Codimension 2 transfer of higher index invariants

Abstract

This paper is devoted to the study of the higher index theory of codimension 2 submanifolds originated by Gromov–Lawson and Hanke–Pape–Schick. The first main result is to construct the ‘codimension 2 transfer’ map from the Higson–Roe analytic surgery exact sequence of a manifold M to that of its codimension 2 submanifold N under some assumptions on homotopy groups. This map sends the primary and secondary higher index invariants of M to those of N. The second is to establish that the codimension 2 transfer map is adjoint to the co-transfer map in cyclic cohomology, defined by the cup product with a group cocycle. This relates the Connes–Moscovici higher index pairing and Lott’s higher $\rho$-number of M with those of N.

Appendices

Appendix A: Secondary external product via pseudo-local C*-algebra

In this appendix, we define the secondary external product in coarse C*-algebra K-theory, the product of the analytic structure set and the K-homology group, in terms of the pseudo-local coarse C*-algebra $D^*(\widetilde{M})^\Gamma$, instead of Yu’s localization algebra studied in [71]. The construction is inspired from the definition of the Kasparov product [35].

Remark A.1

We use the K-theory of $\mathbb {Z}_2$-graded C*-algebras by Van Daele [17] for coarse C*-algebras with Clifford algebra symmetry. In general, for a $\mathbb {Z}_2$-graded (Real) C*-algebra A, its (Real) K-theory ${{\,\mathrm{\textrm{K}}\,}}_1(A)$ is defined to be the set of homotopy classes of (Real) odd self-adjoint unitaries on A. We define the coarse C*-algebras $C^*_{p,q}(\widetilde{M})^\Gamma$, $D^*_{p,q}(\widetilde{M})^\Gamma$, $Q^*_{p,q}(\widetilde{M})^\Gamma$ consisting of operators which is graded-commutative with the action of Clifford algebras. Then ${{\,\mathrm{\textrm{K}}\,}}_1(C^*_{p,q}(\widetilde{M})^\Gamma ) \cong {{\,\mathrm{\textrm{K}}\,}}_{1-p+q}(C^*(\widetilde{M})^\Gamma )$ holds, and the same is also true for $D^*$ and $Q^*$.

For a spin manifold M with $n:=\dim M =8m - q$ (where $q = 0, \ldots , 7$), the Dirac operator on the spinor bundle of $\textrm{Spin}(M) \times _{\textrm{Spin}_n} \Delta _{8m}$ is equipped with an additional symmetry of ${ C}\ell _{0,q}$, and hence determines a K-theory class (cf. 4.8). Similarly, the signature operator on an odd-dimensional manifold determines a complex K-theory class $[\chi (D_M^\textrm{sgn})] \in {{\,\mathrm{\textrm{K}}\,}}_1(Q^*_{0,1}(\widetilde{M})^\Gamma )$ (we refer to [63], Definition and Notation 1.1).

Definition A.2

The external product

$$\begin{aligned} {\cdot } \boxtimes {\cdot } :{{\,\mathrm{\textrm{K}}\,}}_1(D^*_{p_1,q_1}(M_1)^{\Gamma _1}) \otimes {{\,\mathrm{\textrm{K}}\,}}_1(Q^*_{p_2,q_2}(\widetilde{M_2})^{\Gamma _2}) \rightarrow {{\,\mathrm{\textrm{K}}\,}}_1(D^*_{p,q}(\widetilde{M}_1 \times \widetilde{M}_2)^{\Gamma _1 \times \Gamma _2}), \end{aligned}$$

where $p=p_1+p_2$ and $q=q_1+q_2$, is defined as

$$\begin{aligned}{}[F_1] \boxtimes [F_2]:= [ f(F_1 \hat{\otimes } 1 + (1-F_1^2) \hat{\otimes } F_2) ], \end{aligned}$$

where $f(x):=x/|x|$.

This definition makes sense because the operator $F_1 \hat{\otimes } 1 + (1-F_1^2) \hat{\otimes } F_2$ is invertible in $D^*_{p+q}(\widetilde{M}_1 \times \widetilde{M}_2)^{\Gamma _1 \times \Gamma _2}$. Indeed, this is seen as

$$\begin{aligned} (F_1 \hat{\otimes } 1 + (1-F_1^2) \hat{\otimes } F_2)^2 = F_1 ^2 \hat{\otimes } 1 + (1-F_1^2) \hat{\otimes } F_2^2 \ge F_1^2 \hat{\otimes } 1 >0. \end{aligned}$$

This also shows that $[F_1] \boxtimes [F_2]$ is well-defined independent of the choice of representatives $F_1$ and $F_2$.

