))\simeq K^1(Y)\).

\([D_{Y}] \in K_1(Y)\).

\(\mathrm {Ind}^Y(D_{TY}) \in KK^1(C(Y), C^*(TY))\) represented by the ungraded Kasparov C(Y)\(C^*(TY)\) bimodule \((C^*(TY ; S(TY)), \mathrm {multi}, \psi (D_{TY}))\), where multi is the multiplication by C(Y). This is an ungraded version of (2.35).

\(m \in KK(C(Y)\otimes C^*(TY), C^*(TY))\) represented by the Kasparov \(C(Y)\otimes C^*(TY)\)\(C^*(TY)\) bimodule \((C^*(TY), \mathrm {multi} \otimes id_{C^*(TY)}, 0)\).

\([\sigma _Y] \in KK(C^*(TY), \mathbb {C})\).
4.3.2 Twisted signature operators.
Here we explain in the case of twisted signature operators. The argument is parallel to that in the case for twisted \(spin^c\)Dirac operators. Let \((M^{\mathrm {ev}}, \pi : \partial M \rightarrow Y^{\mathrm {odd}})\) be a compact oriented manifold with fibered boundaries equipped with a \(\mathbb {Z}_2\)graded complex vector bundle \(E \rightarrow M\). Assume we are given an element \(Q_\pi \in \mathcal {I}^{\mathrm {sign}}(\pi , E)\). We choose additional data as in Sect. 3.3.2, and define the twisted signature with respect to the fiberwise invertible perturbation, analogously as in the twisted \(spin^c\) Dirac operator case.
Definition 4.24
The relative formula for the twisted signature case is as follows.
Proposition 4.26
Proof
The proof is analogous to that for Proposition 4.20. The factor 2 in the above formula is due to the following observation.
First of all, recall the definition of odd signature operators acting on odd dimensional manifolds ([RW06, Definition and Notation 1]). On an odd dimensional riemannian manifold Y, the essentially selfadjoint operator \(d + d^*\) acting on \(\wedge _{\mathbb {C}}T^*Y\) commutes with the Hodge star \(\tau \), so we define the odd signature operator \(D_Y^{\mathrm {sign}}\) to be the operator \(d + d^*\) restricted to the \(+1\)eigenbundle of \(\tau \). So the total signature operator is isomorphic to the direct sum of two copies of \(D_Y^{\mathrm {sign}}\). We define odd signature operators for Lie groupoids whose dimensions of sfibers are odd dimensional analogously.
So the factor 2 appears in the equation corresponding to (4.22). \(\quad \square \)
5 The Index Pairing

For a \(C^*\)algebra A, the symbol \(\mathcal {M}(A)\) denotes its multiplier algebra.

For a \(C^*\)algebra A and a Hilbert Amodule \(H_A\), the symbols \(\mathcal {B}(H_A)\) and \(\mathcal {K}(H_A)\) denote the \(C^*\)algebras of adjointable operators and compact operators on \(H_A\), respectively.
 For a Euclidean space E, let us denote by \(\mathbb {C}l(E)\) the \(*\)algebra over \(\mathbb {C}\), generated by the elements of E and relationsThis construction applies to Euclidean vector bundles as well.$$\begin{aligned} e = e^* \text{ and } e^2 = e^2\cdot 1 \text{ for } \text{ all } e \in E. \end{aligned}$$

Let us denote by \(\epsilon \in \mathbb {C}l_1 = \mathbb {C}l(\mathbb {R})\) the element corresponding to the unit vector in \(\mathbb {R}\). In other words, this element is a generator of \(\mathbb {C}l_1\), which is odd, selfadjoint and unitary.
5.1 The case of \(spin^c\)Dirac operators

The pair \((M,\pi : \partial M \rightarrow Y)\) is a compact manifold with fibered boundary.

The fiber bundle \(\pi \) is equipped with a pre\(spin^c\) structure \(P'_\pi \).
 (1)We define a \(C^*\)algebra \(\mathcal {A}_\pi \) whose Kgroups fit in the exact sequence(Definition 5.4 and Proposition 5.5). The groups \(K_*(\mathcal {A}_\pi )\) are regarded as Kgroups relative to the boundary pushforward.$$\begin{aligned} \cdots \rightarrow K^{*}(M) \xrightarrow {\pi ! \circ i^*} {K}^{*n}(Y) \rightarrow {K}_{*n}(\mathcal {A}_\pi ) \rightarrow {K}^{*+1}(M) \xrightarrow {\pi ! \circ i^*} \cdots . \end{aligned}$$
 (2)
For a pair \((E, Q_\pi )\) where E is a \(\mathbb {Z}_2\)graded complex vector bundle over M and \(Q_\pi \in \mathcal {I}(P_\pi , E)\), we show that it naturally defines a class \([(E, Q_\pi )] \in K_{n1}(\mathcal {A}_\pi )\) (Lemma 5.7).
 (3)
Assume n is even. For a pair \((P'_M, P'_Y)\) of pre\(spin^c\) structures on TM and TY which satisfies \(P'_M_{\partial M} = \pi ^*P'_Y \oplus P'_\pi \), we show that it naturally defines a class \([(P'_M, P'_Y)] \in KK(\mathcal {A}_\pi , \Sigma ^{\mathring{M}}(G_\Phi ))\) (Definition 5.23).
 (4)We show the equality(Theorem 5.24). This is the desired index pairing formula.$$\begin{aligned} \mathrm {Ind}_\Phi (P'_M, P'_Y, E, Q_\pi ) = [(E, Q_\pi )] \otimes _{\mathcal {A}_\pi } [(P'_M, P'_Y)] \otimes _{\Sigma ^{\mathring{M}}(G_\Phi )} \mathrm {ind}^{\mathring{M}}(G_\Phi ) \in \mathbb {Z}. \end{aligned}$$
 (1)
Assume that the typical fiber of \(\pi \) is odd dimensional. Define \(\chi \in C([\infty , \infty ])\) as \(\chi (x) :=\frac{1}{2}(1 + x/\sqrt{1 + x^2})\). Let \(\Psi (D_\pi )\) denote the \(C^*\)subalgebra of \(\mathcal {B}(L_Y^2(N; S(T^VN)))\) generated by \(\{\chi (D_\pi )\}\), C(N) and \(\mathcal {K}(L_Y^2(N; S(T^VN)))\).
 (2)
Assume that the typical fiber of \(\pi \) is even dimensional. Define the odd function \(\psi \in C([\infty , \infty ])\) by \(\psi (x) := x/\sqrt{1+x^2}\). Let \(\Psi (D_\pi )\) denote the \(\mathbb {Z}_2\)graded \(C^*\)subalgebra of \(\mathcal {B}(L_Y^2(N; S(T^VN)))\) generated by \(\{\psi (D_\pi )\}\), C(N) and \(\mathcal {K}(L_Y^2(N; S(T^VN)))\).
Lemma 5.1
 (1)When the typical fiber of \(\pi \) is odd dimensional, the algebra \(\Psi (D_\pi )\) fits into the exact sequenceThe connecting element of this extension coincides with the class \(\pi _! \in KK^1(C(N), C(Y))\).$$\begin{aligned} 0 \rightarrow \mathcal {K}(L_Y^2(N; S(T^VN))) \rightarrow \Psi (D_\pi ) \rightarrow C(N) \rightarrow 0. \end{aligned}$$
 (2)When the typical fiber of \(\pi \) is even dimensional, the algebra \(\Psi (D_\pi )\) fits into the exact sequence of graded \(C^*\)algebrasThe connecting element of this extension coincides with the class \(\pi _! \in KK(C(N), C(Y))\).$$\begin{aligned} 0 \rightarrow \mathcal {K}(L_Y^2(N; S(T^VN))) \rightarrow \Psi (D_\pi ) \rightarrow C(N)\otimes \mathbb {C}l_1 \rightarrow 0. \end{aligned}$$(5.2)
Proof

