Monoids of moduli spaces of manifolds
Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
Standard
Monoids of moduli spaces of manifolds. / Galatius, Søren; Randal-Williams, Oscar.
I: Geometry & Topology, Bind 14, 2010, s. 1243-1302.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
Harvard
APA
Vancouver
Author
Bibtex
}
RIS
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 [Invent. Math. 130 (1997) 257–275] and Galatius, Madsen, Tillman and Weiss [Acta Math. 202 (2009) 195–239]. There is a category C¿ of closed smooth(d - 1)–manifolds and smooth d–dimensional cobordisms, equipped with generalised orientations specified by a map¿: X ¿ BO(d). The main result of [Acta Math. 202 (2009) 195–239] is a determination of the homotopy type of the classifying space BC¿. The goal of the present paper is a systematic investigation of subcategoriesD¿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.
AB - We study categories of d–dimensional cobordisms from the perspective of Tillmann [Invent. Math. 130 (1997) 257–275] and Galatius, Madsen, Tillman and Weiss [Acta Math. 202 (2009) 195–239]. There is a category C¿ of closed smooth(d - 1)–manifolds and smooth d–dimensional cobordisms, equipped with generalised orientations specified by a map¿: X ¿ BO(d). The main result of [Acta Math. 202 (2009) 195–239] is a determination of the homotopy type of the classifying space BC¿. The goal of the present paper is a systematic investigation of subcategoriesD¿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.
KW - Faculty of Science
U2 - 10.2140/gt.2010.14.1243
DO - 10.2140/gt.2010.14.1243
M3 - Journal article
VL - 14
SP - 1243
EP - 1302
JO - Geometry & Topology
JF - Geometry & Topology
SN - 1465-3060
ER -
ID: 22502949