  In this paper we address the classification problem for locally compact
(n-1)-connected CW-complexes with dimension less or equal than n+2 up to proper
homotopy type. We obtain complete classification theorems in terms of purely
algebraic data in those cases where the representation type of the involved
algebra is finite. For this we define new quadratic functors in controlled
algebra and new homotopy and cohomology invariants in proper homotopy theory.