The element \([\sigma (D_\pi )] \in K^1(C(\mathfrak {S}^*\Gamma ))\). This element coincides with the element in \(KK(\mathbb {C}l_1, C(\mathfrak {S}^*\Gamma ; \mathrm {End}(S(\mathfrak {A}\Gamma )))\) given by the unital \(*\)homomorphism which maps \(\epsilon \in \mathbb {C}l_1\) to \(\sigma (\psi (D_\pi ))\) (see Remark 4.9).

The element \([m] \in KK(C(N)\otimes C(\mathfrak {S}^*\Gamma ), C(\mathfrak {S}^*\Gamma )))\) given by the \(*\)homomorphism \(f \otimes \xi \mapsto f\cdot \xi \).
Remark 5.3
As we work in KKtheory in this section, we only need \(C^*\)algebras to be defined up to KKequivalence. As in Lemma 5.1, in order to define \(C^*\)algebras in terms of operators, we need to fix rigid structures, such as differential \(spin^c\)structures. However, the KKequivalence class is determined by homotopy equivalence class of those structures, such as pre\(spin^c\)structure (c.f. Remark 4.14). In order to simplify the arguments, we often omit this procedure of “choosing a rigid structure, defining algebras and forgetting the structure to get a KKequivalence class”, but the reader should note that we always need such steps.
Definition 5.4
 (1)Assume that the typical fiber of \(\pi \) is odd dimensional. We define \(\mathcal {A}_\pi \) to be the \(C^*\)algebra defined by the pullback (c.f. Remark 5.3)
 (2)Assume that the typical fiber of \(\pi \) is even dimensional. We define \(\mathcal {A}_\pi \) to be the \(\mathbb {Z}_2\)graded \(C^*\)algebra defined by the pullback (c.f. Remark 5.3)
Proposition 5.5
Proof
Lemma 5.7
Let \((M,\pi : \partial M \rightarrow Y)\) be a compact manifold with fibered boundary. Denote \(\Gamma \) the groupoid \(\partial M \times _\pi \partial M \rightrightarrows \partial M\). Assume that \(\pi \) is equipped with a pre\(spin^c\)structure \(P'_\pi \). Let E be a \(\mathbb {Z}_2\)graded complex vector bundle over M. Let \(Q_\pi \in \mathcal {I}(P'_\pi , E)\) (see Remark 4.14). Then the pair \((E, Q_\pi )\) naturally defines a class \([(E, Q_\pi )] \in K_{n1}(\mathcal {A}_\pi )\), where n is the dimension of the fiber of \(\pi \).
Proof
Given a pair \((E,Q_\pi )\) where \(Q_\pi \in \mathcal {I}(P'_\pi , E)\), we define an element in \(K_1(\mathcal {A}_\pi (E))\) as follows. Let us choose a representative \(\tilde{D}_\pi ^E \in \tilde{\mathcal {I}}_{\mathrm {sm}}(D_\pi ^E)\). Since \(\tilde{D}_\pi ^E\) is an invertible family which differs from \(D_\pi ^E\) by a lower order family, \(\psi (\tilde{D}_\pi ^E)\) is an invertible operator in \(\Psi (D_\pi ^E)\). Thus the element \((1_M \hat{\otimes }\epsilon , \psi (\tilde{D}_\pi ^E)) \in \mathcal {A}_\pi (E)\) is invertible. The \(K_1\) class defined by this element does not depend on the choice of \(\tilde{D}_\pi ^E\). By composing with the KKequivalence between \(\mathcal {A}_\pi \) and \(\mathcal {A}_\pi (E)\), we get the element \([(E, Q_\pi )] := [(1_M \hat{\otimes }\epsilon , \psi (\tilde{D}_\pi ^E))] \in K_1(\mathcal {A}_\pi )\). \(\quad \square \)
Now we assume that M is even dimensional and Y is odd dimensional. We construct an element \([(P'_M, P'_Y)] \in KK^n(\mathcal {A}_\pi , \Sigma ^{\mathring{M}}(G_\Phi ))\) for given \(P'_M\), \(P'_Y\) pre\(spin^c\)structures on TM and TY which are compatible with the given fiberwise pre\(spin^c\) structure \(P'_\pi \) at the boundary, i.e., \(P'_M_{\partial M} = \pi ^*P'_Y \oplus P'_\pi \).
The next lemma can be proved in the same way as Lemma 5.1.
Lemma 5.9
Let \((M^{\mathrm {ev}},\pi : \partial M \rightarrow Y^{\mathrm {odd}})\) be a compact manifold with fibered boundary, equipped with a pre\(spin^c\)structure \(P'_\pi \) on \(T^V\partial M\). Let \(P'_M\), \(P'_Y\) be pre\(spin^c\)structures on TM and TY respectively. We assume that the pre\(spin^c\)structures are compatible at the boundary.
Definition 5.10
First we consider a general setting. Suppose we are given a compact manifold Y, and an oriented real vector bundle V over Y. For simplicity we only consider the case where the rank m of this vector bundle V is odd. We consider the Clifford algebra bundle \(\mathbb {C}l(V)\) over V (trivial on each fiber of \(V \rightarrow Y\)) and the \(\mathbb {Z}_2\)graded \(C^*\)algebra \(C_0(V; \mathbb {C}l(V))\), where the algebra structure comes from the pointwise operations and grading comes from the grading on \(\mathbb {C}l(V)\). We consider a variant of the construction of an asymptotic morphism in [GH04, Section 1], which gives a KKequivalence between \(C_0(V; \mathbb {C}l(V))\) and C(Y). We are going to apply the following general construction to the real vector bundle \(V = T^*Y\) in our \(\Phi \)\(spin^c\)setting. The construction below is also used in the next subsection for signature operators.
Lemma 5.11
The map \(\phi \) defines a \(*\)homomorphism.
Proof
Remark 5.12
We would like to treat KKelements arising from asymptotic morphisms directly in our construction, so we do not use the \(C^*\)algebra \(\mathcal {S}\) and consider the above asymptotic morphism \(\phi \), which indeed gives the KKequivalence between \(C_0(\mathbb {R} ; \mathbb {C}l_1) \hat{\otimes }C_0(V; \mathbb {C}l(V))\) and C(Y) (see Proposition 5.15 below).
Lemma 5.14
