The degrees of maps between $(2n-1)$-Poincar\' e complexes

Abstract

In this paper, using exclusively homotopy theoretical methods, we study degrees of maps between $(n-2)$-connected $(2n-1)$-dimensional Poincar\' e complexes which have torsion free integral homology. Necessary and sufficient algebraic conditions for the existence of map degrees between such Poincar\' e complexes are established. We calculate the set of all map degrees between certain two $(n-2)$-connected $(2n-1)$-dimensional torsion free Poincar\'e complexes. For low $n$, using knowledge of possible degrees of self maps, we classify, up to homotopy, torsion free $(n-2)$-connected $(2n-1)$-dimensional Poincar\' e complexes.