Abstract
Quasi-elliptic cohomology is a variant of Tate K-theory. It is the orbifold K-theory of a space of constant loops. For global quotient orbifolds, it can be expressed in terms of equivariant K-theories. In this paper we show how this theory is equipped with power operations. We also prove that the Tate K-theory of symmetric groups modulo a certain transfer ideal classify the finite subgroups of the Tate curve.
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1 Introduction
An elliptic cohomology theory is an even periodic multiplicative generalized cohomology theory whose associated formal group is the formal completion of an elliptic curve. It is an old idea of Witten, as shown in [17], that the elliptic cohomology of a space X is related to the \(\mathbb {T}\)-equivariant K-theory of the free loop space \(LX=\mathbb {C}^{\infty }(S^1, X)\) with the circle \(\mathbb {T}\) acting on LX by rotating loops.
It is surprisingly difficult to make this precise, especially if one wishes to consider equivariant generalization of this construction. In this case the loop space LX with the natural rotation action is a rich orbifold. In this paper we offer a new formulation between the loop space and Tate K-theory via a new theory which we call quasi-elliptic cohomology.
Tate K-theory is the generalized elliptic cohomology associated to the Tate curve. The Tate curve Tate(q) is an elliptic curve over Spec\(\mathbb {Z}((q))\), which is classified as the completion of the algebraic stack of some nice generalized elliptic curves at infinity. A good reference for Tate(q) is Section 2.6 of [1]. We give a sketch of it in Sect. 6.1. The relation between Tate K-theory and string theory is better understood than for most known elliptic cohomology theories. The definition of G-equivariant Tate K-theory for finite groups G is modelled on the loop space of a global quotient orbifold, which is formulated explicitly in Section 2, [10]. Its relation with string theory and loop space makes Tate K-theory itself a distinctive subject to study.
The idea of quasi-elliptic cohomology is motivated by Ganter’s construction of Tate K-theory. It is not an elliptic cohomology but from it we can recover the Tate K-theory. This new theory can be interpreted in a neat form by equivariant K-theories, which makes many constructions on it easier and more natural than those on the Tate K-theories. Some formulations can be generalized to other equivariant cohomology theories. In addition, quasi-elliptic cohomology provides a method that reduces facts such as the classification of geometric structures on the Tate curve into questions in representation theory.
1.1 Loop space
Quasi-elliptic cohomology is modelled on a version of equivariant loop space. For background on orbifolds and Lie groupoids, we refer the readers to Sections 2, 3, [18, 23].
For any compact Lie group G and a manifold X with a smooth G-action, there is a Lie groupoid \(X/\!\!/G\) which is explained in detail in Chapter 11, [6]. Smooth unbased loops in the orbifold \(X/\!\!/G\) carries a lot of structure: on the one hand, it includes loops represented by smooth maps \(\gamma : \mathbb {R}\longrightarrow X\) such that \(\gamma (t+1)=\gamma (t)g\) for some \(g\in G\); other than the group action by the loop group \(LG:=\mathbb {C}^{\infty }(S^1, G)\), the loop space also has the circle action by rotation. Lerman discussed thoroughly in Section 3, [18] that the strict 2-category of Lie groupoids can be embedded into a weak 2-category whose objects are Lie groupoids, 1-morphisms are bibundles and 2-morphisms equivariant diffeomorphisms between bibundles. Thus, the free loop space of an orbifold M is the category of bibundles from the trivial groupoid \(S^1/\!\!/*\) to the Lie groupoid M. We will write
which is discussed in Definition 2.2. In Definition 2.3, we extend \(Loop_1(X/\!\!/G)\) to a groupoid \(Loop_1^{ext}(X/\!\!/G)\) by adding rotations as morphisms.
Especially we are interested in the ghost loops groupoid \(GhLoop(X/\!\!/G)\), which is defined to be the full subgroupoid of \(Loop^{ext}_1(X/\!\!/G)\) consisting of objects \((\pi , f)\) with the image of f contained in a single G-orbit. Ghost loops are introduced by Rezk in his unpublished manuscript [26]. Another reference is Section 2.1.3, [12]. This groupoid has several good properties. They are computed locally in X. For instance, if \(X=U\cup V\) where U and V are G-invariant open subsets, then
So it satisfies a kind of Mayer–Vietoris property. In addition, if H is a closed subgroup of G and X is the quotient space G / H, \(GhLoop(X/\!\!/G)\) is equivalent to \(GhLoop(\text{ pt }/\!\!/H)\). In other words, it has the change-of-group property.
When G is finite, \(GhLoop(X/\!\!/G)\) is isomorphic to the full subgroupoid \(\Lambda (X/\!\!/G)\) of \(Loop^{ext}_1(X/\!\!/G)\) consisting of constant loops. This groupoid \(\Lambda (X/\!\!/G)\) can be regarded as an extended version of the inertia groupoid \(I(X/\!\!/G)\). Please see Definition 3.7 for inertia groupoid.
1.2 Quasi-elliptic cohomology
For any compact orbifold groupoid \(\mathbb {G}\), the orbifold K-theory \(K_{orb}(\mathbb {G})\) is defined to be the Grothendieck ring of isomorphism classes of \(\mathbb {G}\)-vector bundles on \(\mathbb {G}\). In particular, \(K_{orb}(X/\!\!/G)\) is \(K_G(X)\). A reference for orbifold K-theory is Chapter 3, [3] and a reference for equivariant K-theory is [27].
Quasi-elliptic cohomology \(QEll^*(X/\!\!/G)\) is defined to be the orbifold K-theory of a subgroupoid \(\Lambda (X/\!\!/G)\) of \(GhLoop(X/\!\!/G)\) consisting of constant loops. When G is a finite group, \(QEll^*_{G}(X)\) can be expressed in terms of the equivariant K-theory of X and its subspaces as
where \(G_{conj}\) is a set of representatives of G-conjugacy classes in G. The group \(\Lambda _G(\sigma ):= C_G(\sigma )\times \mathbb {R}/\langle (\sigma , -1)\rangle \) acts on the fixed point space \(X^{\sigma }\) by \([g, t]\cdot m=g\cdot m\). In a coming paper by the author [13], we will present the construction of \(QEll^*_{G}(X)\) for any compact Lie group G.
\(QEll_G(X)\) has the structure of a \(\mathbb {Z}[q^{\pm }]\)-algebra. We have
We formulate the K\(\ddot{u}\)nneth map, restriction map, change of group isomorphism and transfer for QEll. In general, if \(H^*\) is an equivariant cohomology theory, then the functor
gives a new equivariant cohomology theory. Moreover, for each global cohomology theory, we can formulate a new global cohomology theory via the ghost loops.
1.3 Power operation
One significant feature of quasi-elliptic cohomology is that it has power operations, which was first observed by Ganter, as shown in [10, 11]. In Sect. 4 we construct the total power operation of quasi-elliptic cohomology. It satisfies the axioms for equivariant power operations that Ganter gave in Definition 4.3 in [9]. For more details, please see Theorem 4.12.
The power operation \(\{\mathbb {P}_n\}_{n\ge 0}\) mixes the power operation in K-theory with the natural operations of dilating and rotating loops. The key point of the construction of the power operation is an intermediate groupoid \(d_{(\underline{g}, \sigma )}(X)\) with \((\underline{g}, \sigma )\in G\wr \Sigma _n\). It is constructed from \(\Lambda (X/\!\!/G)\) and isomorphic to \((X^{\times n})^{(\underline{g}, \sigma )}/\!\!/ \Lambda _{G\wr \Sigma _n}(\underline{g}, \sigma )\). For more details of the construction, please see Sect. 4.2.
We illustrate what this power operation looks like by examples. Let G be the trivial group and X a space. Let \((-)_k\) denote the rescaling map defined in (4.11).
When \(n=2\), \(\mathbb {P}_{(\underline{1}, (1)(1))}(x)=x\boxtimes x\) and \(\mathbb {P}_{(\underline{1}, (12))}(x)=(x)_2\).
When \(n=3\), \(\mathbb {P}_{(\underline{1}, (1)(1)(1))}(x)=x\boxtimes x\boxtimes x\), \(\mathbb {P}_{(\underline{1}, (12)(1))}(x)=(x)_2\boxtimes x\), and \(\mathbb {P}_{(\underline{1}, (123))}(x)=(x)_3\).
In these cases, the number of factors corresponds to the number of cycles in the permutation and the rescaling map corresponds to the length of each cycle. For more examples please see Example 4.13.
For any equivariant cohomology theory \(\{H^*_G( -)\}_G\) with an \(H_{\infty }\)-structure in Ganter’s sense, we can formulate a power operation for the equivariant cohomology theories
in the same way.
In addition, we can formulate the total power operation for the orbifold quasi-elliptic cohomology in the sense of Definition 3.9, [11]. The construction of the power operation is shown in Sect. 5.3.
1.4 Classification of the finite subgroups of the Tate curve
Though the general formulas for the power operations in \(QEll_G\) are complicated, to understand it, it is useful to consider special cases. It is already illuminating to consider the case that X is a point and G is the trivial group, the power operation has a neat form, as shown in Example 4.13. It has a natural interpretation in terms of the Tate elliptic curve.
In Sect. 6.3 applying the power operation we prove that the Tate K-theory of symmetric groups modulo the transfer ideal classifies the finite subgroups of the Tate curve, which is analogous to the principal result in Strickland [28] that the Morava E-theory of the symmetric group \(\Sigma _n\) modulo a certain transfer ideal classifies the power subgroups of rank n of the formal group \(\mathbb {G}_E\).
The finite subgroups of the Tate curve are classified by
First we prove the parallel conclusion for quasi-elliptic cohomology.
Theorem 1.1
where \(\mathcal {I}^{QEll}_{tr}\) is the transfer ideal defined in (6.4) and \(q'\) is the image of q under the power operation \(\mathbb {P}_N\).
Then applying the relationship between \(QEll^*\) and Tate K-theory, we obtain the main theorem.
Theorem 1.2
The Tate K-theory of symmetric groups modulo the transfer ideal \(I^{Tate}_{tr}\) defined in (6.3) classifies finite subgroups of the Tate curve. Explicitly,
where \(q'\) is the image of q under the power operation \(P^{Tate}\) constructed in Definition 5.10, [10].
Moreover, via the isomorphism in Theorem 1.1, we can define a ring homomorphism
as shown in Proposition 6.5. Under the identification (1.2), it extends uniquely to the ring homomorphism
constructed in Section 5.4, [10]. In [14] we construct the universal finite subgroup of the Tate curve via the operation \(\overline{P}_N\).
2 Models for orbifold loops and ghost loops
To understand \(QEll^*_G(X)\), it is essential to understand the orbifold loop space. In this section, we will describe several models for the loop space of \(X{/\!\!/}G\). In Definition 2.2 we discuss \(Loop_1(X/\!\!/G)\) and introduce another model \(Loop_2(X{/\!\!/}G)\) in Definition 2.4.
The groupoid structure of \(Loop_1(X/\!\!/G)\) generalizes \(Map(S^1, X)/\!\!/G\), which is a subgroupoid of it. Other than the G-action, we also consider the rotation by the circle group \(\mathbb {T}\) on the objects and form the groupoids \(Loop_1^{ext}(X/\!\!/G)\) and \(Loop_2^{ext}(X/\!\!/G)\). The groupoid \(Loop_2^{ext}(X/\!\!/G)\) has a skeleton
where each \({_1}\mathcal {L}{_g} X =\text{ Map }_{\mathbb {Z}/l\mathbb {Z}}(\mathbb {R}/l\mathbb {Z}, X)\) with l the order of g is equipped with an evident \(C_G(g)\)-action. \(\mathcal {L} (X/\!\!/G)\) has the same space of objects as the groupoid \(L(X/\!\!/G)\) discussed in Definition 2.3, [21], from which equivariant Tate K-theory is defined. It has richer morphisms. The circle group \(\mathbb {T}\) acts on \(\mathbb {R}/l\mathbb {Z}\) by rotation, and so in principle on the orbifold \({_1}\mathcal {L}_gX\).
The key groupoid \(\Lambda (X/\!\!/G)\) in the construction of quasi-elliptic cohomology is the full subgroupoid of \(\mathcal {L} (X/\!\!/G)\) consisting of the constant loops. In order to unravel the relevant notations in the construction of \(QEll^*_G(X)\), we study the orbifold loop space in Sects. 2.1.2 and 2.1.3.
In Sect. 2.1.1 we define \(Loop_1(X/\!\!/G)\). In Sect. 2.1.2 we interpret the enlarged groupoid \(Loop^{ext}_1(X/\!\!/G)\) and introduce a skeleton \(\mathcal {L}(X/\!\!/G)\) of it. In Sect. 2.1.3 we show the construction of quasi-elliptic cohomology by ghost loops. In Sect. 3.1 we show the representation ring of \(\Lambda _G(g)\). In Sect. 3.2 we introduce the construction of quasi-elliptic cohomology first in terms of orbifold K-theory and then equivariant K-theory. We show the properties of the theory in Sect. 3.3.
2.1 Loop space
2.1.1 Bibundles
A standard reference for groupoids and bibundles is Sections 2 and 3, [18]. For each pair of Lie groupoids \(\mathbb {H}\) and \(\mathbb {G}\), the bibundles from \(\mathbb {H}\) to \(\mathbb {G}\) are defined in Definition 3.25, [18]. The category \(Bibun(\mathbb {H}, \mathbb {G})\) has bibundles from \(\mathbb {H}\) to \(\mathbb {G}\) as the objects and bundle maps as the morphisms.
