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Huan's inertia orbifold

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Contents

Idea

The notion of an inertia orbifold is closely related to that of a free loop space/free loop stack of an orbifold. As such, one may expect there to be an action of the circle group S 1S^1 on inertia orbifolds, corresponding to “rotation of loops”. An explicit component-based modification of the inertia orbifold construction which builds in an S 1S^1-action of sorts (akin to cyclic loop spaces/cyclic loop stacks) was highlighted in Huan 18 (Def. below), following Ganter 07, Def. 2.3 (and, apparently, following suggestions by Charles Rezk).

The orbifold K-theory of these modified inertia orbifolds (“quasi-elliptic cohomology”) is closely related to equivariant elliptic cohomology at the Tate curve of the original orbifold (see also at Tate K-theory).

Definition

In the following

Definition

For gGg \in G any element, with C gGC_g \subset G denoting its centralizer, write

Λ g(C g×)/(g 1,1) \Lambda_g \;\coloneqq\; \big( C_g \times \mathbb{R} \big) \, \big/ \, \underset{ \simeq \mathbb{Z} }{ \underbrace{ \langle (g^{-1},1) \rangle }}

for the Lie group which is the quotient group of the direct product group of GG with the additive Lie group of real numbers by the subgroup (isomorphic to the natural numbers) which is generated from the pair (g,1)G×(g,-1) \in G \times \mathbb{R}.

Hence this sits in a short exact sequence of Lie groups of this form:

11(g 1,1)C g×p gΛ g1. 1 \to \mathbb{Z} \xhookrightarrow{ \; 1 \mapsto (g^{-1},1) \; } C_g \times \mathbb{R} \overset{\;p_g\;}{\twoheadrightarrow} \Lambda_g \to 1 \,.

Remark

For gGg \in G the group action of GG on XX restricts to an action of the centralizer GG on the fixed locus X g=X gX^g = X^{\langle g\rangle}:

C g×X g()()X g. C_g \times X^g \xrightarrow{ (-) \cdot (-) } X^g \,.

Moreover, since gg itself (a) commutes with all element in C gC_g and (b) has trivial action on X gX^g, this lifts to an action of Λ g\Lambda_g (Def. )

(1)Λ g×X g X g ([h,r],x) hx. \array{ \Lambda_g \times X^g &\xrightarrow{\;\;\;}& X^g \\ ( [h,r], x ) &\mapsto& h \cdot x \,. }

The following definition modifies the skeletal presentation of inertia orbifolds:

Definition

Write

Λ S 1𝒳[g]ConjCl(G)X gΛ g \Lambda_{S^1} \mathcal{X} \;\coloneqq\; \underset{[g] \in ConjCl(G)}{\coprod} X^g \!\sslash\! \Lambda_g

for the orbifold which is the disjoint union over the conjugacy classes [g][g] of GG of the global quotient orbifolds of the fixed loci X g=X gX^g = X^{\langle g\rangle} by the group action (1) of the group from Def. .

(Huan 18, Def. 2.14, review in Dove 19, p. 62)

Remark

A similar definition is obtained by restricting Ganter 07, Def. 2.3 to constant loops and to k=1k = 1, which yields

[g]ConjCl(G)X g(C g/g) \simeq \underset{[g] \in ConjCl(G)}{\coprod} X^g \!\sslash\! (C_g/\langle g\rangle)

Properties

Relation to inertia orbifold

The canonical group homomorphism (via Def. )

C g Λ g C g/g h (h,0) [h,n] [h] \array{ C_g &\xhookrightarrow{\;\;}& \Lambda_g &\xrightarrow{\;\;}& C_g/\langle g \rangle \\ h &\mapsto& (h,0) \\ && [h,n] &\mapsto& [h] }

induce canonical morphism from the plain inertia orbifold to Huan’s (Def. ) and Ganter’s orbifolds (Def. ):

Λ𝒳 Λ S 1𝒳 . \array{ \Lambda\mathcal{X} &\xrightarrow{\;\;\;}& \Lambda_{S^1} \mathcal{X} &\xrightarrow{\;\;\;}& \cdots \,. }

References

The notion is highlighted in:

following

followiong, in turn, a similar construction in:

and motivated (as made explicit on p. 63 of Dove 19) by the “rotation condition” on Tate K-theory, due to

following Ganter 07, Def. 3.1.

Streamlined review is in:

Last revised on July 3, 2021 at 16:56:39. See the history of this page for a list of all contributions to it.