David Spivak (University of Oregon)
Derived smooth manifolds
Abstract: Given a smooth manifold $M$ and two submanifolds $A,B\subset M$, the intersection $A\cap B$ need not be a smooth manifold. By Thom's transversality theorem, one can deform $A$ to be transverse to $B$ and take the intersection: the result, written $A\pitchfork B$, will be a smooth manifold. Moreover, if $A$ and $B$ are compact, then there is a cup product formula in cobordism, integral cohomology, etc. of the form $$[A]\smile [B]=[A\pitchfork B],$$ where $[-]$ denotes the cohomology fundamental class. The problem is that $A\pitchfork B$ is not unique, and there is no functorial way to choose transverse intersections for pairs of submanifolds. The goal of the theory of {\em derived manifolds} is to correct this defect. The category of derived manifolds contains the category of manifolds as a full subcategory, is closed under taking intersections of manifolds, and yet has enough structure that every compact derived manifold has a fundamental class. Even if $A,B\subset M$ are not transverse (in which case their intersection can be arbitrarily singular), their intersection $A\times_MB$ will be a derived manifold with $[A\times_MB]=[A\pitchfork B]$, and thus satisfy the above cup product formula. To construct the category of derived manifolds, one imitates the constructions of schemes, but in a smooth and homotopical way. I will begin the talk by explaining this construction. Then I will give some examples and discuss some features of the category of derived manifolds. I will end by sketching the Thom-Pontrjagin argument which implies that compact derived manifolds have fundamental classes. |
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