Wolfgang Heil

Department of Mathematics,
Florida State University,
Tallahassee, FL 32306, USA.

Title: Fico's cats

Abstract: Given a complex $K$, the $K$-$cat$ of a space $M$ is the smallest number $k$ such that $M$ admits a covering by open subsets $W_1 , . . . , W_k$ for which the inclusions $W_i \rightarrow M$ factor homotopically through maps $W_i\rightarrow K \rightarrow M$. When $K$ is a point, this is the classical Lusternik-Schnirelmann category $cat(M )$. We will give an overview of joint work with J. C. Gómez- Larrañaga and F. González-Acuña about the classification of $n$-manifolds $M^n$ of $K$-category 2, when $K = S^1$ , $S^2$ or $P^2$.

Title: Fico's amenable cats

Abstract: We will give an overview of joint work with J. C. Gómez-Larrañaga and F. González-Acuña about the $\cal G$-category of 3-manifolds. The motivation comes from Gromov's Vanishing Theorem, which states that if a closed orientable $n$-manifold $M^n$ admits an open cover by $n$ amenable sets, then the simplicial volume of $M^n$ vanishes. If $n = 3$ it follows from Perelman that in this case $M^3$ is a connected sum of graph manifolds. We say that $M$ has $amenable$-$cat$ $k$, if $k$ is the smallest number $k$ such that $M$ admits a covering by $k$ open amenable sets. More generally, for a given class $\cal G$ of groups, $M$ has ${\cal G}$-$cat \leq k$ if it can be covered by $k$ open subsets such that for each path-component $W$ of the subsets the image of its fundamental group $\pi_1 (W ) \rightarrow \pi_1(M )$ belongs to $\cal G$ . For $n = 3$, $M^3$ has $\cal G$-$cat \leq 4$. We characterize all closed 3-manifolds of amenable-cat $1$, $2$, and $3$ and of $\cal G$-$cat$ $1$, $2$, and $3$ for various classes $\cal G$.