These are notes prepared for a series of lectures at the conference Variétés rationnellement connexes: aspects géométriques et arithmétiques of the Société Mathématique de France held in Strasbourg, France in May 2008.

For each n > 1 and each multiplicative closed set of integers S, we study closed model category structures on the pointed category of topological spaces, where the class of weak equivalences are classes of maps inducing isomorphism on homotopy groups with coefficients in determined torsion abelian groups, in degrees higher than or equal to n. We take coefficients either on all the cyclic groups...

Hitchin proved that if M is a spin manifold with positive scalar curvature, then the A^O-characteristic number a(M) vanishes. Gromov and Lawson conjectured that for a simply connected spin manifold M of dimension > 5, the vanishing of a(M) is sufficient for the existence of a Riemannian metric on M with positive scalar curvature. We prove this conjecture using techniques from stable homotopy th...

This paper develops a formula for the equivariant chain-complex of the universal covering space of a Serre bration's total space in terms of equivariant chain-complexes of the base and ber. The present paper's distinguishing feature is that both the base and ber are allowed to be non-simply-connected: the fundamental group of the total space is thus an extension of that of the base by that of t...

where |B| is the length of second fundamental form. |B|2 = 4|H|2 − 2K, where H is the mean curvature and K is the Gauss curvature. The curvature of minimal surfaces is nonpositive. Then we exam what is still hold for some minimal surface theorems if extending the minimal condition H ≡ 0 to the surfaces with K ≤ 0 and (1). In 2001, F. Xavier [18] has the following theorem: Theorem [Xavier]. Let ...

Every compact smooth simply connected 3-manifold is a twisted 3-sphere, so every smooth simply connected 3-manifold is diffeomorphic to S.