The degrees of maps between $(n-1)$-connected $(2n+1)$-dimensional manifolds or Poincaré complexes and their applications

In this paper, using homotopy theoretical methods we study the degrees of maps between $(n-1)$-connected $(2n+1)$-dimensional Poincaré complexes. Necessary and sufficient algebraic conditions for the existence of mapping degrees between such Poincaré complexes are established. These conditions allow us, up to homotopy, to construct explicitly all maps with a given degree.
As an application of mapping degrees, we consider maps between ${(n-1)}$-connected $(2n+1)$-dimensional Poincaré complexes with degree $\pm 1$, and give a sufficient condition for these to be homotopy equivalences. This resolves a homotopy theoretical analogue of Novikov's question: when is a map of degree $1$ between manifolds a homeomorphism? For low $n$, we classify, up to homotopy, torsion free $(n-1)$-connected $(2n+1)$-dimensional Poincaré complexes.
Bibliography: 29 titles.