Embedded Cobordism Categories and Spaces of Submanifolds

TY - JOUR

T1 - Embedded Cobordism Categories and Spaces of Submanifolds

AU - Randal-Williams, Oscar

PY - 2011

Y1 - 2011

N2 - 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 paper we apply the techniques of spaces of manifolds, as developed by the author and Galatius in [8], to identify the homotopy type of the cobordism category with objects (d -1)-dimensional submanifolds of a fixed background manifold M. There is a description in terms of a space of sections of a bundle over M associated to its tangent bundle. This can be interpreted as a form of Poincaré duality, relating a space of submanifolds of M to a space of functions on M.

AB - 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 paper we apply the techniques of spaces of manifolds, as developed by the author and Galatius in [8], to identify the homotopy type of the cobordism category with objects (d -1)-dimensional submanifolds of a fixed background manifold M. There is a description in terms of a space of sections of a bundle over M associated to its tangent bundle. This can be interpreted as a form of Poincaré duality, relating a space of submanifolds of M to a space of functions on M.

U2 - 10.1093/imrn/rnq072

DO - 10.1093/imrn/rnq072

M3 - Journal article

SN - 1073-7928

VL - 2011

SP - 572

EP - 608

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 3

ER -