Example 2.1
(\(Bibun(S^1/\!\!/*, */\!\!/G)\)) According to the definition, a bibundle from \(S^1/\!\!/*\) to \(*/\!\!/G\) with G a Lie group is a smooth manifold P together with two maps \(\pi : P\longrightarrow S^1\) a smooth principal G-bundle and the constant map \(r: P\longrightarrow *\). So a bibundle in this case is equivalent to a smooth principal G-bundle over \(S^1\). The morphisms in \(Bibun(S^1/\!\!/*, */\!\!/G)\) are bundle isomorphisms.
Definition 2.2
(\(Loop_1(X/\!\!/G)\)) Let G be a Lie group acting smoothly on a manifold X. We use \(Loop_1(X/\!\!/G)\) to denote the category \(Bibun(S^1/\!\!/*, X/\!\!/G)\), which generalizes Example 2.1. Each object consists of a smooth manifold P and two structure maps \(P\buildrel {\pi }\over \longrightarrow S^1\) a smooth principal G-bundle and \(f: P\longrightarrow X\) a G-equivariant map. We use the same symbol P to denote both the object and the smooth manifold when there is no confusion. A morphism is a G-bundle map \(\alpha : P\longrightarrow P'\) making the diagram below commute.
Thus, the morphisms in \(Loop_1(X/\!\!/G)\) from P to \(P'\) are bundle isomorphisms.
Only the G-action on X is considered in \(Loop_1(X/\!\!/G)\). We add the rotations by adding more morphisms into the groupoid.
Definition 2.3
(\(Loop^{ext}_1(X/\!\!/G)\)) Let \(Loop^{ext}_1(X/\!\!/G)\) denote the groupoid with the same objects as \(Loop_1(X/\!\!/G)\). Each morphism consists of the pair \((t, \alpha )\) where \(t\in \mathbb {T}\) is a rotation and \(\alpha \) is a G-bundle map. They make the diagram below commute.
The groupoid \(Loop_1(X/\!\!/G)\) is a subgroupoid of \(Loop^{ext}_1(X/\!\!/G)\).
2.1.2 Another model for orbifold loop space
We give an equivalent description of the groupoids discussed in Sect. 2.1.1. The new models \(Loop_2(X/\!\!/G)\) and \(Loop_2^{ext}(X/\!\!/G)\) are more practicable to compute. We give a skeleton \(\mathcal {L}(X/\!\!/G)\) of \(Loop_2^{ext}(X/\!\!/G)\) when G is finite in Proposition 2.7.
Definition 2.4
(\(Loop_2(X/\!\!/G)\)) Let \(Loop_2(X/\!\!/G)\) denote the groupoid whose objects are \((\sigma , \gamma )\) with \(\sigma \in G\) and \(\gamma : \mathbb {R}\longrightarrow X\) a continuous map such that \(\gamma (s+1)= \gamma (s)\cdot \sigma \), for any \(s\in \mathbb {R}\). A morphism \(\alpha : (\sigma , \gamma )\longrightarrow (\sigma ', \gamma ')\) is a continuous map \(\alpha : \mathbb {R}\longrightarrow G\) satisfying \(\gamma '(s)= \gamma (s)\alpha (s)\). Note that \(\alpha (s)\sigma '=\sigma \alpha (s+1)\), for any \(s\in \mathbb {R}\).
Moreover, we can extend the groupoid \(Loop_2(X/\!\!/G)\) by adding the rotations.
Definition 2.5
(\(Loop^{ext}_2(X/\!\!/G)\))
Let \(Loop^{ext}_2(X/\!\!/G)\) denote the groupoid with the same objects as \(Loop_2(X/\!\!/G)\). A morphism \((\sigma , \gamma )\longrightarrow (\sigma ', \gamma ')\) consists of the pair \((\alpha , t)\) with \(\alpha :\mathbb {R}\longrightarrow G\) a continuous map and \(t\in \mathbb {R}\) satisfying \(\gamma '(s)=\gamma (s-t)\alpha (s-t)\). Note that \((\alpha , t+1)\) and \((\alpha \sigma ', t)\) are the same morphism and each morphism can be represented by a pair \((\alpha , t)\) with \(t\in [0, 1)\).
\(Loop_2(X/\!\!/G)\) is a subgroupoid of \(Loop^{ext}_2(X/\!\!/G)\).
Lemma 2.6
The groupoid \(Loop^{ext}_1(X/\!\!/G)\) is isomorphic to a full subgroupoid of \(Loop^{ext}_2(X/\!\!/G)\).
Proof
Define a functor
by sending an object
to \((\sigma , \gamma )\) with \(\gamma (s):= f([s, e])\) and \(\sigma =\gamma (0)^{-1}\gamma (1)\) and sending a morphism
to \((\alpha , t): (\sigma , \gamma )\longrightarrow (\sigma ', \gamma ')\) with \(\alpha (s):=F([s, e])^{-1}.\)
F is a fully faithful functor but not essentially surjective. \(\square \)
Therefore, the groupoid \(Loop^{ext}_2(X/\!\!/G)\) contains all the information of \(Loop^{ext}_1(X/\!\!/G)\). Next we will show a skeleton of this larger groupoid when G is finite. Before that, we introduce some symbols.
Let \(k\ge 0\) be an integer and g an element in the compact Lie group G. Let \(L^k_{g}G\) denote the twisted loop group
The multiplication of it is defined by
The identity element e is the constant map sending all the real numbers to the identity element of G. We extend this group by adding the rotations. Let \(L^k_{g}G\rtimes \mathbb {T}\) denote the group with elements \((\gamma , t)\), \(\gamma \in L^k_{g}G\) and \(t\in \mathbb {T}\). The multiplication is defined by
The set of constant maps \(\mathbb {R}\longrightarrow G\) in \(L^k_{g}G\) is a subgroup of it, i.e. the centralizer \(C_G(g)\). When G is finite, \(L^k_{g}G=C_G(g)\).
When G is finite, the objects of \(Loop_2(X/\!\!/G)\) can be identified with the space
where
and l is the order of the element g. The cyclic group \(\mathbb {Z}/l\mathbb {Z}\) is isomorphic to the subgroup \(k\mathbb {Z}/kl\mathbb {Z}\) of \(\mathbb {R}/kl\mathbb {Z}\). The isomorphism \(\mathbb {Z}/l\mathbb {Z}\longrightarrow k\mathbb {Z}/kl\mathbb {Z}\) sends the generator [1] corresponding to 1 to the generator [k] of \(k\mathbb {Z}/kl\mathbb {Z}\) corresponding to k. \(k\mathbb {Z}/kl\mathbb {Z}\) acts on \(\mathbb {R}/kl\mathbb {Z}\) by group multiplication. Thus, via the isomorphism, \(\mathbb {Z}/l\mathbb {Z}\) acts on \(\mathbb {R}/kl\mathbb {Z}\). \(\mathbb {Z}/l\mathbb {Z}\) is also isomorphic to the cyclic group \(\langle g\rangle \) by identifying the generater [1] with g. So it acts on X via the G-action on it.
\({_1}\mathcal {L}{_g} X/\!\!/ L^1_gG\) is a full subgroupoid of \(Loop_2(X/\!\!/G)\). Moreover, \({_1}\mathcal {L}{_g} X/\!\!/ L^1_gG\rtimes \mathbb {T}\) is a full subgroupoid of \(Loop^{ext}_2(X/\!\!/G)\) where \(L^k_{g}G\rtimes \mathbb {T}\) acts on \({_k}\mathcal {L}{_g} X\) by
The action by g on \({_k}\mathcal {L}{_g} X\) coincides with that by \(k\in \mathbb {R}\). So we have the isomorphism
where \(\overline{g}\) represents the constant loop \(\mathbb {T}\longrightarrow \{g\}\subseteq G\).
In fact we have already proved Proposition 2.7.
Proposition 2.7
Let G be a finite group. The groupoid
is a skeleton of \(Loop^{ext}_2(X/\!\!/G)\), where the coproduct goes over conjugacy classes in \(\pi _0G\).
Next we show the physical meaning of \(L_{\sigma }^1 G\). Recall that the gauge group of a principal bundle is defined to be the group of its vertical automorphisms. The readers may refer [22] for more details. For a G-bundle \(P\longrightarrow S^1\), let \(L_P G\) denote its gauge group.
We have the well-known facts below.
Lemma 2.8
The principal G-bundles over \(S^1\) are classified up to isomorphism by homotopy classes
Up to isomorphism every principal G-bundle over \(S^1\) is isomorphic to one of the forms \(P_{\sigma }\longrightarrow S^1\) with \(\sigma \in G\) and
A complete collection of isomorphism classes is given by a choice of representatives for each conjugacy class of \(\pi _0G\).
For the gauge group \(L_{P_{\sigma }} G\) we have the conclusion below.
Proposition 2.9
For the bundle \(P_{\sigma }\longrightarrow S^1\), \(L_{P_{\sigma }}G\) is isomorphic to the twisted loop group \(L^1_{\sigma }G\).
Proof
Each automorphism f of the bundle \(P_{\sigma }\longrightarrow S^1\) has the form
for some \(\gamma _f: \mathbb {R}\longrightarrow G\). The morphism is well-defined if and only if \(\gamma _f(s+1)=\sigma ^{-1}\gamma _f(s)\sigma \). So we get a well-defined map
It is a bijection. Moreover, by the property of group action, F sends the identity map to the constant map \(\mathbb {R}\longrightarrow G\text{, } s\mapsto e\), which is the trivial element in \(L^1_{\sigma }G\), and for two automorphisms \(f_1\) and \(f_2\) at the object, \(F(f_1\circ f_2)= \gamma _{f_1}\cdot \gamma _{f_2}\). So \(L_{P_{\sigma }}G\) is isomorphic to \( L^1_{\sigma }G\). \(\square \)
2.1.3 Ghost loops
Let G be a compact Lie group and X a G-space. In this section we introduce a subgroupoid \(GhLoop(X/\!\!/G)\) of \(Loop^{ext}_1(X/\!\!/G)\), which can be computed locally.
Definition 2.10
(Ghost loops) The groupoid of ghost loops is defined to be the full subgroupoid \(GhLoop(X/\!\!/G)\) of \(Loop^{ext}_1(X/\!\!/G)\) consisting of objects \(S^1\leftarrow P\buildrel {\widetilde{\delta }}\over \rightarrow X\) such that \(\widetilde{\delta }(P)\subseteq X\) is contained in a single G-orbit.
For a given \(\sigma \in G\), define the space
We have a corollary of Proposition 2.7 below.
Proposition 2.11
\(GhLoop(X/\!\!/G)\) is equivalent to the groupoid
where the coproduct goes over conjugacy classes in \(\pi _0G\).
Example 2.12
If G is a finite group, it has the discrete topology. In this case, LG consists of constant loops and, thus, is isomorphic to G. The space of objects of \(GhLoop(X/\!\!/G)\) can be identified with X. For \(\sigma \in G\) and any integer k, \(L^k_{\sigma } G\) can be identified with \(C_G(\sigma )\); \(L^k_{\sigma }G\rtimes \mathbb {T}\cong C_G(\sigma )\times \mathbb {R}/\langle (\sigma , -k)\rangle \); and \(GhLoop_{\sigma }(X/\!\!/G)\) can be identified with \(X^{\sigma }\).
Unlike true loops, ghost loops have the property that they can be computed locally, as shown in the lemma below. The proof is left to the readers.
Proposition 2.13
If \(X=U\cup V\) where U and V are G-invariant open subsets, then \(GhLoop(X/\!\!/G)\) is isomorphic to the fibred product of groupoids
Thus, the ghost loop construction satisfies Mayer–Vietoris property. Moreover, it has the change-of-group property.
Proposition 2.14
Let H be a closed subgroup of G. It acts on the space of left cosets G / H by left multiplication. Let \(\text{ pt }\) denote the single point space with the trivial H-action. Then we have the equivalence of topological groupoids between \(Loop^{ext}_1((G/H)/\!\!/G)\) and \(Loop^{ext}_1(\text{ pt }/\!\!/H)\). Especially, there is an equivalence between the groupoids \(GhLoop((G/H)/\!\!/G)\) and \(GhLoop(\text{ pt }/\!\!/ H)\).
Proof
First we define a functor \(F: Loop^{ext}_1((G/H)/\!\!/G)\longrightarrow Loop^{ext}_1(\text{ pt }/\!\!/ H)\) sending an object \(S^1\leftarrow P\buildrel {\widetilde{\delta }}\over \rightarrow G/H\) to \(S^1\leftarrow Q\rightarrow \{eH\}=\text{ pt }\) where \(Q\longrightarrow eH\) is the constant map, and \(Q\longrightarrow S^1\) is the pull back bundle
It sends a morphism
to the morphism
where all the squares are pull-back.
In addition, we can define a functor \(F': Loop^{ext}_1 (\text{ pt }/\!\!/ H) \longrightarrow Loop^{ext}_1((G/H)/\!\!/G)\) sending an object \(S^1\leftarrow Q\rightarrow \text{ pt }\) to \(S^1\leftarrow G\times _HQ\rightarrow G\times _H\text{ pt }=G/H\) and sending a morphism
to
\(F\circ F'\) and \(F'\circ F\) are both identity maps. So the topological groupoids \(Loop^{ext}_1((G/H)/\!\!/G)\) and \(Loop^{ext}_1(\text{ pt }/\!\!/ H)\) are equivalent.
We can prove the equivalence between \(GhLoop((G/H)/\!\!/G)\) and \(GhLoop(\text{ pt }/\!\!/ H)\) in the same way. \(\square \)
Remark 2.15
In general, if \(H^*\) is an equivariant cohomology theory, Proposition 2.14 implies the functor
gives a new equivariant cohomology theory. When \(H^*\) has the change of group isomorphism, so does \(H^*(GhLoop(-))\).