Let g be a function in \(C_0(\mathbb {R})\). For \(t \in (0, \infty )\), consider the functional calculus \(g(t^{1}B_V) \in \mathcal {K}(L^2_Y(V; \mathbb {C}l(V)))\). This family canonically extends to an element in \(T_P\) by setting \(g(t^{1}B_V)_{t = 0} = g(0)P\). We abuse the notation and still denote this element by \(g(t^{1}B_V) \in T_P\).
Proof
The essential point is that \(B_V\) has discrete spectrum with finite multiplicity. The lemma is proved by checking on the generators \(e^{x^2}\) and \(xe^{x^2}\) of \(C_0(\mathbb {R})\), and the computations are essentially the same as in [GH04, Section 1.13].\(\quad \square \)
Proposition 5.15
Proof
On the other hand, by Lemma 5.14, we see that \(\psi (\mathbb {B}_V)\) is a selfadjoint multiplier of \(\mathcal {C}\) which satisfies \(\psi (\mathbb {B}_V)^2  \mathrm {Id}_{\mathcal {C}} \in \mathcal {C}\), and by construction \(\psi (\mathbb {B}_V)\) commutes with the action of C(Y) on \(\mathcal {C}\) by multiplication. So we get the element \([\psi (\mathbb {B}_V)] \in KK(C(Y), \mathcal {C})\). It satisfies \([\psi (\mathbb {B}_V)] \otimes [ev_\infty ^{\mathcal {C}}] = [\psi (X\epsilon \hat{\otimes } 1 + 1 \hat{\otimes }C_V)] \in KK(C(Y), C_0(\mathbb {R} \oplus V; \mathbb {C}l(\mathbb {R} \oplus V)))\) and \([\psi (\mathbb {B}_V)]\otimes [ev_0^{\mathcal {C}}] = id_{C(Y)} \in KK(C(Y), C(Y))\) (this is because \(\psi (t^{1}B_V)_{t = 0} = \psi (0)P= 0\)). Thus we have \( [\psi (X\epsilon \hat{\otimes } 1 + 1 \hat{\otimes }C_V)] \otimes [ev_\infty ^{\mathcal {C}}]^{1} \otimes [ev_0^{\mathcal {C}}] = id_{C(Y)} \in KK(C(Y), C(Y))\). Since the element \( [\psi (X\epsilon \hat{\otimes } 1 + 1 \hat{\otimes }C_V)] \) is the Thom element which is a KKequivalence, we see that \( [ev_0^{\mathcal {C}}] \in KK(\mathcal {C}, C(Y))\) is a KKequivalence, which proves (5.16). \(\quad \square \)
Now, we return to settings of manifolds with fibered boundaries \((M, \pi : \partial M\rightarrow Y)\) equipped with pre\(spin^c\)structures. We assume that M is even dimensional and Y is odd dimensional. We apply the above constructions for \(V := T^*Y\) which is an oriented vector bundle over Y. Let us denote by \(\Gamma \) the groupoid \(\partial M \times _\pi \partial M \rightrightarrows \partial M\).
Definition 5.17
We have a \(*\)homomorphism \(ev_0 := ev_0^{\mathcal {C}} \otimes _{C(Y)} id_{C^*(\Gamma )}: \mathcal {C}\hat{\otimes }C^*(\Gamma ; S(\mathfrak {A}\Gamma )) \rightarrow C^*(\Gamma ; S(\mathfrak {A}\Gamma )) \). This \(*\)homomorphism extends to a \(*\)homomorphism \(\mathcal {D}_\pi \rightarrow \mathcal {A}_\pi \), by sending \( \psi (\mathbb {B}_V \hat{\otimes }1 + 1 \hat{\otimes }D_\pi )\) to \(\psi (D_\pi )\) and by evaluating at \(0 \in [0, \infty ]\) on \(C(M \times [0, \infty ]) \otimes \mathbb {C}l_1\), since \(\psi (0) = 0\) and P is the projection to the kernel of \(B_V\). We also denote this \(*\)homomorphism by \(ev_0\).
On the other hand, using the isomorphism \(C_0(\mathbb {R} \oplus T^*Y; \mathbb {C}l(\mathbb {R} \oplus T^*Y))\simeq C^*(TY \times \mathbb {R}; S(TY \times \mathbb {R}))\), we have a \(*\)homomorphism \(ev_\infty := ev_\infty ^{\mathcal {C}}\otimes _{C(Y)} id_{C^*(\Gamma )} : \mathcal {C}\hat{\otimes }_{C(Y)}C^*(\Gamma ; S(\mathfrak {A}\Gamma )) \rightarrow C^*(TY \times \mathbb {R}; S(TY \times \mathbb {R})) \hat{\otimes }_{C(Y)} C^*(\Gamma , S(\mathfrak {A}\Gamma )) \simeq C^*(G_\Phi _{\partial M}; S(\mathfrak {A}G_\Phi _{\partial M}))\). This \(*\)homomorphism extends to a \(*\)homomorphism \(\mathcal {D}_\pi \rightarrow \mathcal {B}_\pi \) by sending \(\psi (\mathbb {B}_V \hat{\otimes }1 + 1 \hat{\otimes }D_\pi )\) to \(\psi (D_{TY \times \mathbb {R}} \hat{\otimes }1 + 1 \hat{\otimes }D_\pi ) = \psi (D_{\Phi , \partial })\) and evaluation at \(\infty \in [0, \infty ]\) on \(C(M \times [0, \infty ]) \otimes \mathbb {C}l_1\), since \(X\epsilon \hat{\otimes } 1 + 1 \hat{\otimes } C_V\) corresponds to \(D_{TY \times \mathbb {R}}\) under the Fourier transform. We also denote this \(*\)homomorphism by \(ev_\infty \).
Proposition 5.18
Proof
For ease of notations, we drop the coefficient bundle in this proof and write, for example, \(C^*(\Gamma )\) for \(C^*(\Gamma ; S(\mathfrak {A}\Gamma ))\). The commutativity of the diagram (5.19) directly follows from the definition.
Next let us look at the middle column of the diagram (5.19). By the commutativity of the diagram and the five lemma in KKtheory, the above KKequivalence result on the left column implies that the middle vertical arrows are also KKequivalences. Finally, the commutativity of the diagram (5.21) follows from (5.22). \(\quad \square \)
Definition 5.23
Theorem 5.24
Proof
We only prove the Theorem in the case where the dimension of the fiber of \(\pi \) is even. As in Lemma 5.7, a pair \((E, Q_{\Phi , \partial } )\), where \(E \rightarrow M\) is a \(\mathbb {Z}_2\)graded complex vector bundle and \(Q_{\Phi , \partial } \in \mathcal {I}(D_{\Phi , \partial }^E)\), naturally defines a class \([(E, Q_{\Phi , \partial } )] \in K_1(\mathcal {B}_\pi )\). Here we denoted by \(D_{\Phi , \partial }^E\) the \(spin^c\) Dirac operator twisted by E and \(\mathcal {I}(D_{\Phi , \partial }^E)\) is the set of homotopy classes of \(\mathbb {C}l_1\)invertible perturbations of the operator \(D_{\Phi , \partial }^E\) on \(G_\Phi _{\partial }\).
