Duke Mathematical Journal

Derived smooth manifolds

David I. Spivak

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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.

Article information

Duke Math. J., Volume 153, Number 1 (2010), 55-128.

First available in Project Euclid: 28 April 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55N
Secondary: 55N33: Intersection homology and cohomology 18F20: Presheaves and sheaves [See also 14F05, 32C35, 32L10, 54B40, 55N30]


Spivak, David I. Derived smooth manifolds. Duke Math. J. 153 (2010), no. 1, 55--128. doi:10.1215/00127094-2010-021. https://projecteuclid.org/euclid.dmj/1272480932

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