Lemma A.3

For $i=1,2$, let $D_i$ be a $\Gamma _i$-invariant $\mathbb {Z}_2$-graded elliptic first-order differential operator on $M_i$ anticommuting with a $\mathbb {Z}_2$-graded representation of ${ C}\ell _{p_i,q_i}$. Moreover, we assume that $D_1$ is invertible. Then we have

$$\begin{aligned}{}[\chi (D_1)] \boxtimes [\chi (D_2) ] = [\chi (D_1 \hat{\otimes } 1 + 1 \hat{\otimes } D_2)]. \end{aligned}$$

Proof

This is a standard argument in Kasparov theory. We just refer the reader to [3] or [7], Proposition 18.10.1. $\square$

Lemma A.3 shows that

where $\epsilon = 1$ if both $\dim M_1$ and $\dim M_2$ are odd, and otherwise $\epsilon = 0$. In particular, when $M_2=\mathbb {R}$, these equalities means that

$$\begin{aligned} \rho (g_{N\times \mathbb {R}})&= \rho (g_N ) \boxtimes [\mathbb {R}] ,\\ \rho _\textrm{sgn}(f_{N \times \mathbb {R}} )&= 2^{\epsilon } \rho _\textrm{sgn}(f_N ) \boxtimes [\mathbb {R}]_\textrm{sgn}, \end{aligned}$$

where $\epsilon$ is 0 or 1 if $\dim N$ is even or odd. Therefore, the coarse Mayer–Vietoris boundary map

sends the higher $\rho$-invariants of $N \times \mathbb {R}$ as

$$\begin{aligned} \partial _{\textrm{MV}} (\rho (g_{N \times \mathbb {R}}))&= \partial _{\textrm{MV}} (\rho (g_N ) \boxtimes [\mathbb {R}]) = \rho (g_N) \boxtimes \partial _{\textrm{MV}} [\mathbb {R}] = \rho (g_N), \\ \partial _{\textrm{MV}} (\rho _\textrm{sgn}(f_{N \times \mathbb {R}}))&= \partial _{\textrm{MV}} (2^\epsilon \rho (f_N ) \boxtimes [\mathbb {R}]) = 2^\epsilon \rho (f_N) \boxtimes \partial _{\textrm{MV}} [\mathbb {R}] = 2^\epsilon \rho (g_N). \end{aligned}$$

This completes the proof of Lemma 4.12 (3), (4).

Appendix B: Cyclic homology of a crossed product and group homology

In this appendix, we give a more detailed discussion on the proof of Lemma 5.6. We show that the exact sequence of cyclic chain complexes

$$\begin{aligned} 0 \rightarrow \ker \theta \rightarrow {{\,\mathrm{\textrm{CC}}\,}}_*(C(\mathcal {B}, \mathcal {K}_\sigma ) \rtimes _\textrm{alg}\Gamma )[\![v^{\pm 1}]\!] \xrightarrow {\theta } {{\,\mathrm{\textrm{CC}}\,}}_*(\mathbb {C}[\Gamma ])[\![v^{\pm 1}]\!] \rightarrow 0 \end{aligned}$$

(B.1)

is quasi-isomorphic to the exact sequence of chain complexes

$$\begin{aligned} 0 \rightarrow C_\bullet (\Gamma , \Omega ^\bullet _0(\mathcal {B}_0 ) )[\![v^{\pm 1}]\!]&\rightarrow C_\bullet (\Gamma , \Omega ^\bullet (\mathcal {B}) )[\![v^{\pm 1}]\!] \rightarrow C_\bullet (\Gamma , \mathbb {C})[\![v^{\pm 1}]\!] \rightarrow 0 \end{aligned}$$

(B.2)

given in (5.7) (note that the excision property of cyclic homology states that the inclusion ${{\,\mathrm{\textrm{CC}}\,}}_*(C_0(\mathcal {B}_0, \mathcal {K}_\sigma ) \rtimes _\textrm{alg}\Gamma )[\![v^{\pm 1}]\!] \rightarrow \ker \theta$ is quasi-isomorphic). To this end, we construct maps from (B.1) and (B.2) to another exact sequence of complexes

$$\begin{aligned} 0 \rightarrow \underline{\Omega }{}_0(\mathcal {B}_0 \rtimes \Gamma )[u^{\pm 1}]' \rightarrow \underline{\Omega }(\mathcal {B}\rtimes \Gamma )[u^{\pm 1}]' \rightarrow \Omega (\textrm{pt}\rtimes \Gamma )[u^{\pm 1}]' \rightarrow 0. \end{aligned}$$

(B.3)

Here, for a vector space V, $V'$ stands for the algebraic dual vector space ${{\,\mathrm{\textrm{Hom}}\,}}(V,\mathbb {C})$.

