[Paper Review] Orbifolds and stable homotopy groups

Johann Leida|ArXiv.org|May 20, 2005

Homotopy and Cohomology in Algebraic Topology14 references3 citations

TL;DR

This paper introduces intrinsic stable orbifold homotopy groups and extended unstable orbifold homotopy groups using Lie groupoids and fixed point sectors, generalizing classical orbifold homotopy invariants. By leveraging tom Dieck's theorem on equivariant stable homotopy, the authors define Morita-invariant invariants that capture isotropy and fixed point data, proving these groups are orbifold invariants—especially in global quotient cases where isomorphisms between stable homotopy groups of different presentations are established.

ABSTRACT

Lie groupoids generalize transformation groups, and so provide a natural language for studying orbifolds and other noncommutative geometries. In this paper, we investigate a connection between orbifolds and equivariant stable homotopy theory using such groupoids. A different sort of twisted sector, along with a classical theorem of tom Dieck, allows for a natural definition of stable orbifold homotopy groups, and motivates defining extended unstable orbifold homotopy groups generalizing previous definitions.

Motivation & Objective

• To address the insufficiency of classical orbifold homotopy groups in distinguishing orbifolds with isomorphic classifying spaces but different isotropy structures.
• To define stable orbifold homotopy groups that incorporate fixed point data across all isotropy subgroups, inspired by equivariant stable homotopy theory.
• To establish that these new invariants are Morita-invariant, thus truly intrinsic to the orbifold structure.
• To prove that for global quotient orbifolds, the total stable equivariant homotopy group is an orbifold invariant, generalizing results from orbifold K-theory.

Proposed method

• Define the stable orbifold homotopy group as $\varpi^{\mathrm{orb}}_{n}(\mathfrak{X}) := \varpi_{n}(B\widetilde{\mathcal{G}}_{+})$, where $\widetilde{\mathcal{G}}$ is the groupoid of fixed point sectors derived from the orbifold groupoid $\mathcal{G}$.
• Introduce extended unstable orbifold homotopy groups via $\hat{\pi}^{\mathrm{orb}}_{n}(\mathfrak{X},(x,H)) := \pi_{n}(B\widetilde{\mathcal{G}},[x,H])$, using base points in fixed point sectors.
• Apply tom Dieck's isomorphism $\varpi_{n}^{G}(X) \cong \bigoplus_{(H)} \varpi^{W_{G}H}_{n}(EW_{G}H_{+} \wedge X^{H})$ to relate equivariant stable homotopy to orbifold invariants.
• Use the Borel construction $EG \times_G M \simeq B(G \ltimes M)$ to relate classifying spaces of translation groupoids to equivariant homotopy types.
• Construct a homotopy equivalence between $B(G \ltimes M)$ and $EG \times_G M$ via simplicial groupoid models, generalizing Segal's argument.
• Establish isomorphisms between stable homotopy groups of different global quotient presentations of the same orbifold using the vanishing of $d(H)$ for finite groups.

Experimental results

Research questions

• RQ1 Can stable homotopy groups of orbifolds be defined intrinsically using fixed point data, independent of a specific group action or presentation?
• RQ2 To what extent do fixed point sectors and isotropy subgroups determine the homotopy type of an orbifold in a model-theoretic sense?
• RQ3 Are stable equivariant homotopy groups of different global quotient presentations of the same orbifold isomorphic?
• RQ4 How does the new definition of stable orbifold homotopy groups relate to classical orbifold homotopy and equivariant stable homotopy theory?
• RQ5 Can a model structure on the category of orbifolds or Lie groupoids be constructed such that these new homotopy groups detect weak equivalences?

Key findings

• The stable orbifold homotopy groups $\varpi^{\mathrm{orb}}_{n}(\mathfrak{X})$ and extended unstable orbifold homotopy groups $\hat{\pi}^{\mathrm{orb}}_{n}(\mathfrak{X},(x,H))$ are proven to be Morita invariants, hence intrinsic to the orbifold.
• For global quotient orbifolds $\mathfrak{X} = M/G$, the total stable equivariant homotopy group $\varpi^{G}_{\mathrm{tot}}(M_{+})$ is an orbifold invariant.
• In the case of finite group quotients, the isomorphism $\varpi^{G}_{n}(M_{+}) \cong \varpi^{G'}_{n}(M'_{+})$ holds for any two global quotient presentations of the same orbifold $\mathfrak{X}$.
• The construction relies on identifying fixed point sectors with mapping spaces of homomorphisms from finite groups into the groupoid, drawing analogy to stacks and schemes.
• The proof of the isomorphism in the finite group case follows directly from tom Dieck’s theorem and the vanishing of the dimension function $d(H)$ for finite groups.
• The paper suggests that future model structures on $\mathtt{Gpd}$ or $\mathtt{Orb}$ could formalize the notion of homotopy equivalence between orbifolds using these invariants.