The homotopy type of the cobordism category

Galatius, Søren; Madsen, Ib; Tillmann, Ulrike; Weiss, Michael

doi:10.1007/s11511-009-0036-9

Cited by 175 publications

(490 citation statements)

References 20 publications

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“…Define the subcategory C red X ⊂ C X to have the same objects as C X , but (S, ψ) is a morphism in C red X only if each connected component of S has a nonempty outgoing boundary. The following was proved in [4].…”

Section: Stability Of the Space Of Surfacesmentioning

confidence: 82%

“…Our method is to use the results of [4] on cobordism categories and to adapt the methods of McDuff-Segal [11] and Tillmann [13] on group completions of categories. Alternatively, one could use the argument given in [10], section 7.…”

Section: Stability Of the Space Of Surfacesmentioning

confidence: 99%

“…(This is sometimes called the classifying space of the category.) The following was proved in [10], but it is part of a more general theorem about cobordism categories proved in [4]. .…”

Section: Stability Of the Space Of Surfacesmentioning

confidence: 99%

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Surfaces in a background space and the homology of mapping class groups

Cohen¹,

Madsen²

2009

Algebraic Geometry

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Abstract. In this paper we study the topology of the space of Riemann surfaces in a simply connected space X, Sg,n(X, γ). This is the space consisting of triples, (Fg,n, φ, f ), where Fg,n is a Riemann surface of genus g and n-boundary components, φ is a parameterization of the boundary, ∂Fg,n, and f : Fg,n → X is a continuous map that satisfies a boundary condition γ. We prove three theorems about these spaces. Our main theorem is the identification of the stable homology type of the space S∞,n(X; γ), defined to be the limit as the genus g gets large, of the spaces Sg,n(X; γ). Our result about this stable topology is a parameterized version of the theorem of Madsen and Weiss proving a generalization of the Mumford conjecture on the stable cohomology of mapping class groups. Our second result describes a stable range in which the homology of Sg,n(X; γ) is isomorphic to the stable homology. Finally we prove a stability theorem about the homology of mapping class groups with certain families of twisted coefficients. The second and third theorems are generalizations of stability theorems of Harer and Ivanov.

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Section: Stability Of the Space Of Surfacesmentioning

confidence: 82%

Section: Stability Of the Space Of Surfacesmentioning

confidence: 99%

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Surfaces in a background space and the homology of mapping class groups

Cohen¹,

Madsen²

2009

Algebraic Geometry

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“…These categories and their higher dimensional analogues have been studied extensively because of their fundamental relationship with conformal field theories and Mumford's conjecture [27,47].…”

Section: The Props Isomorphismmentioning

confidence: 99%

String topology of classifying spaces

Chataur

Menichi

2012

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Abstract. Let G be a finite group or a compact connected Lie group and let BG be its classifying space. Let LBG := map(S 1 , BG) be the free loop space of BG i.e. the space of continuous maps from the circle S 1 to BG. The purpose of this paper is to study the singular homology H * (LBG) of this loop space. We prove that when taken with coefficients in a field the homology of LBG is a homological conformal field theory. As a byproduct of our main theorem, we get a Batalin-Vilkovisky algebra structure on the cohomology H * (LBG). We also prove an algebraic version of this result by showing that the Hochschild cohomology HH

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“…The mapping class groups N g of nonorientable surfaces are not as widely studied as their counterparts for oriented surfaces, but with Wahl's proof [16] of homological stability for these groups, one can apply the machinery of Madsen and Weiss [12] used to prove the Mumford conjecture or its more concise variant by Galatius, Madsen, Tillman and Weiss [8] to study their stable homology. Together these results show that the homology of N 1 coincides with that of a component of an infinite loop space, 1 0 MTO.2/, which we define in Section 2.2.…”

Section: Introductionmentioning

confidence: 99%

The homology of the stable nonorientable mapping class group

Randal-Williams¹

2008

Algebr. Geom. Topol.

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Combining results of Wahl, Galatius-Madsen-Tillmann-Weiss and Korkmaz one can identify the homotopy-type of the classifying space of the stable non-orientable mapping class group N∞ (after plus-construction). At odd primes p, the Fp-homology coincides with that of Q 0 (H P ∞ + ), but at the prime 2 the result is less clear. We identify the F 2 -homology as a Hopf algebra in terms of the homology of well-known spaces. As an application we tabulate the integral stable homology of N∞ in degrees up to six.As in the oriented case, not all of these cohomology classes have a geometric interpretation. We determine a polynomial subalgebra of H * (N∞; F 2 ) consisting of geometrically-defined characteristic classes.

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The homotopy type of the cobordism category

Cited by 175 publications

References 20 publications

Surfaces in a background space and the homology of mapping class groups

Surfaces in a background space and the homology of mapping class groups

String topology of classifying spaces

The homology of the stable nonorientable mapping class group

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