The topology of line-free subgroups of Banach spaces
by
Tadeusz Dobrowolski
Pittsburg State University and Institute of Mathematics, Polish Academy of Sciences

A line-free subgroup of an infinite-dimensional Banach space is an additive subgroup that contains no line. We are interested in a classification problem of such groups. Surprisingly, the class of line-free groups is sufficiently rich to make this task interesting and nontrivial. The role of the topological dimension will be emphasized. For example we have:(1) Every nondiscrete line-free subgroups of a Hilbert space that is closed in the weak topology is homeomorphic to the complete Erdös space. (2) There exists an infinite-dimensional line-free subgroup of the Hilbert space l₂ which is zero-dimensional in the l₁-topology. It is important to identify line-free groups that admit a manifold structure. Recently, the question of whether a locally connected subgroup of a Banach space carries a manifold structure has been solved in the negative. Namely, there exists a locally connected line-free subgroup of a Hilbert space that is not connected in dimension 1.

Date received: February 19, 1998

Copyright © 1998 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 # caas-92.