Homotopy Trees with Trivial Classifying Ring

In this note we study a certain class of groups w for which the homotopy classification of (rr, m)-complexes is independent of the fc-invariant for small m. 1. Homotopy trees. A (tt, w)-complex is a finite, connected w-dimensional CW-complex such that ttx(X) = m and ttj(X) = 0 for 1 < i < m. The homotopy tree HT(tt, m) is a directed tree whose vertices are homotopv classes of (tt, w)-complexes. A vertex[X] is connected by an edge to a vertex [Y] iff Y ss X V Sm. The problem is to describe HT(it,m) (see [3]). There is an algebraic analog to this problem in the theory of algebraic mtypes. The algebraic w-type H(X) of a (tt, w)-complex A" is a triple T1(X) = (TT,TTm(X),k(X)) where k(X) G Hm+x(Tr,Trm(X)) is the first A:-invariant of X (see [7], [3]). It is proved in [4], that if ttm = Ttm(X) is finitely generated as a 77-module and Hm+x(Tt;ZTt) = 0, then R(iT,m) = Hm+x(Tt,TTm) has the structure of a ring with identity such that the units U(tr,m) of Hm+l(Tr,Ttm) are the projective /V-invariants. Note that Hm+x(it,TTm(X)) s Hm+l(Tt,irm(Y)) for any two (tt, w)-complexes X and Y, by Schanuel's lemma. In §2 we isolate a large class of groups it having H'(tt; Ztt) = 0 for all but one value of /'. Furthermore, there is a homomorphism k: U(tt, m) -» K0Ztt where K0Ztt is the reduced projective class group of the integral group ring Ztt of tt. The kernel of k, denoted SF(tt, m), is precisely the set of units which arise as the kinvariants of (77-, m)-complexes. Definition. The ring R(tt, m) is called the classifying ring of the homotopy tree HT(Tr,m); k: U —> KqZtt, the classifying homomorphism. We say that the classifying ring of HT(tt, m) is trivial if it is isomorphic to the zero ring or the ring of integers Z. In the latter case k: U —> K0Ztt is the zero homomorphism. In order to state the results, we need to define an isomorphism of algebraic w-types. (9,B): U(X) —» T1(Y) is an isomorphism if 6: tt -» tt is a group automorphism, B: Trm(X) -> Trm(Y) is a ^-isomorphism of Tr-modules (B(x ■ y) = 9(x)B(y), for x G Tt,y G Trm(X) and B is bijective) and 6*"'/?*(k(X)) = k(Y) in the following diagram: Hm+X(TT,TTm(X)) JU Hm+X(TT,TTm(Y)9) +*■ /V"+1 (tt, TTm(Y )) where Trm(Y)e is the vr-module with action x*y = 9(x) ■ y (x G tr,y Received by the editors May 9, 1975. AMS (MOS) subject classifications (1970). Primary 55D15, 18H10. © American Mathematical Society 1976 405 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use