3 Quasi-elliptic cohomology \(QEll^*_G\)
Unless otherwise indicated, we assume G is a finite group and X is a G-space in the rest part of the paper. The main references for Sect. 3 are Rezk’s unpublished work [25] and the author’s PhD thesis [12]. The construction of the theory \(QEll^*_G\) for any compact Lie group G will be shown in the paper [13]. In Sect. 3.2 we define \(QEll^*_G\) and prove some of its main properties. Before that we discuss in Sect. 3.1 the complex representation ring of
which is a factor of \(QEll^*_G(\text{ pt })\). We assume familiarity with [5, 27].
3.1 Preliminary: representation ring of \(\Lambda _G(g)\)
Let \(q: \mathbb {T}\longrightarrow U(1)\) be the isomorphism \(t\mapsto e^{2\pi it}\). The complex representation ring \(R\mathbb {T}\) is \(\mathbb {Z}[q^{\pm }]\).
We have an exact sequence
where the first map is \(g\mapsto [g, 0]\) and the second map is
The map \(\pi ^*: R\mathbb {T}\longrightarrow R\Lambda _G(g)\) equips the representation ring \(R\Lambda _G(g)\) the structure as an \(R\mathbb {T}\)-module.
There is a relation between the complex representation ring of \(C_G(g)\) and that of \(\Lambda _G(g)\), which is shown as Lemma 1.2 in [25] and Lemma 2.4.1 in [12].
Lemma 3.1
The \(R\mathbb {T}\)-module \(R\Lambda _G(g)\) with the action defined by \(\pi ^*: R\mathbb {T}\longrightarrow R\Lambda _G(g)\) is a free module.
In particular, there is an \(R\mathbb {T}\)-basis of \(R\Lambda _G(g)\) given by irreducible representations \(\{V_{\lambda }\}\), such that restriction \(V_{\lambda }\mapsto V_{\lambda }|_{C_G(g)}\) to \(C_G(g)\) defines a bijection between \(\{V_{\lambda }\}\) and the set \(\{\lambda \}\) of irreducible representations of \(C_G(g)\).
Proof
Let l be the order of g. Note that \(\Lambda _G(g)\) is isomorphic to
Thus, it is the quotient of the product of two compact Lie groups.
Let \(\lambda : C_G(g)\longrightarrow GL(n, \mathbb {C})\) be an n-dimensional \(C_G(g)\)-representation with representation space V and \(\eta : \mathbb {R}\longrightarrow GL(n, \mathbb {C})\) be a representation of \(\mathbb {R}\) such that \(\lambda (g)\) acts on V via scalar multiplication by \(\eta (1)\). Define a n-dimensional \(\Lambda _G(g)\)-representation \(\lambda \odot _{\mathbb {C}} \eta \) with representation space V by
Any irreducible n-dimensional representation of the quotient group \(\Lambda _G(g)=C_G(g)\times \mathbb {R}/\langle (g, -1)\rangle \) is an irreducible n-dimensional representation of the product \(C_G(g)\times \mathbb {R}\). And any finite dimensional irreducible complex representation of the product of two compact Lie groups is the tensor product of an irreducible representation of each factor. So any irreducible representation of the quotient group \(\Lambda _G(g)\) is the tensor product of an irreducible representation \(\lambda \) of \(C_G(g)\) with representation space V and an irreducible representation \(\eta \) of \(\mathbb {R}\). Any irreducible complex representation \(\eta \) of \(\mathbb {R}\) is one dimensional. So the representation space of \(\lambda \odot _{\mathbb {C}} \eta \) is still V. \(\eta (1)^l=I\). We need \(\eta (1)=\lambda (g)\). So \(\eta (1)=e^{\frac{2\pi ik}{l}}\) for some \(k\in \mathbb {Z}\). So
Any \(m\in \mathbb {Z}\) gives a choice of \(\eta \) in this case. And \(\eta \) is a representation of \(\mathbb {R}/l\mathbb {Z}\cong \mathbb {T}\).
Therefore, we have a bijective correspondence between
-
1.
isomorphism classes of irreducible \(\Lambda _G(g)\)-representation \(\rho \), and
-
2.
isomorphism classes of pairs \((\lambda , \eta )\) where \(\lambda \) is an irreducible \(C_G(g)\)-representation and \(\eta :\mathbb {R}\longrightarrow \mathbb {C}^*\) is a character such that \(\lambda (g)=\eta (1)I\). \(\lambda =\rho |_{C_G(g)}\).
Then as a corollary, the \(R\mathbb {T}\)-module \(R\Lambda _G(g)\) with the \(R\mathbb {T}\)-action defined by \(\pi ^*: R\mathbb {T}\longrightarrow R\Lambda _G(g)\)
\(\pi ^*: R\mathbb {T}\longrightarrow R\Lambda _G(g)\) exhibits \(R\Lambda _G(g)\) as a free \(R\mathbb {T}\)-module. \(\square \)
Remark 3.2
We can make a canonical choice of \(\mathbb {Z}[q^{\pm }]\)-basis for \(R\Lambda _{G}(g)\). For each irreducible G-representation \(\rho : G\longrightarrow Aut(G)\), write \(\rho (\sigma )=e^{2\pi ic}id\) for \(c\in [0,1)\), and set \(\chi _{\rho }(t)=e^{2\pi ict}\). Then the pair \((\rho , \chi _{\rho })\) corresponds to a unique irreducible \(\Lambda _{G}(g)\)-representation
Example 3.3
(\(G=\mathbb {Z}/N\mathbb {Z}\)) Let \(G=\mathbb {Z}/N\mathbb {Z}\) for \(N\ge 1\), and let \(\sigma \in G\). Given an integer \(k\in \mathbb {Z}\) which projects to \(\sigma \in \mathbb {Z}/N\mathbb {Z}\), let \(x_k\) denote the representation of \(\Lambda _G(\sigma )\) defined by
\(R\Lambda _G(\sigma )\) is isomorphic to the ring \(\mathbb {Z}[q^{\pm }, x_k]/(x^N_k-q^k)\).
Example 3.4
(\(G=\Sigma _3\)) \(G=\Sigma _3\) has three conjugacy classes represented by 1, (12), (123) respectively.
\(\Lambda _{\Sigma _3}(1)=\Sigma _3\times \mathbb {T}\), thus, \(R\Lambda _{\Sigma _3}(1)=R\Sigma _3\otimes R\mathbb {T}=\mathbb {Z}[X, Y]/(XY-Y, X^2-1, Y^2-X-Y-1)\otimes \mathbb {Z}[q^{\pm }]\) where X is the sign representation on \(\Sigma _3\) and Y is the standard representation.
\(C_{\Sigma _3}((12))=\langle (12)\rangle =\Sigma _2,\) thus, \(\Lambda _{\Sigma _3}((12))\cong \Lambda _{\Sigma _2}((12)).\) So we have
\(C_{\Sigma _3}(123)=\langle (123)\rangle =\mathbb {Z}/3\mathbb {Z},\) thus, \(\Lambda _{\Sigma _3}((123))\cong \Lambda _{\mathbb {Z}/3\mathbb {Z}}(1).\) So we have
Moreover, we have the conclusion below about the relation between the induced representations \(Ind|^{\Lambda _G(\sigma )}_{\Lambda _H(\sigma )}(-)\) and \(Ind|^{C_G(\sigma )}_{C_H(\sigma )}(-).\)
Lemma 3.5
Let H be a subgroup of G and \(\sigma \) an element of H. Let m denote \([C_G(\sigma ):C_H(\sigma )]\). Let V denote a \(\Lambda _H(\sigma )\)-representation \(\lambda \odot _{\mathbb {C}}\chi \) with \(\lambda \) a \(C_H(\sigma )\)-representation, \(\chi \) a \(\mathbb {R}\)-representation and \(\odot _{\mathbb {C}}\) defined in (3.4).
-
(i)
$$\begin{aligned} res^{\Lambda _G(\sigma )}_{\Lambda _H(\sigma )}(\lambda \odot _{\mathbb {C}}\eta ) =(res^{C_G(\sigma )}_{C_H(\sigma )}\lambda )\odot _{\mathbb {C}}\eta . \end{aligned}$$(3.6)
-
(ii)
The induced representation
$$\begin{aligned} Ind^{\Lambda _G(\sigma )}_{\Lambda _H(\sigma )} (\lambda \odot _{\mathbb {C}}\chi ) \end{aligned}$$is isomorphic to the \(\Lambda _G(\sigma )\)-representation
$$\begin{aligned} (Ind^{C_G(\sigma )}_{C_H(\sigma )}\lambda )\odot _{\mathbb {C}}\chi . \end{aligned}$$Their underlying vector spaces are both \(V^{\oplus m}\).
Thus, the computation of both \(Ind^{\Lambda _G(\sigma )}_{\Lambda _H(\sigma )} (\lambda \odot _{\mathbb {C}}\chi )\) and \(res^{\Lambda _G(\sigma )}_{\Lambda _H(\sigma )}(\lambda \odot _{\mathbb {C}}\eta )\) can be reduced to the computation of representations of finite groups.
The proof is straightforward and left to the readers.
Let k be any integer. Next we describe the relation between
and \(\Lambda _G(g)\), which gives the relation between their representation rings.
There is an exact sequence
where the second map \(\pi _k: \Lambda ^k_G(g)\longrightarrow \mathbb {R}/k\mathbb {Z}\) is \(\pi _k([g, t])= e^{2\pi i t}\).
Let \(q^{\frac{1}{k}}: \mathbb {R}/k\mathbb {Z}\longrightarrow U(1)\) denote the composition
The representation ring \(R(\mathbb {R}/k\mathbb {Z})\) is \(\mathbb {Z}[q^{\pm \frac{1}{k}}]\).
Analogous to Lemma 3.1, we have the conclusion about \(R\Lambda ^k_G(g)\) below.
Lemma 3.6
The map \(\pi ^*_k: R(\mathbb {R}/k\mathbb {Z})\longrightarrow R\Lambda ^k_G(g)\) exhibits it as a free \(\mathbb {Z}[q^{\pm \frac{1}{k}}]\)-module. There is a \(\mathbb {Z}[q^{\pm \frac{1}{k}}]\)-basis of \(R\Lambda ^k_G(g)\) given by irreducible representations \(\{\rho _k\}\) such that the restrictions \(\rho _k|_{C_G(g)}\) of them to \(C_G(g)\) are precisely the \(\mathbb {Z}\)-basis of \(RC_G(g)\) given by irreducible representations.
In other words, any irreducible \(\Lambda ^k_G(g)\)-representation has the form \(\rho \odot _{\mathbb {C}} \chi \) where \(\rho \) is an irreducible representation of \(C_G(g)\), \(\chi :\mathbb {R}/k\mathbb {Z}\longrightarrow GL(n, \mathbb {C})\) such that \(\chi (k)=\rho (g)\), and
\(R\Lambda _G^k(g)\) is a \(\mathbb {Z}[q^{\pm }]\)-module via the inclusion \(\mathbb {Z}[q^{\pm }]\longrightarrow \mathbb {Z}[q^{\pm \frac{1}{k}}]\).
By Lemma 3.6, we can make a \(\mathbb {Z}[q^{\pm \frac{1}{k}}]\)-basis \(\{\rho \odot _{\mathbb {C}}\chi _{\rho , k}\}\) for \(R\Lambda ^k_{G}(g)\) with each \(\rho : G\longrightarrow Aut(G)\) an irreducible G-representation and \(\chi _{\rho , k}(t)= e^{2\pi i\frac{ct}{k}}\) with \(c\in [0,1)\) such that \(\rho (\sigma )=e^{2\pi ic}id\). This collection \(\{\rho \odot _{\mathbb {C}}\chi _{\rho , k}\}\) gives a \(\mathbb {Z}[q^{\pm \frac{1}{k}}]\)-basis of \(R\Lambda ^k_G(g)\).
There is a group isomorphism \(\alpha _k: \Lambda ^k_G(g)\longrightarrow \Lambda _G(g)\) sending [g, t] to \([g, \frac{t}{k}]\). Observe that there is a pullback square of groups
So we have the commutative square of a pushout square in the category of \(\lambda \)-rings.
It gives a canonical isomorphism of \(\lambda \)-rings \(R\Lambda _G(g)\longrightarrow R\Lambda _G^k(g)\) sending q to \(q^{\frac{1}{k}}\). A good reference for \(\lambda \)-rings is Chapters 1 and 2, [29].
3.2 Quasi-elliptic cohomology
In this section we introduce the definition of quasi-elliptic cohomology \(QEll^*_G\) in terms of orbifold K-theory, and then express it via equivariant K-theory. We assume familiarity with [27]. The reader may read Chapter 3 in [3, 23] for a reference of orbifold K-theory.
When G is finite, quasi-elliptic cohomology is defined from the ghost loops in Definition 2.10. By Proposition 2.11 and Example 2.12, we can see the groupoid \(GhLoop(X/\!\!/G)\) is equivalent to the disjoint union of some translation groupoids. Before describing this equivalent groupoid \(\Lambda (X/\!\!/G)\) in detail, we recall what inertia groupoid is. A reference for that is Section 4, [20].
Definition 3.7
Let \(\mathbb {G}\) be a groupoid. The inertia groupoid \(I(\mathbb {G})\) of \(\mathbb {G}\) is defined as follows.
An object a is an arrow in \(\mathbb {G}\) such that its source and target are equal. A morphism v joining two objects a and b is an arrow v in \(\mathbb {G}\) such that
In other words, b is the conjugate of a by v, \(b=v\circ a\circ v^{-1}\).
Let X a G-space.
