On 2-dimensional Ctf-complexes with a Single 2-cell

In this paper we are interested in finite connected 2-dimensional CTF-complexes, each with a single 2-celL We show any two such complexes have the same homotopy type if their fundamental groups are isomorphic. In fact, there is a homotopy equivalence inducing any isomorphism of the fundamental groups. We also study the homotopy factorizations of such spaces into finite sums. In this paper we are interested in finite connected 2-dimensional CPF-complexes with a single 2-cell. Each such CW-complex has the homotopy type of the cellular model C(&) of some finite one-relator presentation & = (x ί9 > } x n :R) of Ξ = πJL. If the single relator R is not a proper power, it is known that the cellular model C{&) is aspherical (see [10], [1], or [4]), hence it is determined up to homotopy type by its fundamental group. If the single relator R is a proper power, C(&) is not aspherical, nevertheless we are able to prove the following: THEOREM 1. Any two finite connected 2-dimensional CW-complexes , each with a single 2-cell, have the same homotopy type if their fundamental groups are isomorphic. In fact there is a homotopy equivalence inducing any isomorphism of the fundamental groups. Our proof makes use of Lyndon's resolution for one-relator groups [10] and some combinatorial results on one-relator groups which can be found in the book by Magnus, Karass, and Solitar [11]. Theorem 1 has these corollaries: COROLLARY 1. Let X and Y be two finite connected 2-dimensional CW-complexes, each with a single 2-cell. Then X ~ Y if X\f L ~Y y M where L and M are finite CW-complexes with isomorphic fundamental groups. Thus X ~Y if and only if X V L ~ Y V L where L is any finite CW-complex. Proof. We have πyX^πJjf^π^Y^πJd. Because all groups involved are finite generated, we can write these as free product of 191