We start with the definition of (B.3). Following ([2], Subsection 2.3), let $\underline{\Omega }{}_\bullet (\mathcal {B})$ denotes the dual cocomplex of $\Omega ^\bullet (\mathcal {B})$, which is equipped with the degree $-1$ differential $d_{\textrm{dR}}$. we define the $\mathbb {Z}$-graded vector space $\underline{\Omega }^\bullet (\mathcal {B}\rtimes \Gamma )[u^{\pm 1}]$, where u is a degree 2 formal symbol, as

$$\begin{aligned}&\underline{\Omega }^{r,s} (\mathcal {B}\rtimes \Gamma ) \\&:= \Big \{ (\omega _{(n)}) \in \prod _{n \ge 0} \underline{\Omega }^r (\mathcal {B}\times \Gamma ^n) \otimes \Omega ^s(\Delta ^n) \mid (\textrm{id}\times \delta ^i )^*\omega _{(n)} = (\delta _i \times \textrm{id})^* \omega _{(n-1)} \Big \}_{\textstyle ,} \end{aligned}$$

where $\delta _i :\mathcal {B}\times \Gamma ^{n} \rightarrow \mathcal {B}\times \Gamma ^{n-1}$ and $\delta ^i :\Delta ^{n-1} \rightarrow \Delta ^{n}$ denote the i-th face maps. We write $\omega (g_1, \ldots , g_n)$ for the restriction of $\omega$ to $\mathcal {B}\times \{ g_1, \ldots , g_n \} \times \Delta ^n$. The twisted simplicial de Rham differential is defined as $d_\Theta := ud_{\textrm{dR}} + d_\Delta + \Theta$, where $d_\Delta$ is the de Rham differential on $\Omega ^s(\Delta ^n)$ and $\Theta \in \underline{\Omega }^{1,2}(\mathcal {B}\rtimes \Gamma )$ is the differential form

$$\begin{aligned} \Theta (g_1, \ldots , g_n):= \sum _{1 \le i \le j \le k} 2\alpha (g_1 \cdots g_i, g_{i+1} \cdots g_j)dt_idt_j, \end{aligned}$$

where $\alpha (g,h) = \sigma (g,h)^{-1}d\sigma (g,h) \in \Omega ^1(\mathcal {B})$. The dual of the short exact sequence $0 \rightarrow \Omega _0^\bullet (\mathcal {B}_0) \rightarrow \Omega ^\bullet (\mathcal {B}) \rightarrow \mathbb {C}\rightarrow 0$ gives rise to

$$\begin{aligned} 0 \rightarrow \underline{\Omega }^\bullet (\textrm{pt}\rtimes \Gamma )[u^{\pm 1}] \rightarrow \underline{\Omega }(\mathcal {B}\rtimes \Gamma )[u^{\pm 1}] \rightarrow \underline{\Omega }{}_0^\bullet (\mathcal {B}_0 \rtimes \Gamma )[u^{\pm 1}] \rightarrow 0, \end{aligned}$$

which is dual to (B.3).

Remark B.4

We review the Packer–Raeburn construction ([51], Theorem 3.4). Let $\mathcal {H}_\sigma$ be the Hilbert bundle $\mathcal {B}\times \ell ^2(\Gamma )$, on which $\Gamma$ acts as

$$\begin{aligned} u_g \xi := ((\gamma _g \otimes \lambda _g) \circ v(g)) \xi \end{aligned}$$

for any $\xi \in C(\mathcal {B}, \mathcal {H}_\sigma )$, where

$$\begin{aligned} v(g):= \mathop {\textrm{diag}}(\sigma (g,h)^*)_{h \in \Gamma } \in C(\mathcal {B}, \mathbb {B}(\ell ^2\Gamma )). \end{aligned}$$

Let $\mathcal {K}_\sigma$ denote the associated compact operator algebra bundle on $\mathcal {B}$. Then, the relation $u_gu_h = \sigma (g,h,x)^* u_{gh}$ holds, i.e., $\{ u_g\}$ is a $\sigma ^*$-twisted unitary representation of the groupoid $\mathcal {B}\rtimes \Gamma$. This implements a twisted $\Gamma$-equivariant Morita equivalence between $(C(\mathcal {B}), \sigma )$ and the untwisted $\Gamma$-C*-algebra $C(\mathcal {B}, \mathcal {K}_\sigma )$. In particular, $C(\mathcal {B}) \rtimes _\sigma \Gamma$ is Morita equivalent to $C(\mathcal {B}, \mathcal {K}_\sigma ) \rtimes \Gamma$.

Lemma B.5

There are homomorphisms from the exact sequences (B.1) and (B.2) to (B.3), which induces isomorphism of homology.

