Title: Discrete Homotopies and the Fundamental Group
Abstract: In joint work with Jay Wilkins, we extend and develop the
method of discrete homotopies introduced earlier in work with Valera
Berestovskii.
As an application, we generalize and strengthen the theorem of Gromov
that the fundamental group of any compact Riemannian manifold of
diameter at most $D$ has a set of generators $g_{1},...,g_{k}$ of
length at most $2D$ and relators of the form $g_{i}g_{m}=g_{j}$.
In particular, we obtain an explicit bound for the number $k$ of
generators in terms of the number of "short loops" at every point and
the number of balls required to cover a given semi-locally simply
connected geodesic space. As a corollary we obtain a fundamental group
finiteness theorem (new even for Riemannian manifolds) that implies
the fundamental group finiteness theorems of Anderson and
Shen-Wei. This theorem is a special case of a theorem for arbitrary
compact geodesic spaces involving algebraic invariants that measure
the fundamental group at a given scale. Central to the proof is the
notion of the homotopy critical spectrum, which is related to the
length and covering spectra, and is completely determined (including
multiplicity) by special closed geodesics called "essential
circles". We will introduce all of the basic concepts, and the talk
should be very accessible to anyone who knows what a metric space is.