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15 May 2010 Derived smooth manifolds
David I. Spivak
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Duke Math. J. 153(1): 55-128 (15 May 2010). DOI: 10.1215/00127094-2010-021

Abstract

We define a simplicial category called the category of derived manifolds. It contains the category of smooth manifolds as a full discrete subcategory, and it is closed under taking arbitrary intersections in a manifold. A derived manifold is a space together with a sheaf of local C-rings that is obtained by patching together homotopy zero sets of smooth functions on Euclidean spaces. We show that derived manifolds come equipped with a stable normal bundle and can be imbedded into Euclidean space. We define a cohomology theory called derived cobordism, and use a Pontrjagin-Thom argument to show that the derived cobordism theory is isomorphic to the classical cobordism theory. This allows us to define fundamental classes in cobordism for all derived manifolds. In particular, the intersection AB of submanifolds A,BX exists on the categorical level in our theory, and a cup product formula [A][B]=[AB] holds, even if the submanifolds are not transverse. One can thus consider the theory of derived manifolds as a categorification of intersection theory.

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David I. Spivak. "Derived smooth manifolds." Duke Math. J. 153 (1) 55 - 128, 15 May 2010. https://doi.org/10.1215/00127094-2010-021

Information

Published: 15 May 2010
First available in Project Euclid: 28 April 2010

zbMATH: 05717812
MathSciNet: MR2641940
Digital Object Identifier: 10.1215/00127094-2010-021

Subjects:
Primary: 55N
Secondary: 18F20, 55N33

Rights: Copyright © 2010 Duke University Press

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Vol.153 • No. 1 • 15 May 2010
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