# 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} }

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

### A relative h-principle via cobordism-like categories

- Mathematics
- 2010

We prove an h-principle with boundary condition for a certain class of topological spaces valued sheaves. The techniques used in the proof come from the study of the homotopy type of the cobordism…

### SINGULAR COBORDISM CATEGORIES

- Mathematics
- 2008

Recently Galatius, Madsen, Tillmann and Weiss identified the homotopy type of the classifying space of the cobordism category of embedded d-dimensional manifolds [7] for each positive integer d.…

### A geometric interpretation of the homotopy groups of the cobordism category

- Mathematics
- 2014

The classifying space of the embedded cobordism category has been identified by Galatius, Tillmann, Madsen and Weiss [6] as the infinite loop space of a certain Thom spectrum. This identifies the set…

### The homotopy type of the topological cobordism category

- Mathematics
- 2018

We define a cobordism category of topological manifolds and prove that if $d \neq 4$ its classifying space is weakly equivalent to $\Omega^{\infty -1} MTTop(d)$, where $MTTop(d)$ is the Thom spectrum…

### SINGULAR COBORDISM CATEGORIES

- Mathematics
- 2010

Recently Galatius, Madsen, Tillmann and Weiss identied the ho- motopy type of the classifying space of the cobordism category of embedded d-dimensional manifolds (9) for each positive integer d.…

### Cobordism Category of Manifolds With Baas-Sullivan Singularities, Part I

- Mathematics
- 2012

For a fixed closed manifold P , we construct a cobordism category of embedded manifolds with Baas-Sullivan singularities modeled on P . Our main theorem identifies the homotopy type of the…

### On Strict Higher Categories and their Application to Cobordism Theory

- Mathematics
- 2015

The main goal of the present thesis is an exposition of the BökstedtMadsen theorem ([BM]), which relates the classifying space of the embedded cobordism category to certain iterated loop spaces of…

### Embedded Cobordism Categories and Spaces of Submanifolds

- Mathematics
- 2010

Galatius, Madsen, Tillmann, and Weiss [7] have identified the homotopy type of the classifying space of the cobordism category with objects (d −1)-dimensional manifolds embedded in ℝ ∞ . In this…

### The equivariant cobordism category

- MathematicsJournal of Topology
- 2021

For a finite group G , we define an equivariant cobordism category CdG . Objects of the category are (d−1) ‐dimensional closed smooth G ‐manifolds and morphisms are smooth d ‐dimensional equivariant…

### An additivity theorem for cobordism categories

- Mathematics
- 2018

We give a new proof of the Genauer fibration sequence, relating the cobordism categories of closed manifolds with cobordism categories of manifolds with boundaries. Unlike the existing proofs, it is…

## References

SHOWING 1-10 OF 21 REFERENCES

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

- Mathematics
- 2008

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

- Mathematics
- 1997

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

- Mathematics
- 1985

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

- Mathematics
- 2004

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

- Mathematics
- 1990

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

- Mathematics
- 2004

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

- Mathematics
- 2004

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

- Mathematics
- 1957

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…