Orbispaces, orthogonal spaces, and the universal compact Lie group

Schwede, Stefan

doi:10.1007/s00209-019-02265-1

Cited by 15 publications

(33 citation statements)

References 20 publications

(42 reference statements)

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“…This follows immediately using [3,Lemma A.6] or the in-depth account [5,Theorem 3.5]. Here, the first model category, Pre(O gl , Top) is the variant of orbispaces used in [8] while the second one, Pre(Orb, Top) is used in [3].…”

Section: Resultsmentioning

confidence: 99%

“…This exposition shows that there is a zig-zag of weak equivalences between the respective mapping spaces which is compatible with the given structures of topologically enriched categories. We deduce that there is a zig-zag of Dwyer-Kan equivalences between the orbit category O gl from [8] and the orbit category Orb from [3]. This implies by [3,5] that the associated presheaf categories are Quillen equivalent via a zig-zag, and as a consequence, all the models for global homotopy theory from both papers are equivalent to each other.…”

Section: Introductionmentioning

confidence: 81%

“…(i) Therefore, the weak homotopy type of O gl (H, G) is i∈I BC(α i ) (see also [8,Remark 2.2]). This agrees with the weak homotopy type of Orb(H, G) by Proposition 2.5.…”

Section: A Reminder On O Gl (H G)mentioning

confidence: 99%

“…In [8], Schwede introduces L-spaces and orthogonal spaces along with his version of orbispaces, together with model category structures on all of these categories, and verifies that they are Quillen equivalent. Therefore, he also describes three models for global spaces.…”

Section: Introductionmentioning

confidence: 99%

“…The orbit category used in [3] is called Orb, and the one from [8] is denoted by O gl . As we will recall, there is significant evidence that the two versions of orbispaces should be equivalent.…”

Section: Introductionmentioning

confidence: 99%

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A comparison of two models of orbispaces

Körschgen

2018

Homology, Homotopy and Applications

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This paper proves that the two homotopy theories for orbispaces given by Gepner and Henriques and by Schwede, respectively, agree by providing a zig-zag of Dwyer-Kan equivalences between the respective topologically enriched index categories. The aforementioned authors establish various models for unstable global homotopy theory with compact Lie group isotropy, and orbispaces serve as a common denominator for their particular approaches. Although the two flavors of orbispaces are expected to agree with each other, a concrete comparison zig-zag has not been known so far. We bridge this gap by providing such a zig-zag which asserts that all those models for unstable global homotopy theory with compact Lie group isotropy which have been described by the authors named above agree with each other.On our way, we provide a result which is of independent interest. For a large class of free actions of a compact Lie group, we prove that the homotopy quotient by the group action is weakly equivalent to the strict quotient. This is a known result under more restrictive conditions, e.g., for free actions on a manifold. We broadly extend these results to all free actions of a compact Lie group on a compactly generated Hausdorff space. A Reminder on O gl

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 81%

“…(i) Therefore, the weak homotopy type of O gl (H, G) is i∈I BC(α i ) (see also [8,Remark 2.2]). This agrees with the weak homotopy type of Orb(H, G) by Proposition 2.5.…”

Section: A Reminder On O Gl (H G)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A comparison of two models of orbispaces

Körschgen

2018

Homology, Homotopy and Applications

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Equivariant Cohomotopy implies orientifold tadpole cancellation

Sati

Schreiber

2020

Journal of Geometry and Physics

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Proper Equivariant Stable Homotopy Theory

Degrijse¹,

Hausmann

Lück

et al. 2023

Memoirs of the AMS

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This monograph introduces a framework for genuine proper equivariant stable homotopy theory for Lie groups. The adjective ‘proper’ alludes to the feature that equivalences are tested on compact subgroups, and that the objects are built from equivariant cells with compact isotropy groups; the adjective ‘genuine’ indicates that the theory comes with appropriate transfers and Wirthmüller isomorphisms, and the resulting equivariant cohomology theories support the analog of an R O ( G ) R O(G) -grading. Our model for genuine proper G G -equivariant stable homotopy theory is the category of orthogonal G G -spectra; the equivalences are those morphisms that induce isomorphisms of equivariant stable homotopy groups for all compact subgroups of G G . This class of π ∗ \pi _* -isomorphisms is part of a symmetric monoidal stable model structure, and the associated tensor triangulated homotopy category is compactly generated. Consequently, every orthogonal G G -spectrum represents an equivariant cohomology theory on the category of G G -spaces. These represented cohomology theories are designed to only depend on the ‘proper G G -homotopy type’, tested by fixed points under all compact subgroups. An important special case of our theory are infinite discrete groups. For these, our genuine equivariant theory is related to finiteness properties in the sense of geometric group theory; for example, the G G -sphere spectrum is a compact object in our triangulated equivariant homotopy category if the universal space for proper G G -actions has a finite G G -CW-model. For discrete groups, the represented equivariant cohomology theories on finite proper G G -CW-complexes admit a more explicit description in terms of parameterized equivariant homotopy theory, suitably stabilized by G G -vector bundles. Via this description, we can identify the previously defined G G -cohomology theories of equivariant stable cohomotopy and equivariant K-theory as cohomology theories represented by specific orthogonal G G -spectra.

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Orbispaces, orthogonal spaces, and the universal compact Lie group

Cited by 15 publications

References 20 publications

A comparison of two models of orbispaces

A comparison of two models of orbispaces

Equivariant Cohomotopy implies orientifold tadpole cancellation

Proper Equivariant Stable Homotopy Theory

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