First we remark that, when we are given a twisting bundle E, it is convenient to use \(C^*\)algebras \(\mathcal {A}_\pi (E)\), \(\mathcal {B}_\pi (E)\) and \(\mathcal {D}_\pi (E)\) which are Morita equivalent to \(\mathcal {A}_\pi \), \(\mathcal {B}_\pi \) and \(\mathcal {D}_\pi \), respectively; the first one appears in (5.8) and the definitions of the other algebras are selfexplanatory. The corresponding element \(\mu \in KK(\mathcal {A}_\pi (E), \mathcal {B}_\pi (E))\) is realized as \([ev_0]^{1}\otimes _{\mathcal {D}_\pi (E)}[ev_\infty ]\) as in Proposition 5.18, and a \(*\)homomorphism \(\iota _E : \mathcal {B}_\pi (E) \rightarrow \Sigma ^{\mathring{M}}(G_\Phi ; S(\mathfrak {A}G_\Phi )\hat{\otimes }E)\) is constructed analogously to Definition 5.10.
5.2 The case of signature operators
In this subsection, we consider the case of signature operators. The arguments are parallel to those in Sect. 5.1. Let \((M, \pi : \partial M \rightarrow Y)\) be a compact manifold with fibered boundaries, with fixed orientation on TM and TY. For simplicity we only consider the case where the fibers of \(\pi \) are even dimensional.
Denote the odd function \(\psi (x) := x/\sqrt{1+x^2}\). Let \(\Psi (D^{\mathrm {sign}}_\pi )\) denote the \(\mathbb {Z}_2\)graded \(C^*\)subalgebra of \(\mathcal {B}(L_Y^2(N; \wedge _{\mathbb {C}}(T^VN)^*))\) generated by \(\{\psi (D^{\mathrm {sign}}_\pi )\}\), C(N) and \(\mathcal {K}(L_Y^2(N; \wedge _{\mathbb {C}}(T^VN)^*))\).
Lemma 5.26
Definition 5.28
(\(\mathcal {A}^{\mathrm {sign}}_\pi \)). Let \((M,\pi : \partial M \rightarrow Y)\) be a compact manifold with fibered boundary. Assume that \(\pi \) is oriented and the fibers are even dimensional. Denote \(i : \partial M \rightarrow M\) the inclusion.
We can prove that this \(C^*\)algebra induces the desired long exact sequence, analogously to Proposition 5.5.
Proposition 5.29
Then analogously to Lemma 5.7, we have the following.
Lemma 5.30
Let \((M,\pi : \partial M \rightarrow Y)\) be a compact manifold with fibered boundary. Let us denote by \(\Gamma \) the groupoid \(\partial M \times _\pi \partial M \rightrightarrows \partial M\). Assume that \(\pi \) is oriented and the fibers are even dimensional. Let E be a \(\mathbb {Z}_2\)graded complex vector bundle over M. Assume we are given an element \(Q_\pi \in \mathcal {I}^{\mathrm {sign}}(\pi , E)\). Then the pair \((E, Q_\pi )\) naturally defines a class \([(E, Q_\pi )] \in K_1(\mathcal {A}^{\mathrm {sign}}_\pi )\).
Now we assume that M is even dimensional and oriented. We construct an element \([M^{\mathrm {sign}}] \in KK(\mathcal {A}^{\mathrm {sign}}_\pi , \Sigma ^{\mathring{M}}(G_\Phi ))\).
Lemma 5.31
Let \((M^{\mathrm {ev}},\pi : \partial M \rightarrow Y^{\mathrm {odd}})\) be a compact manifold with fibered boundary, equipped with orientations on TM and \(T^V\partial M\).
Definition 5.32
We are going to construct a KKelement \(\mu ^{\mathrm {sign}} \in KK(\mathcal {A}_\pi ^{\mathrm {sign}}, \mathcal {B}_\pi ^{\mathrm {sign}})\) analogous to \(\mu \) in Proposition 5.18. Note that in the signature case, this element is not a KKequivalence.
We construct a \(C^*\)algebra \(\mathcal {D}_\pi ^{\mathrm {sign}}\), using the \(C^*\)algebra \(\mathcal {C}\) constructed in the last subsection. Recall that the construction of \(\mathcal {C}\) does not need any \(spin^c\)structure on vector bundle \(V \rightarrow Y\). We apply the constructions in the last subsection for \(V = T^*Y\). Denote \(\Gamma \) the groupoid \( \partial M \times _\pi \partial M \rightrightarrows \partial M\).
Definition 5.33
Analogously to the last subsection, the \(*\)homomorphism \(ev_0^{\mathcal {C}} : \mathcal {C} \rightarrow C(Y)\) induces a \(*\)homomorphism \(ev_0 : \mathcal {C}\hat{\otimes }C^*(\Gamma ; \wedge _{\mathbb {C}}(\mathfrak {A}\Gamma )^*) \rightarrow C^*(\Gamma ; \wedge _{\mathbb {C}}(\mathfrak {A}\Gamma )^*) \). This \(*\)homomorphism extends to a \(*\)homomorphism \(\mathcal {D}^{\mathrm {sign}}_\pi \rightarrow \mathcal {A}^{\mathrm {sign}}_\pi \), by sending \( \psi (\mathbb {B}_V \hat{\otimes }1 + 1 \hat{\otimes }D^{\mathrm {sign}}_\pi )\) to \(\psi (D^{\mathrm {sign}}_\pi )\) and evaluation at 0 on \(C(M \times [0, \infty ]) \otimes \mathbb {C}l_1\). We denote this \(*\)homomorphism by \(ev_0\).
On the other hand, contrary to the last subsection, the \(*\)homomorphism \(ev_\infty ^{\mathcal {C}} : \mathcal {C} \rightarrow C_0(\mathbb {R}; \mathbb {C}l_1) \hat{\otimes } C_0(V ; \mathbb {C}l(V)) \) induces a \(*\)homomorphism \(ev_\infty : \mathcal {C}\hat{\otimes }_{C(Y)}C^*(\Gamma ; \wedge _{\mathbb {C}}(\mathfrak {A}\Gamma )^*) \rightarrow C_0(V\oplus \mathbb {R}; \mathbb {C}l(V\oplus \mathbb {R})) \hat{\otimes }_{C(Y)} C^*(\Gamma , \wedge _{\mathbb {C}}(\mathfrak {A}\Gamma )^*)\), and the range of this homomorphism is not isomorphic to \(C^*(G_\Phi _{\partial M}; \wedge _{\mathbb {C}}(\mathfrak {A}G_\Phi _{\partial M})^*)\). To overcome this difference, we need an intermediate \(C^*\)algebra \(\tilde{\mathcal {B}}_\pi ^{\mathrm {sign}}\).
Definition 5.34
Proposition 5.35
Finally we define the element \([M^{\mathrm {sign}}] \in KK(\mathcal {A}^{\mathrm {sign}}_\pi , \Sigma ^{\mathring{M}}(G_\Phi ))\) as follows.
Definition 5.37
Then, we can describe the \(\Phi \)signature as follows.
Theorem 5.38
Proof
6 The Local Signature
6.1 Settings

