The equivariant cobordism category

Galatius, Søren; Szűcs, Gergely

doi:10.1112/topo.12181

Cited by 4 publications

(3 citation statements)

References 25 publications

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“…The type of factorisation homology we compute in this article is a special case of equivariant factorisation homology for global quotient orbifolds [Wee20]; namely the case of free actions. The general case, which requires additional input data, should give rise to field theories defined as functors out of the bordism category introduced in [GS21]. Hence, our results provide a first steps towards computing this field theory.…”

Section: Introductionmentioning

confidence: 85%

Finite symmetries of quantum character stacks

Keller¹,

Müller²

2021

Preprint

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For a finite group D, we study categorical factorisation homology on oriented surfaces equipped with principal D-bundles, which 'integrates' a (linear) balanced braided category A with D-action over those surfaces. For surfaces with at least one boundary component, we identify the value of factorisation homology with the category of modules over an explicit algebra in A, extending the work of Ben-Zvi, Brochier and Jordan to surfaces with D-bundles. Furthermore, we show that the value of factorisation homology on annuli, boundary conditions, and point defects can be described in terms of equivariant representation theory.Our main example comes from an action of Dynkin diagram automorphisms on representation categories of quantum groups. We show that in this case factorisation homology gives rise to a quantisation of the moduli space of flat twisted bundles.

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Section: Introductionmentioning

confidence: 85%

Finite symmetries of quantum character stacks

Keller¹,

Müller²

2021

Preprint

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“…In the non-parametrised setting, the composition 𝔐 * ,1 → Cob 2 | 𝑆 1 → Cob 2 is known to induce an equivalence after taking classifying spaces; we hope that in a similar way, the understanding of 𝐵Λ𝔐 * ,1 can shed some light on the homotopy type of 𝐵Cob Z 2 𝐵Cob 2 (𝑆 1 ) in future work. In the case of a finite group G, the homotopy type of Cob 𝐺 𝑑 was recently determined by Szűcs and Galatius [9]. In work by Raptis and Steimle [20], parametrised cobordism categories Cob 𝑑 (𝑌 ) featured as a tool to prove index theorems; however, it was not necessary for the scopes of that work to describe the homotopy type of the classifying spaces of these categories.…”

Section: Related Workmentioning

confidence: 99%

“…In the case of a finite group G , the homotopy type of was recently determined by Szűcs and Galatius [9]. In work by Raptis and Steimle [20], parametrised cobordism categories featured as a tool to prove index theorems; however, it was not necessary for the scopes of that work to describe the homotopy type of the classifying spaces of these categories.…”

Section: Introductionmentioning

confidence: 99%

Parametrised moduli spaces of surfaces as infinite loop spaces

Bianchi

Florian

Reinhold

2022

Forum of Mathematics, Sigma

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We study the $E_2$ -algebra $\Lambda \mathfrak {M}_{*,1}:= \coprod _{g\geqslant 0}\Lambda \mathfrak {M}_{g,1}$ consisting of free loop spaces of moduli spaces of Riemann surfaces with one parametrised boundary component, and compute the homotopy type of the group completion $\Omega B\Lambda \mathfrak {M}_{*,1}$ : it is the product of $\Omega ^{\infty }\mathbf {MTSO}(2)$ with a certain free $\Omega ^{\infty }$ -space depending on the family of all boundary-irreducible mapping classes in all mapping class groups $\Gamma _{g,n}$ with $g\geqslant 0$ and $n\geqslant 1$ .

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An Equivariant Generalisation of McDuff–Segal’s Group–Completion Theorem

Hilman

2023

International Mathematics Research Notices

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In this short note, we prove a $G$–equivariant generalisation of McDuff–Segal’s group–completion theorem for finite groups $G$. A new complication regarding genuine equivariant localisations arises and we resolve this by isolating a simple condition on the homotopy groups of $\mathbb{E}_{\infty }$–rings in $G$–spectra. We check that this condition is satisfied when our inputs are a suitable variant of $\mathbb{E}_{\infty }$–monoids in $G$–spaces via the existence of multiplicative norm structures, thus giving a localisation formula for their associated $G$–spherical group rings.

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The equivariant cobordism category

Cited by 4 publications

References 25 publications

Finite symmetries of quantum character stacks

Finite symmetries of quantum character stacks

Parametrised moduli spaces of surfaces as infinite loop spaces

An Equivariant Generalisation of McDuff–Segal’s Group–Completion Theorem

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