Christine Escher, Oregon State University
Title: Odd-dimensional nonnegatively curved GKM-manifolds
Abstract: A long-standing problem in Riemannian geometry is the
topological classification of Riemannian manifolds M with positive or
non-negative sectional curvature. All known results require high
symmetry, in particular the existence of a "large" torus action on M.
In contrast, the so-called GKMk-manifolds are (even dimensional)
manifolds with arbitrary torus actions. Named after Goresky, Kottwitz
and MacPherson GKMk-manifolds satisfy conditions
that allow
a combinatorial description of their equivariant cohomology
rings.
Examples of GKM2-manifolds are complex projective
spaces,
torus manifolds with vanishing odd degree cohomology and
certain homogeneous spaces. In this talk I will give an overview of
known classification results of GKMk-manifolds with
positive and
non-negative curvature, define a notion of an odd
dimensional
GKMk-manifold, and show how to generalize some of the
classification results to odd dimensional GKMk-manifolds.
In particular, I will outline ideas of the proof that for
odd-dimensional, closed, non-negatively curved GKM3-manifolds
both
the equivariant and the ordinary rational cohomology split off the
cohomology of an odd-dimensional sphere.
This is joint work with Oliver Goertsches and
Catherine Searle.