Sobolev mappings, degree, homotopy classes and rational homology spheres

TL;DR

This paper explores the degree theory and homotopy classification of Orlicz-Sobolev mappings between manifolds, revealing conditions under which the degree is well-defined related to the topology of the target manifold.

Contribution

It establishes a precise criterion linking the well-definedness of the degree in Orlicz-Sobolev spaces to the topological properties of the target manifold, especially regarding rational homology spheres and the 4-sphere.

Findings

Degree is well-defined iff the universal cover of N is not a rational homology sphere.

In dimension 4, degree is well-defined iff N is not homeomorphic to S^4.

Characterizes the relationship between Sobolev mappings and topological properties of manifolds.

Abstract

In the paper we investigate the degree and the homotopy theory of Orlicz-Sobolev mappings $W^{1,P}(M,N)$ between manifolds, where the Young function $P$ satisfies a divergence condition and forms a slightly larger space than $W^{1,n}$, $n=\dim M$. In particular, we prove that if $M$ and $N$ are compact oriented manifolds without boundary and $\dim M=\dim N=n$, then the degree is well defined in $W^{1,P}(M,N)$ if and only if the universal cover of $N$ is not a rational homology sphere, and in the case $n=4$, if and only if $N$ is not homeomorphic to $S^4$.