The homotopy type of the cobordism category

By Søren Galatius, I. Madsen, Ulrike Tillmann, Michael Weiss. 2006

258 citations recorded by the source corpus.

The embedded cobordism category under study in this paper generalizes the category of conformal surfaces, introduced by G. Segal in [S2] in order to formalize the concept of field theories. Our main result identifies the homotopy type of the classifying space of the embedded d-dimensional cobordism category for all d. For d = 2, our results lead to a new proof of the generalized Mumford conjecture, somewhat different in spirit from the original one, presented in [MW].

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f7b717130de863848689e31472dca3a96ebf2120

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Citations in the corpus

The Local Structure of Algebraic K-Theory (2012, unknown): “1 (see also [91]).”
Stable moduli spaces of high-dimensional manifolds (2012, methodology): “ for all basepoints and all i≤κ. Thus this is the subcategory on those morphisms which are κ-connected relative to their outgoing boundary. The category C0 θ is the “positive boundary category” as in [GMTW09], where each path component of a cobordism is required to have non-empty outgoing boundary. ”
Homology of the moduli spaces and mapping class groups of framed, r‐Spin and Pin surfaces (2010, background): “This construction is discussed in more depth in [8, 9, 18].”
A note on the (∞,n)–category of cobordisms (2015, unknown): “We sketch the argument, essentially following the one for showing that SubpM,R(8)q is a classifying space for the group of diffeomorphisms of M , and its modifications in [GTMW09] and [Lur09c]: Consider the space EmbppM, pMj1,.”
Semisimplicial spaces (2017, background): “The third goal of this note is to give a proof of the Group-Completion Theorem, which plays a crucial role in [11], [6], and [7].”
A vanishing theorem for characteristic classes of odd-dimensional manifold bundles (2009, background): “The Madsen-Tillmann-Weiss map (or MTW-map) of the bundle f : E → B is the composition (2.4.2) αE := κE ◦ PTf : Σ∞B+ → MTSO(n), which is defined uniquely up to a contractible space of choices.”
Infinite loop spaces and positive scalar curvature (2014, background): “The basic reference for these spectra is [22] and the connection to the index theorem was pointed out in [17].”
Homological stability for the mapping class groups of non-orientable surfaces (2006, background): “Embedding E in B × Rn+2 for some large n and collapsing everything outside a tubular neighborhood gives a map, the so-called Thom-Pontrjagin map, Sn+2 ∧ B → Th(NpE) to the Thom space of the vertical normal bundle of E. Classifying this bundle gives a further map into Th(U⊥n ), the Thom space of the…”
Homological stability for moduli spaces of high dimensional manifolds (2012, methodology): “es satisfying 2 g(W;‘^ W) 3 if in addition is spherical. As in Theorem 1.5 and the discussion following it, Corollary 1.8 concerns a specic map into a component of a path space of BC , which by [GMTW09] may be identied with the basepoint component 1 0 MTof the innite loop space 1MT. 1.4. Other tange”
Resolutions of moduli spaces and homological stability (2009, background): “As is now well known [16, 8, 5, 9], in both cases the stable homology coincides with that of the infinite loop space of a certain Thom spectrum.”