Topology Seminar: Finiteness conditions for diagrams of spaces with applications to algebraic spheres

In the 1960s, Wall developed a theory of finiteness obstructions for CW-complexes. We extend this result to diagrams of spaces and use it to investigate which homotopy G-spheres can be realized on a finite complex. We will conclude with some new examples of finite (non-linear, non-free) G spheres for some small groups G.