Low dimensional algebraic complexes over integral group rings

TL;DR

This paper classifies algebraic 2-complexes over certain groups, solves the realization problem for these complexes, and explores homotopy equivalences of algebraic complexes related to 5-dimensional manifolds.

Contribution

It provides a classification of algebraic 2-complexes over dihedral groups and solves the realization problem for D8, advancing understanding of algebraic complexes over group rings.

Findings

Algebraic 2-complexes over dihedral groups are parametrized by their second homology groups.

The realization problem is solved for the group D8 using Swan's cancellation theorem.

Homotopy equivalences of algebraic complexes for 5-manifolds can be made to be the identity on most terms, with a homological obstruction preventing full identity.

Abstract

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 equivalent 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 possible 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 D8. Let X be a finite geometric 2- complex. Standard isomorphisms and Schanuel's lemma imply that the stable class of pi_2(X) is determined by pi_1(X). We show how pi_3(X) may be calculated similarly. Specifically,we show that as a module over the fundamental group, pi_3(X) is the symmetric part of the module pi_2(X) otimes pi_2(X). As a consequence, we are able to show that when the order of pi_1(X) is odd, the stable class of pi_3(X) is also determined by pi_1(X). Given a closed, connected, orientable 5- dimensional manifold, with finite fundamental 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 a homological obstruction prevents it from being the identity.