Low dimensional algebraic complexes over integral group rings

The realization problem asks: When does an algebraic complex arise, up to homotopy, from a geometric complex In the case of 2- dimensional algebraic complexes, this is equiv alent to the D2 problem, which asks when homological methods can distinguish between 2 and 3 dimensional complexes. We approach the realization problem (and hence the D2 problem) by classifying all pos sible algebraic 2- complexes and showing that they are realized. We show that if a dihedral group has order 2n, then the algebraic complexes over it are parametrized by their second homology groups, which we refer to as algebraic second homotopy groups. A cancellation theorem of Swan ( 11 ), then allows us to solve the realization problem for the group D{dollar}. Let X be a finite geometric 2- complex. Standard isomorphisms give 7r2(Ar) = H2(X Z), as modules over ni(X). Schanuel's lemma may then be used to show that the stable class of n2(X) is determined by k {lcub}X). We show how 7r3(X) maybe calculated similarly. Specif ically, we show that as a module over the fundamental group, (X) = S2{lcub}ir2{lcub}X)), where S2(ir2(X)) denotes the symmetric part of the module 7r2(X) z tt2(X). As a consequence, we are able to show that when the order of n (X) is odd, the stable class of 7r3(X) is also determined by ir {lcub}X). Given a closed, connected, orientable 5- dimensional manifold, with finite fundamen tal group, we may represent it, up to homotopy equivalence, by an algebraic complex. Poincare duality induces a homotopy equivalence between this algebraic complex and its dual. We consider how similar this homotopy equivalence may be made to the identity, (through appropriate choice of algebraic complex). We show that it can be taken to be the identity on 4 of the 6 terms of the chain complex. However, by finding a homological ob struction, we show that in general the homotopy equivalence may not be written as the identity