Let F be an oriented closed even dimensional smooth manifold.

Let G be a subgroup of the orientationpreserving diffeomorphism group \(\text{ Diff }^+(F) \) of F.

Let \(Z \subset BG\) be a subspace of the classifying space of G. A particular case of interest is when Z is the kskeleton of a CWcomplex model of BG for some integer k.
Definition 6.1

The pair \((M, \pi : \partial M \rightarrow Y)\) is a compact oriented manifold with fibered boundaries, and assume that M is even dimensional.

Assume that \(\pi \) is an Ffiber bundle structure with structure group G.

Assume that \(i_* : [Y, Z] \rightarrow [Y, BG]\), induced by the inclusion \(i : Z \rightarrow BG\), is an isomorphism.
Our main theorem of this section is the following.
Theorem 6.2

(vanishing)
For \((M, \pi ) \in S_{F, G, Z}\), if there exist a compact oriented manifold with boundary \((X, \partial X)\) with a fixed diffeomorphism \(\partial X \simeq Y\), and an Ffiber bundle structure \(\pi ' : M \rightarrow X\) with structure group G which satisfies \(\pi '_{\partial M} =\pi \) such that \(i_* : [X, Z] \rightarrow [X, BG]\) is surjective, then we have$$\begin{aligned} \sigma _{Q_Z}(M, \pi )=0. \end{aligned}$$ 
(additivity)
For \((M_0, \pi _0)\) and \((M_1, \pi _1)\) in \(S_{F, G, Z}\), assume that there exists a decomposition \(\partial M_i = \partial M_i^+ \sqcup \partial M_i^\) for \(i = 0, 1\), and there exists an isomorphism of the fiber bundle \(\phi : \pi _0_{\partial M_0^+} \simeq \pi _1_{ \partial M_1^}\). We can form \((M, \pi ) = (M_0, \pi _0)\cup _\phi (M_1, \pi _1) \in S_{F, G, Z}\). Then we have$$\begin{aligned} \sigma _{Q_Z}(M, \pi ) = \sigma _{Q_Z}(M_0, \pi _0) + \sigma _{Q_Z}(M_1, \pi _1). \end{aligned}$$ 
(compatibility with signature)
An oriented even dimensional closed manifold M can be regarded as an element in \(S_{F, G, Z}\). For this element, we have$$\begin{aligned} \sigma _{Q_Z}(M) = \mathrm {Sign}(M). \end{aligned}$$
6.2 The universal index class and the pullback of \(\mathbb {C}l_1\)invertible perturbations
Next we give fundamental remarks on pullbacks of \(\mathbb {C}l_1\)invertible perturbations for signature operators. Suppose that we are given a continuous map \(f : X_0 \rightarrow X_1\) between topological spaces, and a continuous oriented fiber bundle structure \(\pi : M \rightarrow X_1\) with fiber F. The fiberwise signature class, \(\mathrm {Sign}(\pi ) \in K^0(X_1)\), is defined as above. Consider the pullback bundle \(f^*\pi : f^*(M) \rightarrow X_0\). The fiberwise signature class of this bundle satisfies \(\mathrm {Sign}(f^*\pi ) = f^*\mathrm {Sign}(\pi ) \in K^0(X_0)\).
6.3 The local signature
In this subsection, we return to the settings of Sect. 6.1. First we explain the pullback of \(\mathbb {C}l_1\)invertible perturbations by the classifying maps. We cannot apply the procedure explained in the last section directly, because the classifying map is defined up to homotopy.
Proposition 6.5
 (1)Suppose that the universal signature class satisfies \(n \cdot \mathrm {Sign}(F_{\mathrm {univ}}) = 0 \in K^0(BG)\). Then the classifying map \([f] \in [X, BG]\) induces a welldefined affine space homomorphism$$\begin{aligned}{}[f]^* : \mathcal {I}^{\mathrm {sign}}(\pi _{\mathrm {univ}}, \underline{\mathbb {C}^n}) \rightarrow \mathcal {I}^{\mathrm {sign}}(\pi ,\underline{\mathbb {C}^n}). \end{aligned}$$
 (2)Let \(Z \subset BG\) be a subspace and assume that \(i_* : [X, Z] \rightarrow [X, BG]\) induced by the inclusion \(i : Z \rightarrow BG\) is an isomorphism. Also assume that \(n \cdot i^* \mathrm {Sign}(F_{\mathrm {univ}}) = 0 \in K^0(Z)\). Then the classifying map \([f] \in [X, Z]\) induces a welldefined affine space homomorphism$$\begin{aligned}{}[f]^* : \mathcal {I}^{\mathrm {sign}}(\pi _{\mathrm {univ}}_{Z}, \underline{\mathbb {C}^n}) \rightarrow \mathcal {I}^{\mathrm {sign}}(\pi ,\underline{\mathbb {C}^n}). \end{aligned}$$
Proof
Next we prove the case (2). Denote \(\pi _{\mathrm {univ}}^{1}(Z) = \tilde{Z} \subset M_{\mathrm {univ}}\). In this case, by the next Lemma 6.9, we see that we can take a section \(s : X \rightarrow P \times _G \tilde{Z}\), and any choice of section is homotopic to each other. Thus we can apply exactly the same argument as in the case (1) and get the result. \(\quad \square \)
Lemma 6.9
If \(i_* : [X, Z] \simeq [X, BG]\), the space \(\Gamma (X; P \times _G \tilde{Z})\) is nonempty and pathconnected.
Proof
Definition 6.10
We check that this map satisfies the conditions in Theorem 6.2.
Proof of Theorem 6.2
First we prove the vanishing condition. Suppose we are given an element \((M, \pi ) \in S_{F, G, Z}\) such that \(\pi \) extends to an Ffiber bundle structure \(\pi ' : M \rightarrow X\) with structure group G and \(i_* : [X, Z] \rightarrow [X, BG]\) is surjective. Denote the classifying map of \(\pi '\) by \([f'] \in [X, BG]\). Take any lift of \([f']\) to an element of \([\tilde{f}'] \in [X, Z]\), and realize this map as a bundle map \((\tilde{\phi }', \tilde{f}') : (M, X) \rightarrow (\tilde{Z}, Z)\). Then as in (6.4), we can pullback the element \(Q_{Z} \in \mathcal {I}^{\mathrm {sign}}(\pi _{\mathrm {univ}}_Z, \underline{\mathbb {C}^n})\) by the bundle map \((\tilde{\phi }', \tilde{f}')\) to get an element \((\tilde{\phi }', \tilde{f}')^*Q_Z \in \mathcal {I}^{\mathrm {sign}}(\pi ', \underline{\mathbb {C}^n})\). This element restricts to \([f]^*(Q_{Z}) \in \mathcal {I}^{\mathrm {sign}}(\pi , \underline{\mathbb {C}^n})\) at the boundary. Thus applying the vanishing proposition, the signature version of Proposition 4.19, we get the result.
The additivity follows from the gluing formula, the twisted signature version of Proposition 4.18.
The compatibility with signature is obvious by definition.
The Eq. (6.3) follows from the relative formula for \(\Phi \)signature, Proposition 4.26. \(\quad \square \)
7 Examples
In this section, as an application of Theorem 6.2, we consider the following localization problem for singular surface bundles.
Fix a positive integer k and an integer \(g \ge 0\). Let M, X be \(4k, (4k2)\)dimensional closed oriented smooth manifolds, respectively. Let \(\pi : M \rightarrow X\) be a smooth map and \(X = U \cup \cup ^m_{i=1} V_i\) be a partition into compact manifolds with closed boundaries, i.e., U and \(V_i\) are compact manifolds with closed boundaries, and each two of them intersect only on their boundaries. Assume \(\{V_i\}_i\) are disjoint. Denote \(M_U := \pi ^{1}(U)\) and \(M_i := \pi ^{1}(V_i)\). Assume that \(\pi _{M_U} :M_U \rightarrow U\) defines a smooth fiber bundle with fiber \(\Sigma _g\) (closed oriented surface with genus g). Then the localization problem is stated as follows:
Problem 7.1
We call the real number \(\sigma (M_i, V_i, \pi _{M_i}) \) the local signature. The answer to this problem is positive in the case \(g = 0, 1, 2\). However the answer is negative for \(g \ge 3\), since there exists a smooth \(\Sigma _g\)fiber bundle \(M \rightarrow X\) over a closed surface X with \(\text{ Sign }(M) \ne 0\). However, if we assume some structure on the fiber bundle \(\pi _{M_U}\), the answer can be positive. There are some examples of “structures” for which the localization problem has a positive answer, and the local signatures are constructed and calculated in various areas of mathematics, including topology, algebraic geometry and complex analysis. See [AK02] and the introduction of [Sat13] for more detailed survey on this problem. In this paper, we consider the case \(g \ge 2\) and hyperellipticity (Definition 7.5) as the “structure” imposed on the regular part of the fibration. This is the analogue to the setting in [End00], where the case \(k=1\) is considered (note that in [End00] the case \(g = 0, 1\) are also included). We consider the following variant of the Problem 7.1.
Problem 7.2

