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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 $S^1$ on inertia orbifolds, corresponding to “rotation of loops”. An explicit component-based modification of the inertia orbifold construction which builds in an $S^1$-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).
In the following
$G$ is a discrete group;
$\mathcal{X} \;\simeq\; X \!\sslash\! G$ is a good orbifold which is a global quotient orbifold of a smooth manifold $X$ by a smooth proper action of $G$.
For $g \in G$ any element, with $C_g \subset G$ denoting its centralizer, write
for the Lie group which is the quotient group of the direct product group of $G$ with the additive Lie group of real numbers by the subgroup (isomorphic to the natural numbers) which is generated from the pair $(g,-1) \in G \times \mathbb{R}$.
Hence this sits in a short exact sequence of Lie groups of this form:
For $g \in G$ the group action of $G$ on $X$ restricts to an action of the centralizer $G$ on the fixed locus $X^g = X^{\langle g\rangle}$:
Moreover, since $g$ itself (a) commutes with all element in $C_g$ and (b) has trivial action on $X^g$, this lifts to an action of $\Lambda_g$ (Def. )
The following definition modifies the skeletal presentation of inertia orbifolds:
Write
for the orbifold which is the disjoint union over the conjugacy classes $[g]$ of $G$ of the global quotient orbifolds of the fixed loci $X^g = X^{\langle g\rangle}$ by the group action (1) of the group from Def. .
(Huan 18, Def. 2.14, review in Dove 19, p. 62)
A similar definition is obtained by restricting Ganter 07, Def. 2.3 to constant loops and to $k = 1$, which yields
The canonical group homomorphism (via Def. )
induce canonical morphism from the plain inertia orbifold to Huan’s (Def. ) and Ganter’s orbifolds (Def. ):
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:
Thomas Dove, p. 62 in: Twisted Equivariant Tate K-Theory (arXiv:1912.02374)
Zhen Huan, Matthew Spong, Def. 2.1 in: Twisted Quasi-elliptic cohomology and twisted equivariant elliptic cohomology (arXiv:2006.00554)
Last revised on July 3, 2021 at 16:56:39. See the history of this page for a list of all contributions to it.