Example 3.8
The inertia groupoid \(I(X/\!\!/G)\) is the groupoid with
objects: the space \(\coprod \nolimits _{g\in G}X^{g}\)
morphisms: the space \(\coprod \nolimits _{g, g'\in G}C_G(g,g')\times X^g\) where \(C_G(g,g')=\{\sigma \in G|g'\sigma =\sigma g\}\subseteq G.\)
For \(x\in X^g\) and \((\sigma , g)\in C_G(g,g')\times X^g\), \((\sigma , g)(x)=\sigma x\in X^{g'}.\)
Definition 3.9
The groupoid \(\Lambda (X/\!\!/G)\) has the same objects as \(I(X/\!\!/G)\) but richer morphisms
where \(\Lambda _G(g, g')\) is the quotient of \(C_G(g, g')\times \mathbb {R}\) under the equivalence
For an object \(x\in X^g\) and a morphism \(([\sigma , t], g)\in \Lambda _G(g, g')\times X^g\), \(([\sigma , t], g)(x)=\sigma x\in X^{g'}.\) The composition of the morphisms is defined by
Definition 3.10
The quasi-elliptic cohomology \(QEll^*_G(X)\) is defined to be \(K^*_{orb}(GhLoop(X/\!\!/G))\cong K^*_{orb}(\Lambda (X/\!\!/G))\).
We can unravel the definition and express it via equivariant K-theory.
Let \(\sigma \in G\). The fixed point space \(X^{\sigma }\) is a \(C_G(\sigma )\)-space. We can define a \(\Lambda _G(\sigma )\)-action on \(X^{\sigma }\) by
Then we have
Proposition 3.11
where \(G_{conj}\) is a set of representatives of G-conjugacy classes in G.
Thus, for each \(g\in \Lambda _G(g)\), we can define the projection
For the singe point space, we have
We have the ring homomorphism
where \(\pi : \Lambda _G(g)\longrightarrow \mathbb {T}\) is the projection defined in (3.2) and the second is via the collapsing map \(X\longrightarrow \text{ pt }\). So \(QEll_G^*(X)\) is naturally a \(\mathbb {Z}[q^{\pm }]\)-algebra.
3.3 Properties
In this section we discuss some properties of \(QEll^*_G\), including the restriction map, the Künneth map on it, its tensor product and the change-of-group isomorphism.
Since each homomorphism \(\phi : G\longrightarrow H\) induces a well-defined homomorphism \(\phi _{\Lambda }: \Lambda _G(\tau )\longrightarrow \Lambda _H(\phi (\tau ))\) for each \(\tau \) in G, we can get the proposition below directly.
Proposition 3.12
For each homomorphism \(\phi : G\longrightarrow H\), it induces a ring map
characterized by the commutative diagrams
for any \(\tau \in G\). So \(QEll^*_G\) is functorial in G.
Moreover, we can define Künneth map of quasi-elliptic cohomology induced from that on equivariant K-theory.
Let G and H be two finite groups. X is a G-space and Y is a H-space. Let \(\sigma \in G\) and \(\tau \in H\). Let \(\Lambda _G(\sigma )\times _{\mathbb {T}}\Lambda _H(\tau )\) denote the fibered product of the morphisms
It is isomorphic to \(\Lambda _{G\times H}(\sigma , \tau )\) under the correspondence
Consider the composition below
where the first map is the Künneth map of equivariant K-theory, the second is the restriction map and the third is the isomorphism induced by the group isomorphism \(\Lambda _{G\times H}(\sigma , \tau )\cong \Lambda _G(\sigma )\times _{\mathbb {T}}\Lambda _H(\tau )\).
For any \(g\in G\), let 1 denote the trivial line bundle over \(X^g\) and let q denote the line bundle \(1\odot _{\mathbb {C}} q\) over \(X^g\). The map T above sends both \(1\otimes q\) and \(q\otimes 1\) to q. So we get the well-defined map
Definition 3.13
The tensor produce of quasi-elliptic cohomology is defined by
The direct product of the maps defined in (3.15) gives a ring homomorphism
which is the Künneth map of quasi-elliptic cohomology.
By Lemma 3.1 we have
More generally, we have the proposition below.
Proposition 3.14
Let X be a \(G\times H\)-space with trivial H-action and let \(\text{ pt }\) be the single point space with trivial H-action. Then we have
Especially, if G acts trivially on X, we have
Here \(QEll^*(X)\) is \(QEll^*_{\{e\}}(X)=K^*_{\mathbb {T}}(X)\).
Proof
\(\square \)
Proposition 3.15
If G acts freely on X,
Proof
Since G acts freely on X,
Thus, \(QEll^*_G(X)\cong \prod \limits _{\sigma \in G_{conj}}K^*_{\Lambda _G(\sigma )/C_G(\sigma )}(X^{\sigma }/C_G(\sigma ))\cong K^*_{\mathbb {T}}(X/G).\)
Since \(\mathbb {T}\) acts trivially on X, we have \(K^*_{\mathbb {T}}(X/G)=QEll^*_e(X/G)\) by definition. It is isomorphic to \(K^*(X/G)\otimes R\mathbb {T}\). \(\square \)
We also have the change-of-group isomorphism as in equivariant K-theory.
Let H be a subgroup of G and X a H-space. Let \(\phi : H\longrightarrow G\) denote the inclusion homomorphism. The change-of-group map \(\rho ^G_H: QEll^*_G(G\times _HX)\longrightarrow QEll^*_H(X)\) is defined as the composite
where \(\phi ^*\) is the restriction map and \(i: X\longrightarrow G\times _HX\) is the H-equivariant map defined by \(i(x)=[e, x].\)
Proposition 3.16
The change-of-group map
defined in (3.17) is an isomorphism.
Proof
For any \(\tau \in H_{conj}\), there exists a unique \(\sigma _{\tau }\in G_{conj}\) such that \(\tau =g_{\tau }\sigma _{\tau }g_{\tau }^{-1}\) for some \(g_{\tau }\in G\). Consider the maps
The first map is \(\Lambda _G(\tau )\)-equivariant and the second is equivariant with respect to the homomorphism \(c_{g_{\tau }}: \Lambda _{G}(\sigma )\longrightarrow \Lambda _G(\tau )\) sending \([u, t]\mapsto [g_{\tau } u g_{\tau }^{-1}, t]\). Taking a coproduct over all the elements \(\tau \in H_{conj}\) that are conjugate to \(\sigma \in G_{conj}\) in G, we get an isomorphism
which is \(\Lambda _G(\sigma )\)-equivariant with respect to \(c_{g_{\tau }}\). Then we have the map
It is straightforward to check the change-of-group map coincide with the composite
with the second map the change-of-group isomorphism in equivariant K-theory. \(\square \)
4 Power operation
In Sect. 4.2 we define power operations for equivariant quasi-elliptic cohomology \(QEll_G^*(-)\). We show in Theorem 4.12 that they satisfy the axioms that Ganter established in Definition 4.3, [9] for equivariant power operations.
The power operation of quasi-elliptic cohomology is of the form
where \(\mathbb {P}_n\) maps a bundle over the groupoid
to a bundle over
and each \(\mathbb {P}_{(\underline{g},\sigma )}\) maps a bundle over
to a \(\Lambda _{G\wr \Sigma _n}(\underline{g}, \sigma )\)-bundle over the space \((X^{\times n})^{(\underline{g}, \sigma )}/\!\!/\Lambda _{G\wr \Sigma _n}(\underline{g}, \sigma ).\)
We construct each \(\mathbb {P}_{(\underline{g}, \sigma )}\) as the composition below.
where \(k\in \mathbb {Z}\) and \((i_1, \ldots i_k)\) goes over all the k-cycles of \(\sigma \). We explain the first three functors in detail in Sect. 4.2. In Sect. 4.1 we construct the isomorphism \(f_{(\underline{g}, \sigma )}\) between the groupoid
and the groupoid \(d((X/\!\!/G)\wr \Sigma _n)\) constructed in Definition 4.5. With it, it is convenient to construct the explicit formula of the power operation.
4.1 Loop space of symmetric power
4.1.1 The groupoid \(d((X/\!\!/G)\wr \Sigma _n)\)
For an introduction of actions of wreath product \(G\wr \Sigma _n\) on \(X^{\times n}\) and symmetric power \(\mathbb {G}\wr \Sigma _n\) of a groupoid \(\mathbb {G}\), we refer the readers to Section 4.1, [10]. The symmetric power \((X/\!\!/G)\wr \Sigma _n\) is isomorphic to \(X^{\times n}/\!\!/(G\wr \Sigma _n)\).
Before introducing the groupoid \(d((X/\!\!/G)\wr \Sigma _n)\), we need to introduce several ingredients.
Definition 4.1
(\(\Lambda ^k(X/\!\!/G)\)) The groupoid \(\Lambda ^k(X/\!\!/G)\) has the same objects as \(\Lambda (X/\!\!/G)\) but different morphisms
where \(\Lambda ^k_G(g, g')\) is the quotient of \(C_G(g, g')\times \mathbb {R}\) under the equivalence
For an object \(x\in X^g\) and a morphism \(([\sigma , t], g)\in \Lambda ^k_G(g, g')\times X^g\), \(([\sigma , t], g)(x)=\sigma x\in X^{g'}.\) The composition of the morphisms is defined by
Definition 4.2
(Fibred wreath product) The groupoid \(\Lambda ^k(X/\!\!/G)\wr _{\mathbb {T}}\Sigma _N\) is defined to be the subgroupoid of the symmetric power \(\Lambda ^k(X/\!\!/G)\wr \Sigma _N\) with the same objects but only those morphisms
with all the \(t_j\)s having the same image under the quotient map \(\mathbb {R}/k{\mathbb {Z}}\longrightarrow \mathbb {R}/\mathbb {Z}\).
The isotropy group of each object in \(\prod \nolimits _1^N X^g\) is \(\Lambda _G^k(g)\wr _{\mathbb {T}}\Sigma _N\).
Let Y be an H-space.
Definition 4.3
(Fibred product and fibred coproduct) The groupoid
is defined to be the subgroupoid of \(\Lambda ^{k_1}(X/\!\!/G)\wr _{\mathbb {T}}\Sigma _{N_1}\times \Lambda ^{k_2}(Y/\!\!/H)\wr _{\mathbb {T}}\Sigma _{N_2}\) with the same objects but only those morphisms
with all the \(t_{i, j_i}\)s having the same image under the quotient map \(\mathbb {R}/k_i{\mathbb {Z}}\longrightarrow \mathbb {R}/\mathbb {Z}\), for \(i=1, 2\) and \(j_i=1, \ldots N_i\).
The isotropy group of each object in \(\prod \nolimits _1^{N_1} X^g\prod \nolimits _1^{N_2} Y^h\) is
We can define the fibred coproduct \(\big (\Lambda ^{k_1}(X/\!\!/G)\wr _{\mathbb {T}}\Sigma _{N_1}\big )\coprod _{\mathbb {T}}\big (\Lambda ^{k_2}(Y/\!\!/H)\wr _{\mathbb {T}}\Sigma _{N_2}\big )\) in the same way.
Let \(\sigma \in \Sigma _n\) correspond to the partition \(n = \sum \nolimits _kkN_k\), i.e. it has \(N_k\) k-cycles. Assume that for each cycle \((i_1, \ldots i_k)\) of \(\sigma \), \(i_1< i_2\cdots < i_k\).
For \((\underline{g}, \sigma )\in G\wr \Sigma _n\), we consider the orbits of the bundle \(G\times \underline{\underline{n}}\longrightarrow \underline{\underline{n}}\) under the action by \((\underline{g}, \sigma )\). The orbits of \(\underline{\underline{n}}\) under the action by \(\sigma \) corresponds to the cycles in the cycle decomposition of \(\sigma \). The bundle \(G\times \underline{\underline{n}}\longrightarrow \underline{\underline{n}}\) is the disjoint union of the G-bundles
where \((i_1, \ldots i_k)\) goes over all the cycles of \(\sigma \). Each bundle \(G\times \{i_1, \ldots i_k\}\longrightarrow \{i_1, \ldots i_k\}\) is an orbit of \(G\times \underline{\underline{n}}\longrightarrow \underline{\underline{n}}\) under the action by \((\underline{g}, \sigma )\).
Let \(C_G(g, g')\) denote \(\{x\in G| gx=xg'\}.\) Two G-subbundles
are \((\underline{g}, \sigma )\)-equivariant equivalent if and only if \(k=m\) and \(C_G(g_{i_k}\ldots g_{i_1}, g_{j_k}\ldots g_{j_1})\) is nonempty. For each k-cycle \(i=(i_1, \ldots i_k)\) of \(\sigma \), let \(W^{\sigma }_i\) denote the set of all the G-subbundles \(G\times \{j_1, \ldots j_m\}\longrightarrow \{j_1, \ldots j_m\}\) that are \((\underline{g}, \sigma )\)-isomorphic to \(G\times \{i_1, \ldots i_k\}\longrightarrow \{i_1, \ldots i_k\}\). There is a bijection between \(W^{\sigma }_i\) and the set
Let \(M^{\sigma }_{i}\) denote the size of the set \(W^{\sigma }_i\). Let \(\alpha ^i_1, \alpha ^i_2, \ldots \alpha ^i_{M^{\sigma }_i}\) denote all the elements of the set \(W^{\sigma }_i\). Obviously, \(i=(i_1, \ldots i_k)\) is in \(W^{\sigma }_i\). So we can assume it is \(\alpha ^i_1\).
For any k-cycle i and m-cycle j of \(\sigma \), if \(k=m\) and \(C_G(g_{i_k}\ldots g_{i_1}, g_{j_k}\ldots g_{j_1})\) is nonempty, \(W^{\sigma }_i\) and \(W^{\sigma }_j\) are the same set. Otherwise, they are disjoint. The set of all the k-cycles of \(\sigma \) can be divided into the disjoint union of several \(W^{\sigma }_i\)s. We can pick a set of representatives \(\theta _k\) of k-cycles of \(\sigma \) such that the set of k-cycles of \(\sigma \) equals the disjoint union
Definition 4.4
(\(d_{(\underline{g}, \sigma )}(X)\)) The groupoid \(d_{(\underline{g}, \sigma )}(X)\) is defined to be a full subgroupoid of \(\prod \nolimits _{k}\!{_{\mathbb {T}}}\prod \nolimits _{i\in \theta _k}\!{_{\mathbb {T}}}\Lambda ^k(X/\!\!/G)\wr _{\mathbb {T}}\Sigma _{M^{\sigma }_i}\) with objects the points of the space
where the second product goes over all the k-cycles of \(\sigma \).
