Monoids of moduli spaces of manifolds

TY - JOUR

T1 - Monoids of moduli spaces of manifolds

AU - Galatius, Søren

AU - Randal-Williams, Oscar

N1 - Paper id:: 10.2140/gt.2010.14.1243

PY - 2010

Y1 - 2010

N2 - We study categories of d -dimensional cobordisms from the perspective of Tillmann [14] and Galatius, Madsen, Tillman and Weiss [6]. There is a category Cθ of closed smooth (d - 1)-manifolds and smooth d -dimensional cobordisms, equipped with generalised orientations specified by a map θ: X → BO(d). The main result of [6] is a determination of the homotopy type of the classifying space BCθ. The goal of the present paper is a systematic investigation of subcategories D ⊆ Cθ with the property that BD ≃ BCθ 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 θ -structure is the cohomology of the infinite loop space of a certain Thom spectrum MTθ. This was known for certain special θ, using homological stability results; our work is independent of such results and covers many more cases.

AB - We study categories of d -dimensional cobordisms from the perspective of Tillmann [14] and Galatius, Madsen, Tillman and Weiss [6]. There is a category Cθ of closed smooth (d - 1)-manifolds and smooth d -dimensional cobordisms, equipped with generalised orientations specified by a map θ: X → BO(d). The main result of [6] is a determination of the homotopy type of the classifying space BCθ. The goal of the present paper is a systematic investigation of subcategories D ⊆ Cθ with the property that BD ≃ BCθ 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 θ -structure is the cohomology of the infinite loop space of a certain Thom spectrum MTθ. This was known for certain special θ, using homological stability results; our work is independent of such results and covers many more cases.

KW - Faculty of Science

U2 - 10.2140/gt.2010.14.1243

DO - 10.2140/gt.2010.14.1243

M3 - Journal article

SN - 1465-3060

VL - 14

SP - 1243

EP - 1302

JO - Geometry & Topology

JF - Geometry & Topology

ER -