Curvature and Topology

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

DMS-0104077 Stephan Stolz The principal investigator proposes to study the question which manifolds admit Riemannian metrics of positive scalar and Ricci curvature, respectively. Concerning the first question, previous work of the principal investigator shows that these questions boil down to computing abelian groups which depend only on the dimension, the fundamental group and the first two Stiefel-Whitney classes of the manifold these metrics live on. The role these groups play in the existence/classification problem for positive scalar curvature metrics is analogous of the role of Wall's surgery obstruction groups in the existence/classification problem for smooth structures. The principal investigator hopes to gain an algebraic/functorial understanding of these groups by studying various maps that relate these groups with homology/K-theory/connective K-theory of classifying spaces of groups, and with the K-theory of the associated group C*-algebras. Concerning positive Ricci curvature, the investigator is pursuing a proof of his conjecture that the existence of a positive Ricci curvature metric on a spin manifold with vanishing first Pontryagin class implies the vanishing of its Witten genus. This invariant arose from considerations in string theory. Heuristically it is the index of a yet to be rigorously defined "Dirac operator" on the free loop space of this manifold. Mike Hopkins has described a homotopy theoretic way to define the Witten genus of a family of such manifolds which lives in the elliptic cohomology of the parameter space. It is a very interesting challenge to express the elliptic cohomology and the family Witten genus in terms of the objects that string theorists are analysing and thus to give a geometric interpretation of Hopkins construction. These projects fit in the general framework of trying to relate the topology of a manifold (qualitative information about its global shape) and its geometry (quantitative information about its local shape). For 2-dimensional manifolds (like the surface of a ball or a tire), a nice classification has been known for a long time: Two such surfaces have the same topology (that is, they can be deformed into each other if we think of them as being made of thin rubber) if and only if they have the same number of `holes' (the surface of a ball no holes, the surface of a tire or a cup has one hole, and a pretzel has two holes). Moreover, if a surface has `positive curvature' in the sense that the angle sum in each triangle whose edges are geodesics (shortest curves) is larger than 180 degrees, then this surface has the same topology as the surface of a ball. It is a major goal of modern day mathematics to generalize these results to higher dimensional manifolds. For example, our universe is a manifold of dimension 3, Einstein's space-time has dimension 4, and manifolds of dimension 10, respectively, 26 play a crucial role in the theoretical physics of the attempted unification of the four fundamental forces.