Texas Project NExT Journal
Vol. 2 (2004), pp. 1-10.

A Magnus Embedding Theorem for Second Homotopy Modules

Jacqueline Jensen

MSC (2000): Primary 57M20; Secondary 57M05

Key words: Two-dimensional complexes, Presentations

Abstract: Let $G$ be any group, and $\p$ a presentation for the group. Every group presentation gives rise to a connected two-dimensional CW-complex, $X_{\p}$, with fundamental group, $\pi_1 X_{\p} \cong G$, in a standard way. These two-dimensional CW-complexes are called $[G,2]$-complexes. We will restrict our attention to these since all two-dimensional CW-complexes with $\pi_1X \cong G$ are homotopy equivalent to $[G,2]$-complexes. An open problem in low-dimensional topology is the classification of the homotopy type of $[G,2]$-complexes. Some progress has been made on this problem in different contexts. In this paper, we examine the Magnus Embedding Theorem, its application to this problem, and extend it to an embedding of the second homotopy module.

Dept. of Mathematics and Statistics, Sam Houston State Univ., Huntsville, TX 77341
e-mail: jensen@shsu.edu.edu

Received by Editor: May 28, 2004 and, in revised form November 14, 2004.
Posted: December 22, 2004.

Copyright 2004