Quasi-smooth Derived Manifolds

The category Man of smooth manifolds is not closed under arbitrary fiber products; for example the zeroset of a smooth function on a manifold is not necessarily a manifold, and the non-transverse intersection of submanifolds is not a manifold. We describe a category dMan, called the category of derived manifolds with the following properties: 1. dMan contains Man as a full subcategory; 2. dMan is closed under taking zerosets of arbitrary smooth functions (and consequently fiber products over a smooth base); and 3. every compact derived manifold has a fundamental homology class which has the desired properties. In order to accomplish this, we must incorporate homotopy theoretic methods (e.g. model categories or ∞-categories) with basic manifold theory. Jacob Lurie took on a similar project in his thesis, namely incorporating homotopy theory and algebraic geometry. We derive much of our inspiration from that and subsequent works by Lurie. For example, we define a derived manifold to be a structured space that can be covered by principal derived manifolds, much as a derived scheme is a structured space that can be covered by affine derived schemes. We proceed to define a cotangent complex on derived manifolds and use it to show that any derived manifold is isomorphic to the zeroset of a section of a vector bundle E → R . After defining derived manifolds with boundary we begin to explore the notion of derived cobordism over a topological space K. We prove two crucial facts: that any two manifolds which are derived cobordant are in fact (smoothly) cobordant, and that any derived manifold is derived cobordant over K to a smooth manifold. In other words, the cobordism ring agrees with the derived cobordism ring. Finally, we define the fundamental class on a derived manifold X by equating it with the fundamental class of a smooth manifold which is derived cobordant to X . The main attraction of derived manifolds is that they provide a robust way of intersecting submanifolds. That is, the intersection product (as well as other constructions such as the Euler class of a vector bundle) is functorially defined within the category dMan itself, rather than just homologically.