Definition 4.5
(\(d((X/\!\!/G)\wr \Sigma _n)\)) The groupoid \(d((X/\!\!/G)\wr \Sigma _n)\) is defined to be
where \((\underline{g}, \sigma )\) goes over \((G\wr \Sigma _n)_{conj}\).
Proposition 4.6
Each \(d_{(\underline{g}, \sigma )}(X)\) is isomorphic to the translation groupoid
where \(\alpha _j=g_{j_k}\ldots g_{j_1}\) with \(j=(j_1, \ldots j_k)\).
The proof is straightforward.
To study \(K_{orb}(d_{(\underline{g}, \sigma )}(X))\), we start by studying the representation ring of the wreath product
Theorem 4.7 gives all the irreducible representations of a wreath product. It is Theorem Theorem 4.3.34 in [15].
Theorem 4.7
Let \(\{\rho _k\}^N_1\) be a complete family of irreducible representations of G and let \(V_k\) be the corresponding representation space for \(\rho _k\). Let (n) be a partition of n. \((n)=(n_1, \ldots n_N).\) Let \(D_{(n)}\) be the representation
of \(G^{\times N}\) on \(V_1^{\otimes n_1}\otimes \cdots \otimes V_N^{\otimes n_N}.\) Let \(\Sigma _{(n)}=\Sigma _{n_1}\times \cdots \times \Sigma _{n_N}.\)
Let \((D_{(n)})^{\sim }\) be the extension of \(D_{(n)}\) from \(G^{\times n}\) to \(G\wr \Sigma _{(n)}\) defined by
where \(\sigma =\sigma _1\times \cdots \times \sigma _N\) with each \(\sigma _k\in \Sigma _{n_k}\).
Let \(D_{\tau }\) with \(\tau \in R\Sigma _{(n)}\) be the representation of \(G\wr \Sigma _{(n)}\) defined by
Then,
goes over all the irreducible representations of \(G\wr \Sigma _{n}\) nonrepeatedly.
The proof of Theorem 4.8 is analogous to that of Theorem 4.7 in [15], applying Clifford’s theory in [7, 8]. Note that
goes over all the irreducible representations of the fibred product
Theorem 4.8
Let \(\{\rho _k\}^N_1\) be a basis of the \(\mathbb {Z}[q^{\pm }]\)-module \(R\Lambda _G(\sigma )\) and let \(V_k\) be the corresponding representation space for \(\rho _k\). Let (n) be a partition of n. \((n)=(n_1, \ldots n_N).\) Let \(D^{\mathbb {T}}_{(n)}\) be the \(\Lambda _G(\sigma )^{\times _{\mathbb {T}} n}\)-representation
on the space \(V_1^{\otimes n_1}\otimes \cdots \otimes V_N^{\otimes n_N}.\) Let \(\Sigma _{(n)}=\Sigma _{n_1}\times \cdots \times \Sigma _{n_N}.\)
Let \((D^{\mathbb {T}}_{(n)})^{\sim }\) be the extension of \(D_{(n)}\) from \(\Lambda _G(\sigma )^{\times _{\mathbb {T}} n}\) to \(\Lambda _G(\sigma )\wr _{\mathbb {T}}\Sigma _{(n)}\) defined by
where k is from 1 to N and \(\sigma =\sigma _1\times \cdots \times \sigma _N\) with each \(\sigma _k\in \Sigma _{n_k}\).
Let \(D^{\mathbb {T}}_{\tau }\) with \(\tau \in R\Sigma _{(n)}\) be the representation of \(\Lambda _G(\sigma )\wr _{\mathbb {T}}\Sigma _{(n)}\) defined by
Then,
goes over all the irreducible representation nonrepeatedly of \(\Lambda _G(\sigma )\wr _{\mathbb {T}}\Sigma _{n}\).
From Theorem 4.7, the representation ring of each \(\Lambda _G^k(\alpha _j)\wr \Sigma _{M^{\sigma }_j}\) is a \(\mathbb {Z}[q^{\pm \frac{1}{k}}]\)-module. Thus, the representation ring of each \(\Lambda _G^k(\alpha _j)\wr \Sigma _{M^{\sigma }_j}\) is a \(\mathbb {Z}[q^{\pm }]\)-module via the map
The representation ring
is a \(\mathbb {Z}[q^{\pm }]\)-module. So is \(R(\prod \limits _{k}\!{_{\mathbb {T}}}\prod \limits _{j\in \theta _k}\!{_{\mathbb {T}}}\Lambda _G^k(\alpha _j)\wr _{\mathbb {T}}\Sigma _{M^{\sigma }_j}).\)
Moreover, \(K_{orb}(d_{(\underline{g}, \sigma )}(X))\) is a \(\mathbb {Z}[q^{\pm }]\)-module via the map
which is induced by \(X\longrightarrow \text{ pt }\).
4.1.2 The isomorphism \(f_{(\underline{g}, \sigma )}\)
Before we show in Theorem 4.10 that the groupoids \(\Lambda (X^{\times n}/\!\!/(G\wr \Sigma _n))\) and \(d((X/\!\!/G)\wr \Sigma _n)\) are isomorphic, we recall some properties of \(C_{G\wr \Sigma _n}((\underline{g}, \sigma ), (\underline{g}', \sigma '))\).
\((\underline{h}, \tau )\) is in \(C_{G\wr \Sigma _n} ((\underline{g},\sigma ), (\underline{g}',\sigma '))\) if and only if \(\tau \sigma '=\sigma \tau \) and \(g_{\sigma (\tau (i))}h_{\tau (i)}=h_{\tau (\sigma '(i))}g'_{\sigma '(i)},\) \(\forall i.\) We can reinterpret these two conditions. Since \(\tau \in C_{\Sigma _n}(\sigma , \sigma ')\), \(\tau \) maps a k-cycle \(i=(i_1,\ldots i_k)\) of \(\sigma '\) to a k-cycle \(j=(j_1, \ldots j_k)\) of \(\sigma \). \(\tau \) will still used to denote its map on the cycles, such as \(\tau (r)=s\). For each \(l\in \mathbb {Z}/k\mathbb {Z}\), let \(\tau (i_l)=j_{l+m_i}\) where \(m_i\) depends only on \(\tau \) and the cycle i. Then, the second conditions can be expressed as
From this equivalence, we can induce that the element
maps \(g_{j_k}\ldots g_{j_1}\) to \(g'_{i_k}\ldots g'_{i_1}\) by conjugation. In other words,
is an element in \(C_G (g_{j_k}\ldots g_{j_1}, g'_{i_k}\ldots g'_{i_1})\). Thus, \(C_G (g_{j_k}\ldots g_{j_1}, g'_{i_k}\ldots g'_{i_1})\) is nonempty.
First we show each component \((X^{\times n})^{(\underline{g}, \sigma )}/\!\!/\Lambda _{G\wr \Sigma _n}(\underline{g}, \sigma )\) is isomorphic to the groupoid \(d_{(\underline{g}, \sigma )}(X)\). We construct a functor
It sends a point
to
Note that \(x_{i_k}=x_{i_1}g_{i_1}=\cdots =x_{i_{k-1}}g_{i_{k-1}}\ldots g_{i_1}.\)
Let \([(\underline{h}, \tau ), t]\in \Lambda _{G\wr \Sigma _n}(\underline{g}, \sigma )\). Let \(\tau \) send the k-cycle \(i=(i_1, \ldots i_k)\) of \(\sigma \) to a k-cycle \(j=(j_1, \ldots j_k)\) of \(\sigma \) and \(\tau (i_1)=j_{1+m_i}\). We have
where \(\beta ^{\underline{h}, \tau }_{j, i}\) is the symbol defined in (4.8). So \(f_{(\underline{g}, \sigma )}\) maps the morphism \([(\underline{h}, \tau ), t]\) to
where \(\tau |_{W^{\sigma }_i}\) denotes the permutation induced by \(\tau \) on the set \(W^{\sigma }_i=\{\alpha ^i_1, \alpha ^i_2, \ldots \alpha ^i_{M^{\sigma }_i}\}\), \(\tau ^{-1}(j)\) is short for \(\tau ^{-1} (\alpha ^i_j)\) and \(\tau (j_l)=\tau (j)_{l+m_j}\).
It sends the identity map \([(1, \ldots , 1, \text{ Id }), 0]\) to the identity
and preserves composition of morphisms. So it is well-defined.
Theorem 4.9
The two groupoids \((X^{\times n})^{(\underline{g}, \sigma )}/\!\!/\Lambda _{G\wr \Sigma _n}(\underline{g}, \sigma )\) and \(d_{(\underline{g}, \sigma )}(X)\) are isomorphic. Thus, this isomorphism induces a \(\Lambda _{G\wr \Sigma _n}(\underline{g}, \sigma )\)-action on the space
Proof
We construct the inverse functor
of \(f_{(\underline{g}, \sigma )}\). For an object \( \times _k\times _{i\in \theta _k}\nu _{i, k}\) in \(d_{(\underline{g}, \sigma )}(X)\), \(J_{(\underline{g}, \sigma )}(\times _k\times _{i\in \theta _k}\nu _{i, k})=\{\nu _m\}_1^n\) with \(\nu _{i_k}=\nu _{i, k}|_{[0, 1]}\) and \(\nu _{i_s}(t):=\nu _{i, k}(s+t)g_{i_1}^{-1}\ldots g_{i_s}^{-1}.\)
Let
be a morphism in \(d_{(\underline{g}, \sigma )}(X)\). Let t be a representative of the image of \(m'^i_1\) in \(\mathbb {R}/\mathbb {Z}\). Then, each \(m^i_k:=m'^i_k-t\) is an integer.
When we know how \(\tau \in C_{\Sigma _n}(\sigma )\) permutes the cycles of \(\sigma \), whose information is contained in those \(\varrho ^k_i\in \Sigma _{M^{\sigma }_{i}}\), and the numbers \(m^i_1, \ldots m^i_{M^{\sigma }_{i}},\) we can get a unique \(\tau \). Explicitly, for any number \(j_r=1, 2 \ldots n\), if \(j_r\) is in a k-cycle \((j_1, \ldots j_k)\) of \(\sigma \) and it is in the set \(W^{\sigma }_i\), then \(\tau \) maps \(j_r\) to \(\varrho ^k_i(j)_{r+m^i_j}\), i.e. the \(r+m^i_j\)-th element in the cycle \(\varrho ^k_i(j)\) of \(\sigma \).
For any \(a\in W^{\sigma }_i\), \(\forall k\) and i, we want \(u^i_a=\beta ^{\underline{h}, \tau }_{\tau (a), a}\) for some \(\underline{h}\). Thus,
By (4.7) we can get all the other \(h_{\tau (a)_j}\).
It can be checked straightforward that \(J_{(\underline{g}, \sigma )}\) is a well-defined functor. It does not depend on the choice of the representative t.
\(J_{(\underline{g}, \sigma )}\circ f_{(\underline{g}, \sigma )}= \text{ Id }\); \(f_{(\underline{g}, \sigma )}\circ J_{(\underline{g}, \sigma )}= \text{ Id }\). So the conclusion is proved. \(\square \)
Then by Proposition 4.6, we get the main conclusion in Sect. 4.1.
Theorem 4.10
The two groupoids \(\Lambda ((X/\!\!/G)\wr \Sigma _n)\) and \(d((X/\!\!/G)\wr \Sigma _n)\) are isomorphic.
The last conclusion in this section is some properties of the functor \(f_{(\underline{g}, \sigma )}\).
Proposition 4.11
-
(i)
If \(\sigma =(1)\in \Sigma _1\), the morphism \(f_{(g, (1))}\) is the identity map on \(X^g/\!\!/\Lambda _{G}(g)\).
-
(ii)
Let \((\underline{g}, \sigma )\in G\wr \Sigma _n\) and \((\underline{h}, \tau )\in G\wr \Sigma _m\). The groupoids
$$\begin{aligned} (X^{\times n})^{(\underline{g}, \sigma )}/\!\!/\Lambda _{G\wr \Sigma _n}(\underline{g}, \sigma )\times _{\mathbb {T}}(X^{\times m})^{(\underline{h}, \tau )}/\!\!/\Lambda _{G\wr \Sigma _m}(\underline{h}, \tau ) \end{aligned}$$and
$$\begin{aligned} (X^{\times (n+m)})^{(\underline{g}, \underline{h} \sigma \tau )}/\!\!/\Lambda _{G\wr \Sigma _{n+m}}(\underline{g}, \underline{h}, \sigma \tau ) \end{aligned}$$are isomorphic.
-
(iii)
\(f_{(\underline{g}, \sigma )}\) preserves Cartesian product of loops. The following diagram of groupoids commutes.
Proof
(i) is indicated in the proof of Theorem 4.9.
(ii) We can define a functor \(\Phi \) from
to \((X^{\times (n+m)})^{(\underline{g}, \underline{h} \sigma \tau )}/\!\!/\Lambda _{G\wr \Sigma _{n+m}}(\underline{g}, \underline{h}, \sigma \tau )\) sending an object \((x_1, x_2)\) to \((x_1, x_2)\) and a morphism \(([\alpha ,t],[\beta , t])\) to \([\alpha , \beta , t].\) It is straightforward to check \(\Phi \) is an isomorphism between the groupoids.
