Acta Mathematica

The homotopy type of the cobordism category

Søren Galatius, Ib Madsen, Ulrike Tillmann, and Michael Weiss

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The embedded cobordism category under study in this paper generalizes the category of conformal surfaces, introduced by G. Segal in [S2] in order to formalize the concept of field theories. Our main result identifies the homotopy type of the classifying space of the embedded d-dimensional cobordism category for all d. For d = 2, our results lead to a new proof of the generalized Mumford conjecture, somewhat different in spirit from the original one, presented in [MW].


S. Galatius was partially supported by NSF grant DMS-0505740 and the Clay Institute.

M. Weiss was partially supported by the Royal Society and the Mittag-Leffler Institute.

Article information

Acta Math., Volume 202, Number 2 (2009), 195-239.

Received: 30 April 2007
Revised: 30 September 2008
First available in Project Euclid: 31 January 2017

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2009 © Institut Mittag-Leffler


Galatius, Søren; Madsen, Ib; Tillmann, Ulrike; Weiss, Michael. The homotopy type of the cobordism category. Acta Math. 202 (2009), no. 2, 195--239. doi:10.1007/s11511-009-0036-9.

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