Derived smooth manifolds

By David I. Spivak. 2008

71 citations recorded by the source corpus.

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 {\em categorification} of intersection theory.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/60a0f7d71fe70feec412afe78080b4ff215b0089

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

On manifolds with corners (2009, methodology): “Even if one exists, from the point of view of derived differential geometry [16], it is in some sense the ‘wrong answer’.”
Virtual fundamental classes for moduli spaces of sheaves on Calabi-Yau four-folds (2015, methodology): “3, and Spivak [32], Borisov and Noël [7, 8] and Joyce [21–23,25] for §2.”
An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves (2013, background): “A good notion of a “derived smooth manifold” exists (due to Spivak [75], Borisov and Noel [10] and Joyce [50; 49]), and it is reasonable to expect that a parallel topological theory exists as well.”