Collapses, products and LC manifolds

Locally Collapsed 3-Manifolds

We prove that a 3-dimensional compact Riemannian manifold which is locally collapsed, with respect to a lower curvature bound, is a graph manifold. This theorem was stated by Perelman and was used in his proof of the geometrization conjecture.

A note on collapse, entropy, and vanishing of the Yamabe invariant of symplectic 4-manifolds

We make use of $\mathcal{F}$-structures and technology developed by Paternain - Petean to compute minimal entropy, minimal volume, and Yamabe invariant of symplectic 4-manifolds, as well as to study their collapse with sectional curvature bounded from below. À la Gompf, we show that these invariants vanish on symplectic 4-manifolds that realize any given finitely presented group as their fundamental group. We extend to the symplectic realm a result of LeBrun which relates the Kodaira dimension with the Yamabe invariant of compact complex surfaces.

Half dimensional collapse of ends of manifolds of nonpositive curvature

This paper accomplishes two things. First, we construct a geometric analog of the rational Tits building for general noncompact, complete, finite volume $n$-manifolds $M$ of bounded nonpositive curvature. Second, we prove that this analog has dimension less than $\lfloor n/2\rfloor$.

在截面曲率下界和正里奇曲率上界下的坍缩至不可光滑流形

Collapse to nonsmoothable manifolds under lower sectional and positive Ricci curvature bounds

构造截面曲率有下界、正里奇曲率下界的坍缩序列，极限为不可光滑流形，并研究其拓扑与几何性质。

Dehn surgeries on knots in product manifolds

We show that if a surgery on a knot in a product sutured manifold yields the same product sutured manifold, then this knot is a 0-- or 1--crossing knot. The proof uses techniques from sutured manifold theory.

Decomposing manifolds into Cartesian products

The decomposability of a Cartesian product of two nondecomposable manifolds into products of lower dimensional manifolds is studied. For 3-manifolds we obtain an analog of a result due to Borsuk for surfaces, and in higher dimensions we show that similar analogs do not exist unless one imposes further restrictions such as simple connectivity.