Let \(F = \Sigma _g\), a closed oriented closed 2dimensional manifold of genus g.

Denote by \(\mathrm {MCG}^+(F)\) the orientationpreserving mapping class group of F. Let \(G=\mathrm {Diff}^H_g := p^{1}(H_g) \subset \mathrm {Diff}^+(\Sigma _g)\), where \(H_g \subset \mathrm {MCG}^+(G)\) is the hyperelliptic mapping class group (Definition 7.3) and \(p : \mathrm {Diff}^+(F) \rightarrow \mathrm {MCG}^+(F)\) is the quotient map.
Definition 7.3
Remark 7.4
If \(g = 0, 1, 2\), \(\mathrm {MCG}^+(\Sigma _g) = H_g\). But in the case \(g \ge 3\), \(H_g\) is a subgroup of infinite index in \(\mathrm {MCG}^+(\Sigma _g)\).
Definition 7.5
Let X be a topological space. A \(\Sigma _g\)fiber bundle \(\pi : M \rightarrow X\) with structure group \(\mathrm {Diff}^H_g\) is called a hyperelliptic fiber bundle.
We have the following facts about the groups \(H_g\) and \(\mathrm {Diff}_g^H\).
Fact 7.6
 (1)
The rational group cohomology of \(H_g\) satisfies \(H^i(H_g; \mathbb {Q}) = 0\) for all \(i \ge 1\) ([Kaw97]).
 (2)
For \(g \ge 2\), the unit component of \(\mathrm {Diff}_g^H\) is contractible. In particular we have a homotopy equivalence between \(BH_g\) and \(B\mathrm {Diff}_g^H\) ([EE67]).
 (3)
For all \(g \ge 0\), \(H_g\) is of type \(FP_{\infty }\). That is, \(BH_g\) has a realization as a CWcomplex whose mskeleton are finite for all \(m \ge 0\).
It wellknown that the mapping class group of an oriented compact surface of genus g with s punctures and n boundary components is of type \(FP_\infty \) (For example see [Luc05]). This case can be seen by noting that an extension of a type \(FP_\infty \) group by a type \(FP_\infty \) group is also of type \(FP_\infty \), and that we have an extension by Birman–Hilden theorem (see [Kaw97, equation (2.1)])$$\begin{aligned} 0 \rightarrow \mathbb {Z}_2 \rightarrow H_g \rightarrow \pi _0\mathrm {Diff}^+(S^2, \{(2g+2)\text{ punctures }\}) \rightarrow 1. \end{aligned}$$
Remark 7.7
The reason for assuming \(g \ge 2\) in Problem 7.2 comes from Fact 7.6 (2). For \(g =0\) the inclusion \(SO(3) \rightarrow \mathrm {Diff}^H_0(= \mathrm {Diff}^+(S^2))\) is a homotopy equivalence, and for \(g = 1\) the unit component of \(\mathrm {Diff}^H_1 = \mathrm {Diff}^+(T^2)\) is homotopy equivalent to \(S^1 \times S^1\) ([EE67]). These groups have torsion in group cohomology, so the argument below does not work in these cases.
From now on, we fix an integer \(g \ge 2\). From Fact 7.6 (2) and (3), we see that \(B\text{ Diff }_g^H\) has a realization as a CWcomplex whose mskeleton are finite for all \(m \ge 0\). We fix such a realization and denote its mskeleton by \(Z_{g,m}\).
Lemma 7.8
For \(g \ge 2\), the universal fiberwise signature class for hyperelliptic fiber bundle, \(Sign(F_{\mathrm {univ}} ) \in K^0(B\mathrm {Diff}_g^H)\), maps to \(0 \in K^0(B\mathrm {Diff}_g^H; \mathbb {Q})\) under the canonical homomorphism \(K^0(B\mathrm {Diff}_g^H) \rightarrow K^0(B\mathrm {Diff}_g^H; \mathbb {Q})\). Here the symbol \(K^0\) denotes the representable Ktheory.
Proof
We have the rational Chern character isomorphism \(Ch : K^{0} (BH_g; \mathbb {Q}) \simeq H^{\mathrm {ev}}(H_g ; \mathbb {Q})\). Using the Fact 7.6 (1) and (2), we have \(K^0(B\mathrm {Diff}^H_g; \mathbb {Q}) \simeq K^0(BH_g; \mathbb {Q}) \simeq \mathbb {Q}\) and this isomorphism is given by a map \(* \rightarrow B\mathrm {Diff}^H_g\). Since the manifold \(\Sigma _g\) is two dimensional, the virtual rank of the class \(\mathrm {Sign}(F_{\mathrm {univ}}) \in K^0(B\mathrm {Diff}^H_g)\) is 0. Thus we have \(\mathrm {Sign}(F_{\mathrm {univ}}) = 0 \in K^0(B\mathrm {Diff}^H_g; \mathbb {Q})\) and the lemma follows. \(\quad \square \)
For each m, we also denote the restriction of the class \(\mathrm {Sign}(F_{\mathrm {univ}})\) to \(Z_{g, m}\) by \(\mathrm {Sign}(F_{\mathrm {univ}}) \in K^0(Z_{g, m})\). By Lemma 7.8, we have \(\mathrm {Sign}(F_{\mathrm {univ}}) = 0 \in K^0(Z_{g, m}; \mathbb {Q})\). Since \(Z_{g, m}\) is compact, we have \(K^0(Z_{g, m}; \mathbb {Q}) \simeq K^0(Z_{g, m}) \otimes \mathbb {Q}\), so the class \(\mathrm {Sign}(F_{\mathrm {univ}}) \) is of finite order in \(K^0(Z_{g, m}) \).
Definition 7.9
For each positive integer m, let \(n_{g, m}\) denote the order of the class \(\mathrm {Sign}(F_{\mathrm {univ}}) \in K^0(Z_{g, m}) \). i.e., \(n_{g, m}\) is the smallest positive integer satisfying \(n_{g, m} \cdot \mathrm {Sign}(F_{\mathrm {univ}}) = 0 \in K^0(Z_{g, m}) \).
We are in the situation where Theorem 6.2 applies.
Definition 7.10

