Homotopy trees for periodic groups

Abstract

0-+Z-+Z7T-+ Ch_2 —> Ch_3 —*•• •—^C^—»Z7r -^Z—>0 where each Ct is a finitely-generated free 7r-module. Acccording to [7], every finite periodic group of minimal period k has a minimal free period h = pk for some integer p > 0. A convenient listing of all finite periodic groups is given in [9]. DEFINITION. A (TT, m)-complex is a finite, connected m-dimensional CW complex with fundamental group TT whose universal cover is (m l)-connected. Let HT(TT, m) denote the set of homotopy types of (TT, ra)-complexes. This set may be described as a directed tree with one vertex for each homotopy class [X] of (TT, m)-complexes having the homotopy type of X the vertex [X] is connected by an edge to vertex [Y] provided Y has the homotopy type of the sum V S of with the m-sphere S. HT(n, m) is connected by [11, Theorem 14] and clearly contains no circuits. The purpose of this note is to announce a complete description of the homotopy tree HT(rr, m) for certain periodic 7r and for m = ik9ik- (/ > 0). Full details and a description for any periodic n will appear elsewhere. Before stating the theorem, we need two more pieces of notation. Let Z* be the units of the ring Zn of integers modulo n. Then Autfc 7r = {p G Z*13a G Aut TT B0L*k(l) = p where a*: #*(ir, Z) > H (ir, Z)}. Let K0Zn be the reduced projective class group of the integral group ring ZTT of TT. Define a homomorphism Zn —• K0ZTT by v(p) = class of the projective left ideal (p',N) of ZTT generated by any integer p ' G p and Af = XxEi1Tx. v is well defined by [7, Lemma 6.1].