## Duke Mathematical Journal

### Derived smooth manifolds

David I. Spivak

#### 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 $C^\infty$-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 $A\cap B$ of submanifolds $A,B\subset X$ exists on the categorical level in our theory, and a cup product formula $[A]\smile[B]=[A\cap B]$ 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

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

Dates
First available in Project Euclid: 28 April 2010

https://projecteuclid.org/euclid.dmj/1272480932

Digital Object Identifier
doi:10.1215/00127094-2010-021

Mathematical Reviews number (MathSciNet)
MR2641940

Zentralblatt MATH identifier
05717812

#### Citation

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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