Curvature and Topology

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DMS-0104077Stephan Stolz The principal investigator proposes to study the questionwhich manifolds admit Riemannian metrics of positive scalarand Ricci curvature, respectively. Concerning thefirst question, previous work of the principalinvestigator shows that these questions boil down tocomputing abelian groups which depend only on thedimension, the fundamental group and the first twoStiefel-Whitney classes of the manifold these metricslive on. The role these groups play in the existence/classification problem for positive scalarcurvature metrics is analogous of the role of Wall'ssurgery obstruction groups in the existence/classificationproblem 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 theassociated 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 itsWitten 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 thefamily 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). For2-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 ofthem 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.