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

TL;DR

This paper investigates the degrees of maps between specific high-dimensional Poincaré complexes using homotopy theory, establishing algebraic conditions for map degrees and classifying complexes for low dimensions.

Contribution

It provides necessary and sufficient algebraic conditions for the existence of map degrees and classifies certain Poincaré complexes up to homotopy.

Findings

Established algebraic conditions for map degrees.

Calculated all map degrees between specific complexes.

Classified complexes up to homotopy for low dimensions.

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.