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The Hempel-Roeling Conjecture
by
Cameron McA. Gordon
University of Texas at Austin
In the early 1970's, in an unpublished manuscript ``Free factors of handlebody groups'', Hempel and Roeling raised the following question. Let F be a free group of rank n, and W a set of m elements of F, 1 <= m <= n, such that for any subset W0 of W, containing, say, k elements, 1 <= k <= m, the quotient group F/<W0> is free of rank n-k. (We shall say that such a subset W is HR.) Note that if W is primitive (i.e., part of a basis-up-to-conjugacy of F) then W is HR.
Question (Hempel-Roeling). If W is HR, is it primitive?
The case m=1 is a classical result of Whitehead, and Hempel and Roeling showed that the question also has an affirmative answer in the case m=n=2.
The 3-dimensional topological analog of killing elements in a free group of rank n is attaching 2-handles to a handlebody of genus n, and this was the motivation of Hempel and Roeling. Although their algebraic question is still open, we show that this topological analog has an affirmative answer.
Theorem. Let C be a set of disjoint simple loops in the boundary of a handlebody X, such that attaching 2-handles to X along the loops in any subset C' of C gives a handlebody. Then C is primitive, in the sense that it is geometrically dual to some set of disjoint disks in X.
Date received: June 28, 1999
Copyright © 1999 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 # cadc-12.