(iii) The proof is left to the readers. \(\square \)
4.2 Total power operation of \(QEll^*_G\)
In this section we construct the total power operations for quasi-elliptic cohomology and give its explicit formula in (4.17). We show in Theorem 4.12 that they satisfy the axioms that Ganter concluded in Definition 4.3, [9] for equivariant power operation.
We explain each map in the formula (4.1) and (4.2). The functor \(U: \Lambda ^1_{(\underline{g}, \sigma )}(X)\longrightarrow \Lambda (X/\!\!/G)\) is defined in (4.10). The pullback \(( \; )^{\Lambda }_k\) is defined in (4.12). The external product \(\boxtimes \) is explained in (4.16). The fourth is the pullback by \(f_{(\underline{g}, \sigma )}\).
The Functor U
For each \((\underline{g}, \sigma )\in G\wr \Sigma _n\), \(r\in \mathbb {Z}\), let \(\Lambda ^r_{(\underline{g}, \sigma )}(X)\) denote the groupoid with objects
where \((i_1, \ldots i_k)\) goes over all the k-cycles of \(\sigma \), and with morphisms
where \((i_1, \ldots i_k)\) and \((j_1, \ldots j_k)\) go over all the k-cycles of \(\sigma \) respectively. It may not be a subgroupoid of \(\Lambda ^r(X/\!\!/G)\) because there may be cycles \((i_1, \ldots i_k)\) and \((j_1, \ldots j_m)\) such that
Let
denote the functor sending x in the component \(X^{g_{i_k}\ldots g_{i_1}}\) to the x in the component \(X^{g_{i_k}\ldots g_{i_1}}\) of \(\Lambda (X/\!\!/ G)\), and send each morphism
to
In the case that \(g_{i_k}\ldots g_{i_1}\) and \(g_{j_k}\ldots g_{j_1}\) are equal, ([h, t], x) is an arrow inside a single connected component.
The Functor \((\;)_k\)
For each integer k, there is a functor of groupoids \(( \; )_k: \Lambda ^k(X/\!\!/G)\longrightarrow \Lambda (X/\!\!/G)\) sending an object x to x and a morphism \(([h, t_0], x) \text{ to } ([h, \frac{t_0}{k}], x).\) The composition \((( \; )_k)_r=( \; )_{kr}.\)
The functor \((\;)_k\) gives a well-defined map
by pullback of bundles. We still use the symbol \((\;)_k\) to denote it when there is no confusion. For any \(\Lambda (X/\!\!/G)\)-vector bundle \(\mathcal {V}\), \(S^1\) acts on \((\mathcal {V})_k\) via
If \(\mathcal {V}\) has the decomposition \(\mathcal {V}=\bigoplus \nolimits _{j\in \mathbb {Z}}V_jq^j,\) then
The Functor \(( \; )_k^{\Lambda }\)
Let \(\Lambda ^{var}_{(\underline{g}, \sigma )}(X)\) be the groupoid with the same objects as \(\Lambda ^1_{(\underline{g}, \sigma )}(X)\) and morphisms
where \((i_1, \ldots i_k)\) and \((j_1, \ldots j_k)\) go over all the k-cycles of \(\sigma \) respectively.
We can define a similar functor
that is identity on objects and sends each \([g, t]\in \Lambda ^k_G(g_{i_k}\ldots g_{i_1}, g_{j_k}\ldots g_{j_1})\) to \([g, \frac{t}{k}]\in \Lambda ^1_G(g_{i_k}\ldots g_{i_1}, g_{j_k}\ldots g_{j_1})\). We use the same symbol \(( \; )_k^{\Lambda }\) to denote the pull back
The external product \(\boxtimes \)
Let Y an H-space, \((\underline{g}, \sigma )\in G\wr \Sigma _n\) and \((\underline{h}, \tau )\in G\wr \Sigma _m\).
Each \(K^*_{orb}(d_{(\underline{g}, \sigma )}(X))\) is a \(\mathbb {Z}[q^{\pm }]\)-algebra, as shown in Sect. 4.1.1. The external product in the theory \(K^*_{orb}(d_{(\underline{g}, \sigma )}(-))\) is defined to be the tensor product of \(\mathbb {Z}[q^{\pm }]\)-algebras. The fibred product \(d_{(\underline{g}, \sigma )}(X)\times _{\mathbb {T}}d_{(\underline{h}, \tau )}(X)\) has the same objects as \(d_{(\underline{g}, \underline{h}, \sigma \tau )}(X)\) and is a subgroupoid of it.
So we have the Künneth map
It is compatible with the Künneth map (3.15) of the quasi-elliptic cohomology in the sense that the diagram below commutes.
where the horizontal maps are Künneth maps.
If we have a vector bundle \(E=\coprod \limits _k\coprod \limits _{(i_1, \ldots i_k)}E_{g_{i_k}\ldots g_{i_1}}\) over \(\Lambda ^1_{(\underline{g}, \sigma )}(X)\), the external product
is a vector bundler over \(d_{(\underline{g}, \sigma )}(X)\). This defines a map
Composing all the functors as in (4.1) and (4.2), we get the explicit formula of \(\mathbb {P}_{(\underline{g}, \sigma )}\)
\(\mathbb {P}_{(\underline{g}, \sigma )}\) is natural. If \((\underline{g}, \sigma )\) and \((\underline{h}, \tau )\) are conjugate in \(G\wr \Sigma _n\), \(\mathbb {P}_{(\underline{g}, \sigma )}(\mathcal {V})\) and \(\mathbb {P}_{(\underline{h}, \tau )}(\mathcal {V})\) are isomorphic.
Theorem 4.12
The family of maps
satisfy
-
(i)
\(\mathbb {P}_1=\)Id, \(\mathbb {P}_0(x)=1\).
-
(ii)
Let \(x\in QEll^*_G(X)\), \((\underline{g}, \sigma )\in G\wr \Sigma _n\) and \((\underline{h}, \tau )\in G\wr \Sigma _m\). The external product of two power operations
$$\begin{aligned} \mathbb {P}_{(\underline{g}, \sigma )}(x)\boxtimes \mathbb {P}_{(\underline{h}, \tau )}(x)=res|^{\Lambda _{G\wr \Sigma _{m+n}}(\underline{g},\underline{h}; \sigma \tau )} _{\Lambda _{G\wr \Sigma _n}(\underline{g}, \sigma )\times _{\mathbb {T}}\Lambda _{G\wr \Sigma _m}(\underline{h}, \tau )} \mathbb {P}_{(\underline{g},\underline{h}; \sigma \tau )}(x). \end{aligned}$$ -
(iii)
The composition of two power operations is
$$\begin{aligned} \mathbb {P}_{((\underline{\underline{h}, \tau }); \sigma )}(\mathbb {P}_m(x))=res|^{\Lambda _{G\wr \Sigma _{mn}}(\underline{\underline{h}}, (\underline{\tau }, \sigma ))} _{\Lambda _{(G\wr \Sigma _m)\wr \Sigma _n}((\underline{\underline{h}, \tau }); \sigma )}\mathbb {P}_{(\underline{\underline{h}}, (\underline{\tau }, \sigma ))}(x) \end{aligned}$$where \((\underline{\underline{h}, \tau })\in (G\wr \Sigma _m)^{\times n}\), and \(\sigma \in \Sigma _n\). \((\underline{\tau }, \sigma )\) is in \(\Sigma _m\wr \Sigma _n\), thus, can be viewed as an element in \(\Sigma _{mn}\).
-
(iv)
\(\mathbb {P}\) preserves external product. For \(\underline{(g, h)}=((g_1, h_1), \ldots (g_n, h_n))\in (G\times H)^{\times n}\), \(\sigma \in \Sigma _n\),
$$\begin{aligned} \mathbb {P}_{(\underline{(g, h)},\sigma )}(x\boxtimes y)=res|^{\Lambda _{G\wr \Sigma _n}(\underline{g}, \sigma )\times _{\mathbb {T}}\Lambda _{H\wr \Sigma _n}(\underline{h}, \sigma )} _{\Lambda _{(G\times H)\wr \Sigma _n}(\underline{(g, h)}, \sigma )}\mathbb {P}_{(\underline{g}, \sigma )}(x)\boxtimes \mathbb {P}_{(\underline{h}, \sigma )}(y). \end{aligned}$$
Proof
We check each one respectively.
-
(i)
When \(n=1\), all the cycles of a permutation is 1-cycle. \(( \; )_1\) and the homeomorphism \(f_{(g, (1))}\) are both identity maps. Directly from the formula (4.17), \(\mathbb {P}_1(x)=x.\)
-
(ii)
$$\begin{aligned}&\mathbb {P}_{(\underline{g}, \sigma )}(x)\boxtimes \mathbb {P}_{(\underline{h}, \tau )}(x)\\&\quad =f^*_{(\underline{g}, \sigma )}(\boxtimes _{k}\boxtimes _{(i_1,\ldots i_k)}(x_{g_{i_k}\ldots g_{i_1}})_k)\boxtimes f^*_{(\underline{h}, \tau )}(\boxtimes _{j}\boxtimes _{(r_1,\ldots r_j)}(x_{h_{r_j}\ldots h_{r_1}})_j)\\&\quad =res|^{\Lambda _{G\wr \Sigma _{m+n}}(\underline{g},\underline{h}; \sigma \tau )} _{\Lambda _{G\wr \Sigma _n}(\underline{g}, \sigma )\times _{\mathbb {T}}\Lambda _{G\wr \Sigma _m}(\underline{h}, \tau )} f^*_{(\underline{g}, \underline{h}; \sigma \tau )}((\boxtimes _{k}\boxtimes _{(i_1,\ldots i_k)}(x_{g_{i_k}\ldots g_{i_1}})_k)\\&\quad \boxtimes (\boxtimes _{j}\boxtimes _{(r_1,\ldots r_j)}(x_{h_{r_j}\ldots h_{r_1}})_j)).\end{aligned}$$
where \((i_1,\ldots i_k)\) goes over all the k-cycles of \(\sigma \) and \((r_1,\ldots r_j)\) goes over all the j-cycles of \(\tau \) and \(( )_k\) is the map cited in (4.11). The second step is from Proposition 4.11 (iii).
$$\begin{aligned} f^*_{(\underline{g}, \underline{h}; \sigma \tau )}((\boxtimes _{k}\boxtimes _{(i_1,\ldots i_k)}(x_{g_{i_k}\ldots g_{i_1}})_k)\boxtimes (\boxtimes _{j}\boxtimes _{(r_1,\ldots r_j)}(x_{h_{r_j}\ldots h_{r_1}})_j)) \end{aligned}$$is exactly
$$\begin{aligned} \mathbb {P}_{(\underline{g},\underline{h}; \sigma \tau )}(x). \end{aligned}$$ -
(iii)
Recall that for an element \((\underline{\tau }, \sigma )=(\tau _1, \ldots \tau _n, \sigma )\in \Sigma _{mn}\), it acts on the set with mn elements
$$\begin{aligned} \{(i, j)| 1\le i\le n, 1\le j \le m\} \end{aligned}$$in this way:
$$\begin{aligned} (\underline{\tau }, \sigma )\cdot (i, j)=(\sigma (i), \tau _{\sigma (i)}(j)). \end{aligned}$$That also shows how to view it as an element in \(\Sigma _{mn}\).
Then for any integer q,
To find all the cycles of \((\underline{\tau }, \sigma )\) is exactly to find all the orbits of the action by \((\underline{\tau }, \sigma )\). If i belongs to an s-cycle of \(\sigma \) and j belongs to a r-cycle of \(\tau _{\sigma ^s(i)}\tau _{\sigma ^{s-1}(i)}\ldots \tau _{\sigma (i)}\), then the orbit containing (i, j) has sr elements by (4.18). In other words, \((i_1, \ldots i_s)\) is an s-cycle of \(\sigma \) and \((j_1, \ldots j_r)\) is a r-cycle of \(\tau :=\tau _{i_s}\ldots \tau _{i_1}\) if and only if
is an sr-cycle of \((\underline{\tau }, \sigma )\).
where
where \((i_1, \ldots i_k)\) goes over all the k-cycles of \(\sigma \in \Sigma _m\) and \((j_1, \ldots j_r)\) goes over all the r-cycles of \(\tau _{i_k}\ldots \tau _{i_1}\in \Sigma _n\). The last step is by Proposition 4.11 in [10].
is the same space as \(\mathbb {P}_{(\underline{\underline{h}}, (\underline{\tau }, \sigma ))}(x)\), but the action is restricted by
(iv)We have
where \((i_1, \ldots i_k)\) goes over all the k-cycles of \(\sigma \) and \((r_1, \ldots r_j)\) goes over all the j-cycles of \(\sigma \). It equals to
\(\square \)
Example 4.13
Let G be the trivial group and X a space. Let \(\sigma \in \Sigma _n\). Then \(QEll^*_G(X)= K^*_{\mathbb {T}}(X)\). The functor \(f_{(\underline{1}, \sigma )}\) gives the homeomorphism
where the second direct product goes over all the k-cycles of \(\sigma \). By (4.17), the power operation is
When \(n=2\), \(\mathbb {P}_{(\underline{1}, (1)(1))}(x)=x\boxtimes x\) and \(\mathbb {P}_{(\underline{1}, (12))}(x)=(x)_2\).
When \(n=3\), \(\mathbb {P}_{(\underline{1}, (1)(1)(1))}(x)=x\boxtimes x\boxtimes x\), \(\mathbb {P}_{(\underline{1}, (12)(1))}(x)=(x)_2\boxtimes x\), and \(\mathbb {P}_{(\underline{1}, (123))}(x)=(x)_3\).
