Normal Smoothings for Smooth Cube Manifolds

We prove that smooth cube manifolds have normal smooth structures.Comment: The paper "Pinched Smooth Hyperbolization" [arXiv:1110.6374] has been divided in parts. "Normal Smoothings for Smooth Cube Manifolds" is one of the part

Uniqueness and Stability in Inverse Spectral Problems for Collapsing Manifolds

We consider a geometric inverse problems associated with interior measurements: Assume that on a closed Riemannian manifold $(M, h)$ we can make measurements of the point values of the heat kernel on some open subset $U \subset M$. Can these measurements be used to determine the whole manifold $M$ and metric $h$ on it? In this paper we analyze the stability of this reconstruction in a class of $n$-dimensional manifolds which may collapse to lower dimensions. In the Euclidean space, stability results for inverse problems for partial differential operators need considerations of operators with non-smooth coefficients. Indeed, operators with smooth coefficients can approximate those with non-smooth ones. For geometric inverse problems, we can encounter a similar phenomenon: to understand stability of the solution of inverse problems for smooth manifolds, we should study the question of uniqueness for the limiting non-smooth case. Moreover, it is well-known, that a sequence of smooth $n$-dimensional manifolds can collapse to a non-smooth space of lower dimension. To analyze the stability of inverse problem in a class of smooth manifolds with bounded sectional curvature and diameter, we study properties of the spaces which occur as limits of these collapsed manifolds and study uniqueness of inverse problems on collapsed manifolds. Combining these, we obtain stability results for inverse problems in the class of smooth manifolds with bounded sectional curvature and diameter

Derived Smooth Manifolds

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