| Abstract: |
In previous work with Madsen-Tillmann-Weiss we constructed a cobordism category C, whose objects are closed (d-1)-manifolds and whose morphisms are compact d-dimensional cobordisms, and determined the homotopy type of its classifying space BC in terms of a certain Thom spectrum. There is also a version where all manifolds are endowed with a tangential structure, such as an orientation, a spin structure, etc. In my talk I will present joint work with Randal-Williams, centered around the following problem: How to find a subcategory D of C such that BD is homotopy equivalent to BC, with D as small as possible. In dimension 2 we can in most cases find such a D with just one object (i.e. a topological monoid), with homotopy commutative composition. I will explain this result and what it is good for. This is joint work with O. Randal-Williams.
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