On macroscopic dimension of non-spin 4-manifolds

We prove that for 4-manifolds [Formula: see text] with residually finite fundamental group and non-spin universal covering [Formula: see text], the inequality [Formula: see text] implies the inequality [Formula: see text]. This allows us to complete the proof of Gromov’s Conjecture for 4-manifolds with abelian fundamental group.

Geometric and spectral estimates based on spectral Ricci curvature assumptions

We obtain a Bonnet–Myers theorem under a spectral condition: a closed Riemannian {(M^{n},g)} manifold for which the lowest eigenvalue of the Ricci tensor ρ is such that the Schrödinger operator {\Delta+(n-2)\rho} is positive has finite fundamental group. Further, as a continuation of our earlier results, we obtain isoperimetric inequalities from Kato-type conditions on the Ricci curvature. We also obtain the Kato condition for the Ricci curvature under purely geometric assumptions.

Sur l'existence du sch\'ema en groupes fondametal

Let $S$ be a Dedekind scheme, $X$ a connected $S$-scheme locally of finite type and $x\in X(S)$ a section. The aim of the present paper is to establish the existence of the fundamental group scheme of $X$, when $X$ has reduced fibers or when $X$ is normal. We also prove the existence of a group scheme, that we will call the quasi-finite fundamental group scheme of $X$ at $x$, which classifies all the quasi-finite torsors over $X$, pointed over $x$. We define Galois torsors, which play in this context a role similar to the one of Galois covers in the theory of \'etale fundamental group. Comment: in French. Final version (finally!)

On complete Yamabe solitons

In this paper we study an extension of Yamabe solitons for inequalities. We show that a Riemannian complete non-compact shrinking Yamabe soliton (M,g,V,λ) has finite fundamental group, provided that the scalar curvature is strictly bounded above byλ. Furthermore, an instance of illustrating the sharpness of this inequality is given. We also mention that the fundamental group of the sphere bundleSMis finite.