The Mittag-Leffler condition and curves of trivial shape
by
Piotr Minc
Auburn University

An inverse sequence \Pi = (\Pi_n, p_nⁿ⁺¹) of groups satisfies the Mittag-Leffler condition provided that for each n there exists a k > n such that p_n^k(\Pi_k)=p_n^m(\Pi_m) for each m > k. For each curve X represented as the inverse limit of an inverse sequence of graphs (X_n, f_nⁿ⁺¹), one can consider the induced inverse system of fundamental groups \pi₁(X)=(\pi₁(X_n), f_nⁿ⁺¹_#). \pi₁(X) does not satisfy the Mittag-Leffler condition for the Case-Chamberlain example, solenoids and other non-movable curves. However, the Mittag-Leffler condition is always trivially satisfied if the shape of X is zero. During the talk, we will discuss a combinatorial extension of the fundamental group in the category of graphs and simplicial maps. For this extension, the corresponding Mittag-Leffler condition will not always be satisfied on continua of trivial shape. We will show in this context that some tree-like curves, including the Ingram example, have properties strikingly similar to those of solenoids.

Date received: April 13, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caem-10.