The pair \((M, \pi : \partial M \rightarrow Y)\) is a compact oriented manifold with fibered boundaries, and M is 4kdimensional.

The fiber bundle \(\pi : \partial M \rightarrow Y\) is a hyperelliptic fiber bundle with fiber \(\Sigma _g\).
For \((M, \pi : \partial M \rightarrow Y) \in \mathcal {S}_{g, k}\), Y is a \((4k3)\)dimensional manifold. Thus we have \([Y, Z_{g, 4k2}] \simeq [Y, B\mathrm {Diff}_g^H]\). We see that \(\mathcal {S}_{g, k} \subset S_{\Sigma _g,\mathrm {Diff}_g^H , Z_{g, 4k2}}\). We apply Theorem 6.2 to the case \(F = \Sigma _g\), \(G = \mathrm {Diff}_g^H\), \(Z=Z_{g, 4k2}\), and \(n= n_{g, 4k2}\). Note that we have \(K^{1}(Z_{g, 4k2}) \otimes \mathbb {Q} = 0\) because of Fact 7.6, (1) and (2). Thus, choosing any element \(Q_{Z_{4k2}} \in \mathcal {I}^{\mathrm {sign}}(\pi _{\mathrm {univ}}_{Z_{4k2}}, \underline{\mathbb {C}^n})\), we get the same map \(\sigma _{Q_{Z_{4k2}}}\) by (6.3) in Theorem 6.2. So we set \(\sigma := \sigma _{Q_{Z_{4k2}}}\).
Corollary 7.11

