On Rational Homotopy of Four-manifolds

Cyclic group actions on manifolds from deformations of rational homotopy types

This paper nishes the series 12, 16, 17] of papers that aim to show the existence of non-trivial group actions (\symmetries") on certain classes of manifolds. More speciically, we ask whether there is a semifree smooth action of the circle group T = S 1 { resp., a non-trivial action of a cyclic group Z=p; p a prime { on a given manifold X with a xed point set of a given rational homotopy type F...

Chern Numbers and Oriented Homotopy Type

1. Statement of results. This note announces the solution of the following problem, a more precise version of which appears in §2: (1) Determine the linear combinations of rational Chern numbers (of almost-complex manifolds) that are invariants of oriented homotopy type (of almost-complex manifolds). Milnor, who posed the problem, conjectured that every such homotopy invariant could be expresse...

Symmetries on Manifolds, Deformations and Rational Homotopy: a Survey

m at h . D G ] 1 3 Se p 20 02 Curvature , diameter , and quotient manifolds

This paper gives improved counterexamples to a question by Grove ([11], 5.7). The question was whether for each positive integer n and real number D, the simply connected closed Riemannian n-manifolds M with sectional curvature ≥ −1 and diameter ≤ D fall into only finitely many rational homotopy types. This was suggested by Gromov's theorem which bounds the Betti numbers of M in terms of n and ...

m at h . A T ] 2 9 Ju n 20 01 RATIONAL HOMOTOPY THEORY AND NONNEGATIVE CURVATURE

In this note , we answer positively a question by Belegradek and Kapovitch[2] about the relation between rational homotopy theory and a problem in Riemannian geometry which asks that total spaces of which vector bundles over compact nonnegative curved manifolds admit (complete) metrics with nonnegative curvature.