When \(n=4\), \(\mathbb {P}_{(\underline{1}, (1)(1)(1)(1))}(x)= x\boxtimes x\boxtimes x\boxtimes x\), \(\mathbb {P}_{(\underline{1}, (12))}(x)=(x)_2\boxtimes x\boxtimes x\), \(\mathbb {P}_{(\underline{1}, (123))}(x)=(x)_3\boxtimes x\), \(\mathbb {P}_{(\underline{1}, (1234))}(x)=(x)_4\), and \(\mathbb {P}_{(\underline{1}, (12)(34))}(x)=(x)_2\boxtimes (x)_2.\) Note that there is a \(\Sigma _2\)-action permuting the two \((x)_2\) in \(\mathbb {P}_{(\underline{1}, (12)(34))}(x)\).
Remark 4.14
We have the relation between equivariant Tate K-theory and quasi-elliptic cohomology
It extends uniquely to a power operation for Tate K-theory
which is the stringy power operation \(P^{string}_n\) constructed in Definition 5.10, [10]. It is elliptic in the sense of [2].
5 Orbifold quasi-elliptic cohomology and its power operation
The elliptic cohomology of orbifolds involves a rich interaction between the orbifold structure and the elliptic curve. Ganter explores this interaction in the case of the Tate curve in [11], describing \(K_{Tate}\) for an orbifold X in terms of the equivariant K-theory and the groupoid structure of X.
In Sect. 5.1 we give a description of orbifold quasi-elliptic cohomology. In Sect. 5.2 we discuss the inertia groupoid of symmetric power and the groupoids needed for the construction of the power operation in Sect. 5.3.
5.1 Definition
We have two ways to define orbifold quasi-elliptic cohomology. The first one is motivated by Ganter’s definition of orbifold Tate K-theory in Section 2, [11]. The other one is a generalization of the definition of quasi-elliptic cohomology in Sect. 3.2.
We consider the category of groupoids \(\mathcal {G}pd\) as a 2-category with small topological groupoids as the objects and with
This 2-category is different from that in Section 3 [18]. Let \(\mathcal {G}pd^{cen}\) denote the 2-category of centers of groupoids defined in Section 2, [11]. Ganter constructed in Example 2.3 [11] a 2-functor for any \(k\in \mathbb {Z}\)
where \(\xi ^k\) is the center element of the inertia groupoid I(X) sending (x, g) to \((x, g^k)\). We use \(\xi \) to denote \(\xi ^1\).
Let \(\text{ pt }/\!\!/\mathbb {R}\times _{1\sim \xi }I(X)\) denote the groupoid
with \(\sim \) generated by \(1\sim \xi \).
Definition 5.1
For any topological groupoid X, the quasi-elliptic cohomology \(QEll^*(X)\) is the orbifold K-theory
In other words, for a topological groupoid X, QEll(X) is defined to be a subring of \(K_{orb}(X)\llbracket q^{\pm \frac{1}{|\xi |}} \rrbracket \) that is the Grothendieck group of finite sums
satisfying:
In the global quotient case,
In addition, for any topological groupoid X, we can also consider the category
and formulate \(Loop_1^{ext}(X)\) by adding the rotation action by circle, as the construction in Sect. 2.1.2. Afterwards we can construct the subgroupoid \(\Lambda (X)\) of \(Loop_1^{ext}(X)\) consisting of the constant loops, which is isomorphic to \(\text{ pt }/\!\!/\mathbb {R}\times _{1\sim \xi }I(X)\). So in this way we give an equivalent definition of orbifold quasi-elliptic cohomology.
5.2 Symmetric powers of orbifolds and its inertia groupoid
In this section we introduce the groupoids necessary for the construction of the power operation. In Lemmas 5.3, 5.4 and 5.5 we show the relation between them.
For groupoids like \(\text{ pt }/\!\!/\mathbb {R}\times _{k\sim \xi }X\), instead of the total symmetric power (Definition 3.1, [11]) \(S(\text{ pt }/\!\!/\mathbb {R}\times _{k\sim \xi }X)\), we consider a subgroupoid
of it.
Definition 5.2
(The groupoid \(S^R(\text{ pt }/\!\!/\mathbb {R}\times _{k\sim \xi }X)\)) Let
be the functor sending all the objects to the single point, and an arrow
to
Let \(\times _{\mathbb {R}}(\text{ pt }/\!\!/\mathbb {R}\times _{k\sim \xi }X)\) denote the limit of the diagram of groupoids
Let
denote the limit of n morphisms \(\rho _k\)s. It inherits a \(\Sigma _n\)-action on it by permutation from that on the product \((\text{ pt }/\!\!/\mathbb {R}\times _{k\sim \xi }X)^{\times n}\).
Let \(S_n^R(\text{ pt }/\!\!/\mathbb {R}\times _{k\sim \xi }X)\) denote the groupoid with the same objects as
and morphisms of the form \(([g_1, t_1], \ldots [g_n, t_n]; \sigma )\) with \(([g_1, t_1], \ldots [g_n, t_n])\) a morphism in \(\times ^n_{\mathbb {R}}(\text{ pt }/\!\!/\mathbb {R}\times _{k\sim \xi }X)\) and \(\sigma \in \Sigma _n\). This new groupoid \(S_n^R(\text{ pt }/\!\!/\mathbb {R}\times _{k\sim \xi }X)\) is a subgroupoid of
Define
The triple
is a symmetric monoid where \(*\) is the concatenation and the unit \((\;)\) is the unique object in \(X\wr \Sigma _0\). \(S^R(\text{ pt }/\!\!/\mathbb {R}\times _{k\sim \xi }X)\) is the symmetric product that we will use to formulate the power operation.
Lemma 5.3
Let \(\Phi _k(X)\) denote the groupoid in Definition 3.3, [11], and \(\phi _k\in \)
Center\((\Phi _k)\) denote the restriction of \(S_k(\xi )\) to \(\Phi _k\). For each integer \(k\ge 1\), there is an equivalence between
and the groupoid \(\text{ pt }/\!\!/\mathbb {R}\times _{1\sim \xi ^{\frac{1}{k}}}I(X)[\xi ^{\frac{1}{k}}]\) which identifies \(\phi _k\) with \(\xi ^{\frac{1}{k}}\). Here \(\xi ^{\frac{1}{k}}\) is an added element such that the composition of k \(\xi ^{\frac{1}{k}}\)s is \(\xi \).
Proof
We can define a functor
by sending an object \((\underline{x}, \underline{g}, (1 2 \ldots k))\) to \((x_1, g_k\ldots g_1)\) and sending a morphism \([\underline{h}, (1 2\ldots k)^m, t]\) to
Recall \(h_kg^{-1}_{1-m}\ldots g^{-1}_{k-1}g^{-1}_k\) conjugates \(g_k\ldots g_1\) to itself. It is the element
defined in (4.8). The functor \(A_k\) is an isomorphism, as implied in the proof of Theorem 4.9. \(\square \)
Let \(\Phi (X):=\coprod \limits _{k\ge 1}\Phi _k(X)\). Let \(\phi :=\coprod \limits _{k\ge 1}\phi _k\in \text{ Center }(\Phi )\) denote the restriction of \(S(\xi )\) to \(\Phi \).
Theorem 4.9 can be reinterpreted as Lemma 5.4.
Lemma 5.4
The groupoid \(S^R(\coprod \limits _k\text{ pt }/\!\!/\mathbb {R}\times _{1\sim \xi ^{\frac{1}{k}}}I(X)[\xi ^{\frac{1}{k}}])\) is equivalent to
The proof is similar to that of Theorem 4.9.
Lemma 5.5
We have an equivalence of groupoids
which is natural in X and satisfies
Proof
Let I be the inclusion
Let \(\epsilon \) be the counit of the adjunction \((S,*, ( \; ))\dashv \text{ forget }\). Let Q denote the composition
Let \(Q^R\) be the restriction of Q to the subgroupoid \(S^R(\text{ pt }/\!\!/\mathbb {R}\times _{1\sim \phi }\Phi (X))\), i.e. the composition
The essential image of I consists exactly of the indecomposable objects of \(\text{ pt }/\!\!/\mathbb {R}\times _{1\sim S(\xi )} I(S(X))\), thus, both Q and \(Q^R\) are essentially surjective.
Q is not fully faithful but \(Q^R\) is. This is why we need the product \(S^R\) instead of S.\(\square \)
5.3 Power operation for orbifold quasi-elliptic cohomology
In this section we construct the total power operation for the orbifold quasi-elliptic cohomology
in (5.6), which satisfy the axioms that Ganter formulated in Definition 3.9, [11] for power operations for orbifold theories. The power operation we constructed in Sect. 4.2 is a special case of it for G-spaces.
Example 5.6
We can construct Atiyah’s power operation for orbifold quasi-elliptic cohomology.
Let V be an orbifold vector bundle over the orbifold
thus, V represents an element in QEll(X). Then
is an orbifold vector bundle over
So \(P_n(V)\) is in \(QEll^*(S(X))\).
\(P=(P_n)_{n\ge 0}\) satisfies the axioms of a total power operation.
Before the construction of the power operation of QEll, we introduce several maps necessary for the construction of the power operation.
Let X be an orbifold groupoid and \(k\ge 1\) an integer. We define the map
and
The functor
defined in (4.11) is a special local case of \(s_k\) when X is a G-space and \((\underline{g}, \sigma )\) is fixed.
Let \(\theta : QEll(X)\longrightarrow K_{orb}(\text{ pt }/\!\!/\mathbb {R}\times _{1\sim \phi }\Phi (X))\) be the additive operation whose k-th component is \(A_k^*\circ s_k\), where \(A_k\) is the equivalence defined in Lemma 5.3.
Now we are ready to define the total power operation \(P^{Ell}\) of \(QEll^*\) as the composition below:
Theorem 5.7
\(P^{Ell}\) satisfies the axioms of a total power operation in Definition 3.9 [11].
Proof
From the definition of \(P^{Ell}\), we can see it is a well-defined natural transformation \(QEll\Rightarrow QEll\circ S\) and is a comodule over the comonad \((-)\circ S\).
In addition, the functor \(\theta \) has the property of additivity
The power operation P defined in Example 5.6 has the exponential property. Therefore, \(P^{Ell}\) has the exponential property. So \(P^{Ell}\) is a total power operation. \(\square \)
Remark 5.8
Let \(X/\!\!/G\) be a quotient orbifold. The power operation we construct in Sect. 4.1 for quotient orbifolds is in fact the one below.
\(K^*_{orb}(S^R(\coprod \limits _k \text{ pt }/\!\!/\mathbb {R}\times _{1\sim \xi ^{\frac{1}{k}}}I(X/\!\!/G)[\xi ^{\frac{1}{k}}]))\buildrel {J^*}\over \longrightarrow QEll^*(S(X/\!\!/G))\) where J is constructed from the functors \(J_{(\underline{g}, \sigma )}\) in the proof of Theorem 4.9.
For global quotient orbifolds, \(P^{Ell}\) and \(\mathbb {P}\) are the same up to isomorphism. The diagram
commutes. The vertical maps \(\coprod \nolimits _k A_k^*\) and \(S^R(\coprod \nolimits _k A_k^*)\) are both equivalences of groupoids. The horizontal maps are the power operation defined in Example 5.6.
6 Finite subgroups of the Tate curve
Strickland showed in [28] that the quotient of the Morava E-theory of the symmetric group by a certain transfer ideal can be identified with the product of rings \(\prod \nolimits _{k\ge 0}R_k\) where each \(R_k\) classifies subgroup-schemes of degree \(p^k\) in the formal group associated to \(E^0\mathbb {C}P^{\infty }\). In this section we prove similar conclusions for Tate K-theory and quasi-elliptic cohomology. The main conclusion for Sect. 6 is Theorem 6.4.
6.1 Background
In this section we introduce the Tate curve and its finite subgroups. The main references are Section 2.6, [1] and Sections 8.7, 8.8, [16].
An elliptic curve over the complex numbers \(\mathbb {C}\) is a connected Riemann surface, i.e. a connected compact 1-dimensional complex manifold, of genus 1. By the uniformization theorem every elliptic curve over \(\mathbb {C}\) is analytically isomorphic to a 1-dimensional complex torus, and can be expressed as
with \(q\in \mathbb {C}\) and \(0<|q|<1\), where \(\mathbb {C}^*\) is the multiplicative group \(\mathbb {C}\backslash \{0\}.\)
The Tate curve Tate(q) is the elliptic curve
whose coefficients are given by the formal power series in \(\mathbb {Z}((q))\)
Before we talk about the torsion part of Tate(q), we recall a smooth one-dimensional commutative group scheme T over \(\mathbb {Z}[q^{\pm }]\). It sits in a short exact sequence of group-schemes over \(\mathbb {Z}[q^{\pm }]\)
The N-torsion points T[N] of it is the disjoint union of N schemes \(T_0[N]\), \(\ldots \) \(T_{N-1}[N]\), where
It fits into a short exact sequence
The canonical extension structure on T(N) is compatible with an alternating paring of \(\mathbb {Z}[q^{\pm }]\)-group schemes \(e_N: T(N)\times T(N)\longrightarrow \mu _N\) in the sense that
We have the conclusion below, which is Theorem 8.7.5, [16].
Theorem 6.1
There exists a faithfully flat \(\mathbb {Z}[q^{\pm }]\)-algebra R, an elliptic curve E / R, and an isomorphism of ind-group-schemes over R
such that for every \(N\ge 1\), the isomorphism on N-division points \(T[N]\otimes R\buildrel \sim \over \longrightarrow E[N]\) is compatible with \(e_N\)-pairings.
