Texas Project NExT Journal

Vol. 2 (2004), pp. 1-10.

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*A Magnus Embedding Theorem for Second Homotopy
Modules*

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**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