nLab Huan's inertia orbifold

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 $S^1$ 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).

Definition

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 $G$.

Definition

For $g \in G$ any element, with $C_g \subset G$ denoting its centralizer, write

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

for the Lie group which is the quotient group of the direct product group of $G$ 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:

$$1 \to \mathbb{Z} \xhookrightarrow{ \; 1 \mapsto (g^{-1},1) \; } C_g \times \mathbb{R} \overset{\;p_g\;}{\twoheadrightarrow} \Lambda_g \to 1 \,.$$

Remark

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

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

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. )

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

The following definition modifies the skeletal presentation of inertia orbifolds:

Definition

Write

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

for the orbifold which is the disjoint union over the conjugacy classes $[g]$ 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)

Remark

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

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

Properties

Relation to inertia orbifold

The canonical group homomorphism (via Def. )

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

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

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

Related concepts

• inertia orbifold, free loop orbifold

• cyclic loop space, cyclic loop stack

• Tate K-theory, equivariant elliptic cohomology

• transchromatic character

References

The notion is highlighted in:

• Zhen Huan, Def. 2.14 in: Quasi-Elliptic Cohomology I, Advances in Mathematics, Volume 337, 15 October 2018, Pages 107-138 (arXiv:1805.06305, doi:10.1016/j.aim.2018.08.007)

following

• Zhen Huan, Exp. 2.1.14 in: Quasi-elliptic cohomology, 2017 (hdl:2142/97268)

following, in turn

• Charles Rezk, Quasi-Elliptic Cohomology (2014) [pdf]

  (publicly unavailable until upload here, in 2023)

which follows, finally, a similar construction in:

• Nora Ganter, Def. 2.3 in: Stringy power operations in Tate K-theory (arXiv:math/0701565)

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

• Nora Ganter, Def. 2.6 in: Power operations in orbifold Tate K-theory, Homology Homotopy Appl. Volume 15, Number 1 (2013), 313-342. (arXiv:1301.2754, euclid:hha/1383943680)

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:

• Hisham Sati, Urs Schreiber: Cyclification of Orbifolds, Comm. Math. Phys. (2023) [arXiv:2212.13836, talk]