Non-zero degree maps between $2n$-manifolds

TL;DR

This paper provides a computable criterion for when a homomorphism between cohomology groups of certain 2n-manifolds can be realized by a map of a specified degree, with special results for 4-manifolds.

Contribution

It introduces Thom-Pontrjagin-based conditions for realizing homomorphisms as maps of given degree between 2n-manifolds, especially characterizing degree 1 maps in 4-dimensions.

Findings

Necessary and sufficient conditions for degree k maps between 2n-manifolds.

Each (n-1)-connected 2n-manifold admits selfmaps of degree greater than 1.

Characterization of degree 1 maps between 4-manifolds via intersection forms.

Abstract

Thom-Pontrjagin constructions are used to give a computable necessary and sufficient condition when a homomorphism $\phi : H^n(L;Z)\to H^n(M;Z)$ can be realized by a map $f:M\to L$ of degree $k$ for closed $(n-1)$-connected $2n$-manifolds $M$ and $L$, $n>1$. A corollary is that each $(n-1)$-connected $2n$-manifold admits selfmaps of degree larger than 1, $n>1$. In the most interesting case of dimension 4, with the additional surgery arguments we give a necessary and sufficient condition for the existence of a degree $k$ map from a closed orientable 4-manifold $M$ to a closed simply connected 4-manifold $L$ in terms of their intersection forms, in particular there is a map $f:M\to L$ of degree 1 if and only if the intersection form of $L$ is isomorphic to a direct summand of that of $M$.