MATH COLLOQUIUM

Department of Mathematical Sciences

University of Wisconsin-Milwaukee

Euler Characteristic of Aspherical Manifolds

Boris Okun

Dynamics/Topology Interview Candidate

Vanderbilt University

Abstract

The Euler Characteristic Conjecture, usually attributed to H. Hopf, states that the Euler characteristic of a 2n-dimensional aspherical manifold M²ⁿ satisfies the inequality $(-1)^n \chi(M^{2n}) \ge 0$.This would follow from Singer Conjecture that $\ell_2$-homology of an aspherical manifold vanishes, except possibly in the middle dimension. In a joint work with Mike Davis we calculate $\ell_2$-homology of (some) right-angled Coxeter groups. In particular, we show that Singer Conjecture holds for Coxeter group manifolds of dimension $\le 4$, and it follows that the Euler Characteristic Conjecture holds for nonpositively curved cubical manifolds of dimension $\le 4$. A surprising corollary of this calculation is an explicit lower bound for the genus of a graph.

Tuesday, February 20, 2001, 3:30 pm,

REFRESHMENTS: Math Lounge EMS E495B, 3:00 pm