(vanishing)
For \((M, \pi : \partial M \rightarrow Y) \in \mathcal {S}_{g, k}\), assume that \(\pi \) extends to an oriented hyperelliptic \(\Sigma _g\)fiber bundle structure \(\pi ' : (M, \partial M) \rightarrow ( X, \partial X)\) preserving boundaries. Here X is an oriented smooth compact oriented \((4k2)\)dimensional manifold and an orientation preserving diffeomorphism \(\partial X \simeq Y\) is fixed. Then we have$$\begin{aligned} \sigma (M, \pi )=0. \end{aligned}$$ 
(additivity)
For \((M_0, \pi _0)\) and \((M_1, \pi _1)\) in \(\mathcal {S}_{g, k}\), assume that there exists a decomposition \(\partial M_i = \partial M_i^+ \sqcup \partial M_i^\) for \(i = 0, 1\), and there exists an isomorphism of the fiber bundle \(\phi : \pi _0_{\partial M_0^+} \simeq \pi _1_{ \partial M_1^}\). We form the union \((M, \pi ) = (M_0, \pi _0)\cup _\phi (M_1, \pi _1) \in \mathcal {S}_{g,k}\). Then we have$$\begin{aligned} \sigma (M, \pi ) = \sigma (M_0, \pi _0) + \sigma (M_1, \pi _1). \end{aligned}$$ 
(compatibility with signature)
An oriented 4kdimensional closed manifold M can be regarded as an element in \(\mathcal {S}_{g, k}\). For this element we have$$\begin{aligned} \sigma (M) = \mathrm {Sign}(M). \end{aligned}$$
Remark 7.12
We remark that Corollary 7.11 “solves” the localization problem, Problem 7.2, in the sense that we have shown the existence of local signature function. However, this construction is abstract and does not give an explicit formula for the local signature. In contrast, in [End00] the author provides an explicit formula for the local signature in the case \(k=1\). In order to find applications of the above results, we would definitely need to find an explicit formula. To proceed further, we need more geometric insight to signature class and their invertible perturbations on mapping class groups. In future works, the author hopes to investigate more on this aspect.
Footnotes
 1.
Given a unital graded \(C^*\)algebra A and an odd selfadjoint unitary operator \(u \in A\), we can construct a unital graded \(*\)homomorphism \(\mathbb {C}l_1 \rightarrow A\) by sending the generator \(\epsilon \) to u. The \(K_1\) class of this element, \([u] \in K_1(A) \simeq KK(\mathbb {C}l_1, A)\) is defined to be the class given by this graded \(*\)homomorphism. The space of odd selfadjoint invertible elements on A retracts to the space of odd selfadjoint unitary elements, so an odd selfadjoint invertible element also defines the class in \(K_1(A)\) this way.
Notes
Acknowledgements
This paper was written for the author’s master’s thesis. The author would like to thank her supervisor Yasuyuki Kawahigashi for his support and encouragement. She also would like to thank Georges Skandalis, Mikio Furuta, and Yosuke Kubota for fruitful advice and discussions. This work is supported by Leading Graduate Course for Frontiers of Mathematical Sciences and Physics, MEXT, Japan.
References
 [AK02]Ashikaga, T., Konno, K.: Global and local properties of pencils of algebraic curves. Algebraic Geometry 2000, Azumino (Hotaka), vol. 36, pp. 1–49. Advanced Studies in Pure Mathematics, Mathematical Society of Japan, Tokyo (2002)Google Scholar
 [ALMP12]Albin, P., Leichtnam, E., Mazzeo, R., Piazza, P.: The signature package on Witt spaces. Ann. Schi. Ec. Norm. Super. (4) 45(2), 241–310 (2012)MathSciNetCrossRefGoogle Scholar
 [AS04]Atiyah, M., Segal, G.: Twisted \(K\)theory. Ukr. Mat. Bull. 1(3), 291–334 (2004)MathSciNetzbMATHGoogle Scholar
 [BC89]Bismut, J.M., Cheeger, J.: \(\eta \)invariants and their adiabatic limits. J. Am. Math. Soc. 2(1), 33–70 (1989)MathSciNetzbMATHGoogle Scholar
 [Bla98]Blackadar, B.: \(K\)Theory for Operator Algebras, Mathematical Sciences Research Institute Publications, vol. 5, 2nd edn. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
 [Con94]Connes, A.: Noncommutative Geometry. Academic Press Inc, San Diego (1994)zbMATHGoogle Scholar
 [CS84]Connes, A., Skandalis, G.: The longitudinal index theorem for foliations. Publ. R.I.M.S. Kyoto Univ. 20, 1139–1183 (1984)MathSciNetCrossRefGoogle Scholar
 [Deb01]Debord, C.: Holonomy groupoids of singular foliations. J. Differ. Geom. 58(3), 467–500 (2001)MathSciNetCrossRefGoogle Scholar
 [DL10]Debord, C., Lescure, J.M.: Index Theory and Groupoids. Geometric and Topological Methods for Quantum Field Theory, pp. 86–158. Cambridge University Press, Cambridge (2010)zbMATHGoogle Scholar
 [DLR15]Debord, C., Lescure, J.M., Rochon, F.: Pseudodifferential operators on manifolds with fibered corners. Ann. Inst. Fourier (Grenoble) 65(4), 1799–1880 (2015)MathSciNetCrossRefGoogle Scholar
 [DS17]Debord, C., Skandalis, G.: Blowup constructions for Lie groupoids and a Boutet de Monvel type calculus. Preprint. arXiv:1705.09588
 [End00]Endo, H.: Meyer’s signature cocycle and hyperelliptic fibrations. Math. Ann. 316(2), 237–257 (2000)MathSciNetCrossRefGoogle Scholar
 [EE67]Earle, C.J., Eells, J.: The diffeomorphism group of a compact Riemann surface. Bull. Am. Math. Soc. 73, 557–559 (1967)MathSciNetCrossRefGoogle Scholar
 [Fur99]Furuta, M.: Surface bundles and local signatures. Topological Studies around Riemann Surfaces, pp. 47–53 (1999) (in Japanese) Google Scholar
 [GH04]Guentner, E., Higson, N.: Group \(C^*\)algebras and \(K\)theory. In: Noncommutative Geometry, pp. 137–251, Lecture Notes in Mathematics Jond, p. 2004. CIME/CIME Foundation Subseries. Springer, Berlin (1831)Google Scholar
 [HR00]Higson, N., Roe, J.: Analytic \(K\)Homology. Oxford Mathematical Monographs. Oxford Science Publications. Oxford University Press, Oxford (2000)zbMATHGoogle Scholar
 [Kas88]Kasparov, G.G.: Equivariant \(KK\)theory and the Novikov conjecture. Invent. Math. 91(1), 147–201 (1988)ADSMathSciNetCrossRefGoogle Scholar
 [Kaw97]Kawazumi, N.: Homology of hyperelliptic mapping class groups for surfaces. Topol. Appl. 76(3), 203–216 (1997)MathSciNetCrossRefGoogle Scholar
 [Luc05]Luck, L.: Survey on classifying spaces for families of subgroups. In: Infinite Groups: Geometric, Combinatorial and Dynamical Aspects, vol. 248, pp. 269–322, Progress in Mathematical. Birkhauser, Basel, (2005)Google Scholar
 [LM89]Lawson, H.B., Michelsohn, M.L.: Spin geometry. Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ (1989)Google Scholar
 [LMP06]Leichtnam, E., Mazzeo, R., Piazza, P.: The index of Dirac operators on manifolds with fibered boundaries. Bull. Belg. Math. Soc. Simon Stevin 13(5), 845–855 (2006)MathSciNetzbMATHGoogle Scholar
 [LN01]Lauter, R., Nistor, V.: Analysis of geometric operators on open manifolds: a groupoid approach. In: Quantization of Singular Symplectic Quotients, vol. 198, pp. 181–229, Progress in Mathematics. Birkhauser, Basel (2001)CrossRefGoogle Scholar
 [Mat96]Matsumoto, Y.: Lefschetz fibrations of genus two—a topological approach. Topology and Teichmuller Spaces (Katinkulta, 1995), pp. 123–148, World Scientific Publishing, River Edge, NJ (1996)Google Scholar
 [Maz91]Mazzeo, R.: Elliptic theory of differential edge operators. I. Commun. Partial Differ. Equ. 16(10), 1615–1664 (1991)MathSciNetCrossRefGoogle Scholar
 [Mel93]Melrose, R.: The Atiyah–Patodi–Singer Index Theorem. Research Notes in Mathematics, vol. 4. A K Peters Ltd, Wellesley (1993)CrossRefGoogle Scholar
 [Mon99]Monthubert, B.: Pseudodifferential calculus on manifolds with corners and groupoids. Proc. Am. Math. Soc. 127(10), 2871–2881 (1999)MathSciNetCrossRefGoogle Scholar
 [MM98]Mazzeo, R., Melrose, R.: Pseudodifferential operators on manifolds with fibered boundaries, "Mikio Sato: A great Japanese mathematician of the twentieth century". Asian J. Math. 2(4), 833–866 (1998)MathSciNetCrossRefGoogle Scholar
 [MP97]Melrose, R., Piazza, P.: An index theorem for families of Dirac operators on odddimensional manifolds with boundary. J. Differ. Geom. 46(2), 287–334 (1997)MathSciNetCrossRefGoogle Scholar
 [MR06]Melrose, R., Rochon, F.: Index in \(K\)theory for families of fibered cusp operators. KTheory 37(12), 25–104 (2006)MathSciNetCrossRefGoogle Scholar
 [Nis00]Nistor, V.: Groupoids and the integration of Lie algebroids. J. Math. Soc. Jpn. 52(4), 847–868 (2000)MathSciNetCrossRefGoogle Scholar
 [NWX99]Nistor, V., Weinstein, A., Xu, P.: Pseudodifferential operators on differential groupoids. Pac. J. Math. 189(1), 117–152 (1999)MathSciNetCrossRefGoogle Scholar
 [PZ19]Piazza, P., Zenobi, V.F.: Singular spaces, groupoids and metrics of positive scalar curvature. J. Geom. Phys. 137, 87–123 (2019)ADSMathSciNetCrossRefGoogle Scholar
 [RW06]Rosenberg, J., Weinberger, S.: The signature operator at \(2\). Topology 45(1), 47–63 (2006)MathSciNetCrossRefGoogle Scholar
 [Sat13]Sato, M.: A local signature for fibered \(4\)manifolds with a finite group action. Tohoku Math. J. (2) 65(4), 545–568 (2013)MathSciNetCrossRefGoogle Scholar
 [Vas06]Vassout, S.: Unbounded pseudodifferential calculus on Lie groupoids. J. Funct. Anal. 236(1), 161–200 (2006)MathSciNetCrossRefGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.