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

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Abstract

In this paper, using homotopy theoretical methods we study the degrees of maps between $(n-1)$-connected $(2n+1)$-dimensional Poincare complexes. Necessary and sufficient algebraic conditions for the existence of mapping degrees between such Poincare 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 Poincare 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 Poincare complexes. Bibliography: 29 titles.

About the authors

Jelena Grbić

University of Southampton

Email: J.Grbic@soton.ac.uk
PhD, Professor

Aleksandar Vučić

University of Belgrade, Faculty of Mathematics

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