Proof

The pairing $\langle \cdot , \cdot \rangle$ of $C_\bullet (\Gamma , \Omega ^\bullet (\mathcal {B}))[\![v ^{\pm 1} ]\!]$ and $\underline{\Omega } (\mathcal {B}\rtimes \Gamma )^\bullet [u^{\pm 1}]$ is defined by

$$\begin{aligned} \left\langle \sum _{g_1, \ldots , g_m} \xi _{g_1, \ldots , g_m} v^k, (\omega _{(n)}) u^l \right\rangle := \delta _{k,-l} \sum _{g_1, \ldots , g_m} \int _{\mathcal {B}\times \Delta ^n} \xi _{g_1, \ldots , g_m} \wedge \omega _{g_1, \ldots , g_m}. \end{aligned}$$

This pairing satisfy $\langle \xi , ud _{\textrm{dR}}\omega \rangle = \langle vd_{\textrm{dR}} \xi , \omega \rangle$, $\langle \xi , d_{\Delta } \omega \rangle = \langle \delta _\Gamma \xi , \omega \rangle$ and $\langle \xi , \Theta \omega \rangle = \langle \Theta \xi , \omega \rangle$, and that the following diagram commutes;

Moreover the left and the right vertical maps induce the isomorphism of homology. Indeed, the homology groups of two complexes at the left (resp. the right) are both isomorphic to the group homology $H_{[*-1]}(\pi ; \mathbb {C})$ (resp. $H_{[*]}(\Gamma , \mathbb {C})$).

In [2], Angel constructed a pairing of (B.1) and (B.3) as a variation of the JLO character. Although the space $\mathcal {B}$ is not a manifold, the same definition also works in our setting. Moreover, due to the 1-dimensionality of $\mathcal {B}$, the construction is partly simplified.

The simplicial connection and curvature forms of $\mathcal {H}_\sigma$ is defined by gluing the connections and curvature forms

$$\begin{aligned} \nabla _u^k(g_1, \ldots , g_k)&:= d_{\textrm{dR}} + u^{-1}d_\Delta + t_1 A(g_1) + t_2 A(g_1g_2) + \cdots + t_k A(g_1 \cdots g_k), \\ \vartheta _{(k)}(g_1,\ldots ,g_k)&:= \sum _{1 \le i \le j \le k} \alpha (g_1 \cdots g_i, g_{i+1} \cdots g_j)(t_idt_j -t_jdt_i). \end{aligned}$$

The JLO homomorphism

$$\begin{aligned} \mathcal {T}_{\textrm{JLO}} :(C_\bullet (\Gamma , {{\,\mathrm{\textrm{CC}}\,}}_\bullet (C^\infty (\mathcal {B}, \mathcal {K}_\sigma )))[\![v^{\pm 1}]\!], b + uB+\delta _\Gamma ') \rightarrow (\underline{\Omega }^\bullet (\mathcal {B}\rtimes \Gamma )[u^{\pm 1}]', d_\Theta ) \end{aligned}$$

is defined as

$$\begin{aligned}&\langle (\omega _{(n)}) u^k, \mathcal {T}_{\textrm{JLO}}(((\tilde{a}_0 \otimes \cdots \otimes a_p),(g_1, \ldots , g_q))v^l) \rangle \\&\quad = \delta _{k,l} \int _{\mathcal {B}\times \Delta ^q} \omega _{g_1, \cdots , g_q} \wedge \int _{\Delta ^p} \textrm{Tr}(\tilde{a_0}e^{s_0 \vartheta _{(q)}}(\nabla _u^q a_1)e^{s_1 \vartheta _{(q)}} \cdots e^{s_p \vartheta _{(q)} } (\nabla _u^q a_q)) ds_1 \cdots ds_q. \end{aligned}$$

It is shown in the same way as [2, Theorem 5.5] that it is a chain map.

Moreover, in [2, Subsection 5.3] (we also refer to [46, 2.5]), a quasi-isomorphism

$$\begin{aligned} \Psi _A :(C_\bullet (\Gamma , {{\,\mathrm{\textrm{CC}}\,}}_\bullet (A))[u^{\pm 1}], b + uB + \delta _\Gamma ' ) \rightarrow ({{\,\mathrm{\textrm{CC}}\,}}_\bullet (A \rtimes _\textrm{alg}\Gamma )[u^{\pm 1}], b + uB) \end{aligned}$$

is constructed for any $\Gamma$-algebra A. By the construction, this $\Psi _A$ is functorial. Therefore, $\Psi _A \circ \mathcal {T}_{\textrm{JLO}}$ gives rise to a commutative diagram

Since the left and the right vertical maps are both isomorphic, this finishes the proof of the lemma. $\square$