Homotopy groups and twisted homology of arrangements

TL;DR

This paper explores the relationship between higher homotopy groups and twisted homology in arrangements, revealing a dichotomy that generalizes previous results and contrasts with classical homology mappings.

Contribution

It generalizes existing results on homotopy and homology of arrangements, highlighting a dichotomy between twisted and untwisted cases.

Findings

Higher homotopy groups map onto non-resonant homology in some arrangements.

The Hurewicz map to untwisted homology is always zero in degrees greater than one.

The work generalizes previous results and clarifies the dichotomy in homotopy-homology relationships.

Abstract

Recent work of M. Yoshinaga shows that in some instances certain higher homotopy groups of arrangements map onto non-resonant homology. This is in contrast to the usual Hurewicz map to untwisted homology, which is always the zero homomorphism in degree greater than one. In this work we examine this dichotomy, generalizing both results.