Thus, we have the unique isomorphism of ind-group-schemes on \(\mathbb {Z}((q))\)
The isomorphism is compatible with the canonical extension structure: for each \(N\ge 1\),
Therefore, Tate(q)[N] is isomorphic to the disjoint union
In addition, we have the question how to classify all the finite subgroups of Tate(q). As shown in Proposition 6.5.1, [16], the ring \(O_{Sub_n}\) that classifies subgroups of Tate(q) of order n exists. To give a description of it, first we describe the isogenies for the analytic Tate curve over \(\mathbb {C}\).
Let (d, e) be a pair of positive integers such that \(N=de\) and \(q'\) a nonzero complex number such that \(q^d=q'^e\). The map
is well-defined since \(\psi _d(q^{\mathbb {Z}})\subseteq q'^{\mathbb {Z}}\). The kernel of \(\psi _d\) is
where \(\mu _d\) is a d-th primitive root of 1 and \(q^{\frac{1}{e}}\) is a e-th primitive root of q. Its order is N. In fact
gives all the subgroups of \(\mathbb {C}^*/q^{\mathbb {Z}}\) of order N.
Proposition 6.2
For each pair of number (d, e), there exists an isogeny
of the elliptic curves over \(O_{Sub_n}\) such that its kernel is the universal subgroup.
We have
Moreover, we have the conclusion below.
Proposition 6.3
The finite subgroups of the Tate curve are the kernels of isogenies.
6.2 Formulas for induction
Before the main conclusion, we introduce the induction formula for quasi-elliptic cohomology. The induction formula for Tate K-theory is constructed in Section 2.3.3, [11].
Let \(H\subseteq G\) be an inclusion of finite groups and X be a G-space. Then we have the inclusion of the groupoids
Let \(a'=\prod \nolimits _{\sigma \in H_{conj}}a'_{\sigma }\) be an element in \(QEll_H(X)=\prod \nolimits _{\sigma \in H_{conj}}K_{\Lambda _H(\sigma )}(X^{\sigma })\) where \(\sigma \) goes over all the conjugacy classes in H. The finite covering map
is defined by sending an object \((\sigma , [g, x])\) to \((\sigma , gx)\) and a morphism \(([g', t], (\sigma , [g, x]))\) to \(([g', t], (gx, \sigma ))\). The transfer of quasi-elliptic cohomology
is defined to be the composition
where the first map is the change-of-group isomorphism and the second is the finite covering.
Thus
where r goes over a set of representatives of \((G/H)^{g}\), in other words, \(r^{-1}gr\) goes over a set of representatives of conjugacy classes in H conjugate to g in G.
There is another way to describe the transfer, which is shown in Rezk’s unpublished work [25] for quasi-elliptic cohomology. The transfer of Tate K-theory can be described similarly.
6.3 The main theorem
Theorem 6.4 gives a classification of finite subgroups of the Tate curve and a similar conclusion for the quasi-elliptic cohomology. We prove it in this section by representation theory. We assume the readers are familiar with the transfer ideal \(I_{tr}\) of equivariant K-theory. References for that include Chapter II, [19] and Section 1.8, [24].
Let N be an integer. Analogous to the transfer ideal \(I_{tr}\) of equivariant K-theory, we can define the transfer ideal for Tate K-theory
where \(I^G_H\) is the transfer map of \(K_{Tate}\) along \(H\hookrightarrow G\) defined in Proposition 2.23, [11], and the transfer ideal for quasi-elliptic cohomology
with \(\mathcal {I}^G_H\) the transfer map of QEll along \(H\hookrightarrow G\) defined in (6.1).
Theorem 6.4
The Tate K-theory of symmetric groups modulo the transfer ideal \(I^{Tate}_{tr}\) classifies the finite subgroups of the Tate curve. Explicitly,
where \(q'\) is the image of q under the power operation \(P^{Tate}\) constructed in Definition 3.15, [11]. The product goes over all the ordered pairs of positive integers (d, e) such that \(N=de\).
We have the analogous conclusion for quasi-elliptic cohomology.
where \(q'\) is the image of q under the power operation \(\mathbb {P}_N\) constructed in Sect. 4.2. The product goes over all the ordered pairs of positive integers (d, e) such that \(N=de\).
We show the proof of (6.6). The proof of (6.5) is similar.
Proof of (6.6)
We divide the elements in \(\Sigma _N\) into two cases.
Case I
The decomposition of \(\sigma \) has cycles of different length. For example, the element
is in this case and \((1\;2)(3\;4)(5\;6)\), \((1 \; 2\;3\;4\;5)(6\;7 \; 8 \; 9\;10)\) are not.
Most elements in \(\Sigma _N\) belong to Case I. \(\sigma \) is not in this case if and only if it consists of cycles of the same length, such as \((1 \; 2)(3 \; 4)\), \((1 \; 2 \; 3)\), 1, \((1 \; 2 \; 3)(4 \; 5 \; 6)\).
For those \(\sigma \) that belong to Case I, \(\Lambda _{\Sigma _N}(\sigma )=\Lambda _{\Sigma _r\times \Sigma _{N-r}}(\sigma )\), so \(Ind^{\Lambda _{\Sigma _N}(\sigma )}_{\Lambda _{\Sigma _r\times \Sigma _{N-r}}(\sigma )}\) is the identity map, so \(K_{\Lambda _{\Sigma _N}(\sigma )}(\text{ pt })\) is equal to \(Ind^{\Lambda _{\Sigma _N}(\sigma )}_{\Lambda _{\Sigma _r\times \Sigma _{N-r}}(\sigma )} K_{\Lambda _{\Sigma _r\times \Sigma _{N-r}}(\sigma )}(\text{ pt })\). Thus, the summand corresponding to \(\sigma \) in \(QEll(\text{ pt }/\!\!/\Sigma _N)\) is completely cancelled.
Case II
\(\sigma \) consists of cycles of the same length. In other words, it consists of d e-cycles with \(N=de\).
The centralizer \(C_{\Sigma _N}(\sigma )\cong C_e\wr \Sigma _d\), where \(C_e=\mathbb {Z}/e\mathbb {Z}\) is the cyclic group with order e. We have
is the subgroup of \(\Lambda _{\Sigma _e}(12\ldots e)\wr \Sigma _d\) with elements of the form
\(K_{\Lambda _{\Sigma _N}(\sigma )}(\text{ pt })\) is the representation ring \(R\Lambda _{\Sigma _N}(\sigma )\). According to Theorem 4.8, as a \(\mathbb {Z}[q^{\pm }]\)-module, it has the basis
where for each \(a\in \mathbb {Z}\), \(q^{\frac{a}{e}}: \Lambda _{C_e}((12\ldots e))\longrightarrow U(1)\) is the map
Namely, it is the map \(x_1^a\) in the sense of Example 3.3.
For each partition (d) of d, if it has more than one cycle, \(\Sigma _{(d)}\) is a subgroup of some \(\Sigma _{d_1}\times \Sigma _{d-d_1}\) for some positive integer \(0<d_1<d\). So for each
with \(r\ge 2\), it is equal to
by the property of induced representation. Note that
So
is in \(K_{\Lambda _{\Sigma _{d_1e}\times \Sigma _{N-d_1e}}(\sigma )}(\text{ pt })\), Thus, each base element with \(r\ge 2\) is contained in the transfer ideal.
When \(r=1\), consider
with \(\tau \in R\Sigma _d\). As indicated in Proposition 1.1 and Corollary 1.5 in [4], each \(\tau \), except the trivial representation of \(\Sigma _d\), can be induced from a representation \(\tau '\) in some \(R(\Sigma _i\times \Sigma _{d-i})\) with \(d>i>0\).
Claim The representation
is isomorphic to
which is
To prove this, we consider a set \(\{\tau _{\alpha }\}_{\alpha \in \Sigma _d/\Sigma _i\times \Sigma _{d-i}}\) of coset representatives. Then
is a set of coset representatives of
Let W be a representation space of \(\Lambda _{\Sigma _{e}}(12\ldots e)\wr _{\mathbb {T}}(\Sigma _i\times \Sigma _{d-i})\), Then
is the direct product of \([\Sigma _d :\Sigma _i\times \Sigma _{d-i}]\) copies of W. For any element
and each \(\alpha \in \Sigma _d/\Sigma _i\times \Sigma _{d-i}\), there is a unique \(\alpha '\in \Sigma _d/\Sigma _i\times \Sigma _{d-i}\) and a unique
such that \(H\eta _{\alpha }=\eta _{\alpha '}J_{\alpha }\). Note that
is a permutation of
So \((q^{\frac{a_1}{e}})^{\otimes _{\mathbb {Z}[q^{\pm }]}d}(g'_1, \ldots g'_d)= (q^{\frac{a_1}{e}})^{\otimes _{\mathbb {Z}[q^{\pm }]}d}(g_1, \ldots g_d).\) In addition, \(\beta \tau _{\alpha }=\tau _{\alpha '}\gamma _{\alpha }\). Let
be an element in
We have
So the claim is proved.
Since
contains all the irreducible representation of \(\Sigma _d\) except the trivial representation, which is corresponding to the partition (d), thus, by the claim, \(K_{\Lambda _{\Sigma _N}(\sigma )}(\text{ pt })\) modulo the image of the transfer, is a \(\mathbb {Z}[q^{\pm }]\)-module generated by the equivalent classes represented by
For any a, \((q^{\frac{a}{e}})^{\otimes _{\mathbb {Z}[q^{\pm }]}d}\) is \((q^{\frac{1}{e}})^{\otimes _{\mathbb {Z}[q^{\pm }]}d}\) to the a-th power. Note that, by (4.17), \((q^{\frac{1}{e}})^{\otimes _{\mathbb {Z}[q^{\pm }]}d}\) is
To get the isomorphism (6.6), consider a map
by sending q to q and x to \(q'\), which is a well-defined \(\mathbb {Z}[q^{\pm }]\)-homomorphism.
Since \(q'^e= q^d\), \(K_{\Lambda _{\Sigma _N}(\sigma )}(\text{ pt })/\mathcal {I}^{QEll}_{tr}\) is a \(\mathbb {Z}[q^{\pm }]\)-module generated by
So any element in it can be expressed as
where each \(f_j(q)\) is in the polynomial ring \(\mathbb {Z}[q^{\pm }]\). It is the image of
in \(\mathbb {Z}[q^{\pm }][x]\). So \(\Psi \) is surjective.
Then we study its kernel. If
is in \(\mathcal {I}^{QEll}_{tr}\), then it is in \(\mathbb {Z}[q^{\pm }]\). So we can assume \(F=0\).
For each element \([(a_1, \ldots a_d; \beta ), t]\) in \(\Lambda _{\Sigma _N}(\sigma )\) with \((a_1, \ldots a_d; \beta )\in C_{\Sigma _N}(\sigma ),\)
Let
be the complex-valued function in the variable t. Let \(\alpha \) denote the number \(e^{\frac{2\pi i}{e}}\). The integers
go over \(0, 1, \ldots e-1\). Consider the e equations
In other words,
The determinant of the Vandermonde matrix
is
When \(\alpha =e^{\frac{2\pi i}{e}}\), each \((\alpha ^{k}-\alpha ^{j})\) in the product (6.11) is nonzero, so for any e, the determinant is nonzero and the matrix is non-singular. So we get \(F_j(t)=0\) for any \(t\in \mathbb {R}\) and \(j=0, 1, 2, \ldots e-1\).
So each \(f_j(q)\) in F is the zero polynomial.
The kernel of \(\Psi \) is the ideal generated by \(q'^e-q^d\). \(\square \)
From the power operation of quasi-elliptic cohomology, we can construct a new operation for quasi-elliptic cohomology.
Proposition 6.5
The composition
defines a ring homomorphism, where res is the restriction map by the inclusion
diag is the diagonal map
and the last map is the isomorphism (6.6).
Proof
Let \(V=\prod \limits _{g\in G_{conj}}V_{g}\in QEll_G(X)\). Apply the explicit formula of the power operation in (4.17), the composition \(diag^*\circ res\circ \mathbb {P}_N\) sends V to
where \((i_1, \ldots i_k)\) goes over all the k-cycles of \(\sigma \), and the tensor products are those of the \(\mathbb {Z}[q^{\pm }]\)-algebras. Then, as shown in the proof of (6.6), after taking the quotient by the transfer ideal \(\mathcal {I}^{QEll}_{tr}\), all the factors in \(diag^*\circ res\circ \mathbb {P}_N(V)\) are cancelled except those corresponding to the elements in \({\Sigma _N}_{conj}\) with cycles of the same length. For the factor corresponding to the element \(\sigma \in {\Sigma _N}_{conj}\) with d e-cycles and \(de=N\), the nontrivial part is \(V_{g^e, d}\otimes _{\mathbb {Z}[q^{\pm }]} q'_{d, e}\) where \(V_{g^e, d}\) is the fixed point space of \(V_{g^e}^{\otimes _{\mathbb {Z}[q, q^{-1}]}d}\) by the permutations \(\Sigma _d\) and \(q'_{d, e}= \mathbb {P}_{\sigma }(q) = (q^{\frac{1}{e}})^{\otimes _{\mathbb {Z}[q, q^{-1}]}d}\).
Thus,
Let V, W be two elements in \(QEll_G(X)\). We have
Similarly,
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Acknowledgements
I would like to thank Charles Rezk who is always a wonderful advisor and very inspiring teacher. Most of this work was guided by him and it is a great experience to work with him. I would also like to thank Matthew Ando. We had many mathematical discussions, which are also important for my work. At last, I would like to thank the editors and the referee for spending time on reading this work and for their constructive and deep suggestion.
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Communicated by Chuck Weibel.
Z. Huan was partially supported by NSF Grant DMS-1406121.
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Huan, Z. Quasi-elliptic cohomology and its power operations. J. Homotopy Relat. Struct. 13, 715–767 (2018). https://doi.org/10.1007/s40062-018-0201-y
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DOI: https://doi.org/10.1007/s40062-018-0201-y