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THEOREM 1. Let K be C1 finite 2-dimensioncll C W-complex with 7r1 ( K ) finite clnd clbelicln. Then every element of the Whitehecld group of K is recllizclble C1S the torsion of cl self-homotopy equiscllence on K. 1 HEOREM 2. Homotopy equiscllence clnd simple homotopy equiscllence clre the sclme for finite 2-dimensioncll CW-complexes with finite clbelicm fundclmentcll groups. O. Introduction. It is known that there exist finite n-complexes for all n > 2 which are homotopy equivalent but not simply homotopy equivalent. In this paper, we show that in dimension 2 homotopy type and simple homotopy type are the same when the fundamental group of the finite 2-dimensional complexes is finite and abelian. The technique used is to show that all the elements of the Whitehead group of a complex K are realizable as torsions of self-equivalences on K. 1. An important example. Dyer and Sieradski [DS] showed in 1973 that two 2-dimensional CW-complexes whose fundamental group was Zn were homotopy equivalent if and only if they were simple homotopy equivalent. The next case to consider would be Zn X Zm We want to show that any 2-dimensional complex with fundamental group Zn X Zm realizes all of its Whitehead group as torsions of self-equivalences. (t(K) denotes the group of self-homotopy equivalences of K.) THEOREM 1.1. Let K be the standard 2-complex of the presentation P= { a, b | a n, bm, [a, b] }. Then every element of Wh(K ) is realizable as ( f ) for some T E ((K) To prove the above theorem, we need the following three lemmas. LEMMA 1 .2. Suppose G is a group with generating set { ai g i E I } for some index set I. Let A: Z(G) > Z be the augmentation map (i.e., the ring homomorphism taking g to 1 for each g E G). Then each element @ E kerA is of the form @-, f i ( ai-1) for some elements f 1 E Z( G ) . PROOF. See [F, p. 549]. Received by the editors June 1, 1984 and, in revised form, February 18, 1985. Presented to the Society in Louisville, Kentucky, on January 28, 1984. 1980 Mclthemcltics Subject Clclssificcltion. Primary 57M20, 57Q10. t1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page

Simple homotopy type of finite 2-complexes with finite abelian fundamental group