Title:

A periodic monogenic resolution

In this thesis we examine the R(2) − D(2) problem for CW complexes with fundamental group isomorphic to the very nonabelian general affine group J = GA(1, F5) of order 20. In particular, since J is finite of cohomological period 8, it admits a periodic free resolution of finitely generated modules. We devote our efforts towards showing the existence of a diagonal free resolution of Z over Λ = Z[J] of period 8 via a neat decomposition of the syzygies Ωn(Z) (classes of modules stably isomorphic to certain kernels), which occur in a decomposed form in the constituent infinite monogenic resolutions. A traditional strategy for constructing a diagonal resolution would entail considering extensions of all possible submodules after extensive trial and error techniques. However, we side step this method by considering a more intellectual line, namely the kernels K(i) of the generators �(i) : Λ → θi of the row submodules θi ⊂ T4(Z, 5). We describe these kernels as distinct extensions of indecomposable modules and prove that each K(i) is in fact congruent to a quotient module of Λ. Moreover, we also examine the monogenicity of certain K(i) and we show how they relate to the Ωn(Z) . A diagonal resolution would significantly simplify group cohomology calculations Hn Λ(J; Z) ∼= Extn Λ (Z, Z) with coefficients in Z. Moreover, detailed knowledge of the free resolution is an essential step towards solving the R(2)− D(2)problem positively for J. The D(2)problem asks if a threedimensional CW complex is homotopy equivalent to a twodimensional CW complex provided H3(X, ˜ Z) = H3(X; B) = 0 for all coefficient systems B. Johnson proved that this problem is equivalent to a purely algebraic problem he called the R(2)problem.
