Derived Smooth Manifolds

Details Derived Smooth Manifolds Derived Smooth Manifolds

2008-10-29

arxiv.org/abs/0810.5174v3

Mathematics

Algebraic Topology

57 pages. Reformulation of author's PhD thesis. To appear in Duke Math J.

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Derived Smooth Manifolds does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Derived Smooth Manifolds, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Derived Smooth Manifolds will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-71459