Vitali Kapovitch

Nilpotency, almost nonnegative curvature, and the gradient flow on Alexandrov spaces

We show that almost nonnegatively curved m-dimensional manifolds are, up to finite cover, nilpotent spaces in the sense of homotopy theory and have C(m)-nilpotent fundamental groups. We also show that up to a finite cover almost nonnegatively curved manifolds are fiber bundles with simply connected fibers over nilmanifolds.

Topological obstructions to nonnegative curvature

Abstract. We find new obstructions to the existence of complete Riemannian metric of nonnegative sectional curvature on manifolds with infinite fundamental groups. In particular, we construct many examples of vector bundles whose total spaces admit no nonnegatively curved metrics.

Non-negative pinching, moduli spaces and bundles with infinitely many souls

We show that in each dimension

Finiteness theorems for nonnegatively curved vector bundles

n\ge 10

On the torsion in the center conjecture

there exist infinite sequences of homotopy equivalent but mutually non-homeomorphic closed simply connected Riemannian

Perelman’s Stability Theorem

n

REGULARITY OF LIMITS OF NONCOLLAPSING SEQUENCES OF MANIFOLDS

-manifolds with

Biquotients with Singly Generated Rational Cohomology

0\le \sec\le 1

Obstructions to nonnegative curvature and rational homotopy theory

, positive Ricci curvature and uniformly bounded diameter. We also construct open manifolds of fixed diffeomorphism type which admit infinitely many complete nonnegatively pinched metrics with souls of bounded diameter such that the souls are mutually non-homeomorphic. Finally, we construct examples of noncompact manifolds whose moduli spaces of complete metrics with

Structure of fundamental groups of manifolds with Ricci curvature bounded below

\sec\ge 0