# The homotopy type of the cobordism category

@article{Galatius2006TheHT,
title={The homotopy type of the cobordism category},
author={S{\o}ren Galatius and Ib Henning Madsen and Ulrike Tillmann and Michael Weiss},
journal={Acta Mathematica},
year={2006},
volume={202},
pages={195-239}
}
• Published 10 May 2006
• Mathematics
• Acta Mathematica
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].
158 Citations

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## References

SHOWING 1-10 OF 21 REFERENCES

### Homological stability for the mapping class groups of non-orientable surfaces

We prove that the homology of the mapping class groups of non-orientable surfaces stabilizes with the genus of the surface. Combining our result with recent work of Madsen and Weiss, we obtain that

### On the homotopy of the stable mapping class group

Abstract. By considering all surfaces and their mapping class groups at once, it is shown that the classifying space of the stable mapping class group after plus construction, BΓ∞+, has the homotopy

### Stability of the homology of the mapping class groups of orientable surfaces

The mapping class group of F = Fgs r is F = rgs = wo(A) where A is the topological group of orientation preserving diffeomorphisms of F which are the identity on dF and fix the s punctures. When r =

### HOMOLOGY FIBRATIONS AND ” GROUP-COMPLETION

We give a proof of the Jardine-Tillmann generalized group completion theorem. It is much in the spirit of the original homology fibration approach by McDuff and Segal, but follows a modern treatment

### Stability of the homology of the moduli spaces of Riemann surfaces with spin structure

Recently, due largely to its importance in fermionic string theory, there has been much interest in the moduli spaces ~/gl-e] of Riemann surfaces of genus g with spin structure of Arf invariant e e

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

• Mathematics
• 2008
In this paper we study the topology of the space of Riemann surfaces in a simply connected space X, Sg,n(X,). This is the space consisting of triples, (Fg,n,�,f), where Fg,n is a Riemann surface of

### An infinite loop space structure on the nerve of spin bordism categories

In this paper, we exhibit an infinite loop space structure on the nerve of certain spin bordism 2-categories and compare it with the classifying space of suitably stabilized spin mapping class

### The stable moduli space of Riemann surfaces: Mumford's conjecture

• Mathematics
• 2002
D.Mumford conjectured in (30) that the rational cohomology of the stable moduli space of Riemann surfaces is a polynomial algebra generated by certain classes i of di- mension 2i. For the purpose of

### Characteristic Classes

Let (P,M,G) be a principle fibre bundle over M with group G, connection ω and quotient map π. Recall that for all p ∈ P the Lie algebra G is identified with VpP := Kerπp∗ via the derivative of lp : G

### THE GEOMETRIC REALIZATION OF A SEMI-SIMPLICIAL COMPLEX

homology and homotopy groups. The terminology for semi-simplicial complexes will follow John Moore [7]. In particular the face and degeneracy maps of K will be denoted by d: Kn Kn-1 and si:K. -> Kn+1