  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.
