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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 on inertia orbifolds, corresponding to “rotation of loops”. An explicit component-based modification of the inertia orbifold construction which builds in an -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
is a discrete group;
is a good orbifold which is a global quotient orbifold of a smooth manifold by a smooth proper action of .
For any element, with denoting its centralizer, write
for the Lie group which is the quotient group of the direct product group of with the additive Lie group of real numbers by the subgroup (isomorphic to the natural numbers) which is generated from the pair .
Hence this sits in a short exact sequence of Lie groups of this form:
For the group action of on restricts to an action of the centralizer on the fixed locus :
Moreover, since itself (a) commutes with all element in and (b) has trivial action on , this lifts to an action of (Def. )
The following definition modifies the skeletal presentation of inertia orbifolds:
Write
for the orbifold which is the disjoint union over the conjugacy classes of of the global quotient orbifolds of the fixed loci 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 , 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
following, in turn
Charles Rezk, Quasi-Elliptic Cohomology (2014) [pdf]
(publicly unavailable until upload here, in 2023)
which follows, finally, 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)
Characterization in cohesive homotopy theory:
Last revised on November 3, 2023 at 06:51:54. See the history of this page for a list of all contributions to it.