Monoids of moduli spaces of manifolds

Abstract

We study categories of d -dimensional cobordisms from the perspective of Tillmann [14] and Galatius, Madsen, Tillman and Weiss [6]. There is a category C  of closed smooth .d 1/-manifolds and smooth d -dimensional cobordisms, equipped with generalised orientations specified by a map ÂW X ! BO.d /. The main result of [6] is a determination of the homotopy type of the classifying space BC  . The goal of the present paper is a systematic investigation of subcategories D  C  with the property that BD ' BC  , the smaller such D the better.We prove that in most cases of interest, D can be chosen to be a homotopy commutative monoid. As a consequence we prove that the stable cohomology of many moduli spaces of surfaces with  -structure is the cohomology of the infinite loop space of a certain Thom spectrum MT . This was known for certain special  , using homological stability results; our work is independent of such results and covers many more cases. 57R90, 57R15, 57R56, 55P47

On the infinite loop space structure of the cobordism category

“…This will also serve to fix notation and language. In section 3 and 4 we review the proof of theorem 1 of [5]. In section 5 we will construct Γ-space models for the spectra ψ θ and M T θ(d) and in section 6 we will show that these Γ-spaces are equivalent.…”

“…The topological space Ψ θ (R n ) has as underlying set pairs (M, l), where M is a ddimensional smooth manifold without boundary which is closed as a subset of R n and l : M → X is a θ-structure. We refer to [5] and [4] for a description of the topology. We will also in general suppress the tangential structure from the notation.…”

“…This section contains a brief review of the main theorem of [6] as proven in [5]. Recall first the construction of the Madsen-Tillmann spectrum …”

“…We will show that the equivalence of [6] actually extends to an equivalence of infinite loop spaces with the above mentioned infinite loop space structures. In more detail, our proof will rely on certain spaces of manifolds introduced by Galatius and Randal-Williams [5], which form an Ω-spectrum denoted here by ψ θ . Using these spaces, they obtain a new proof of the result of [6], which we record as the following theorem.…”

Spaces of graphs and surfaces: on the work of Søren Galatius

“…We note here that in [17] a proof of Madsen and Weiss's theorem in the form stated above is given that no longer uses homology stability. Indeed, Galatius and Randal-Williams show that the inclusion of the monoid n BAutW n into the whole category induces a homotopy equivalence on classifying spaces after group completion (i.e., applying ΩB).…”