Circle-valued Morse theory for complex hyperplane arrangements

TL;DR

This paper develops Morse theory for the complement of complex hyperplane arrangements, revealing its homotopy type and computing Novikov homology for various fundamental group homomorphisms.

Contribution

It introduces circle-valued Morse theory for hyperplane arrangement complements and characterizes their homotopy type and Novikov homology.

Findings

M has the homotopy type of a space fibered over a circle with attached n-dimensional cells.

Computed Novikov homology for a broad class of fundamental group homomorphisms.

Established a connection between Morse theory and the topology of hyperplane arrangement complements.

Abstract

Let A be an essential complex hyperplane arrangement in an n-dimensional complex vector space V. Let H denote the union of the hyperplanes, and M denote the complement to H in V. We develop the real-valued and circle-valued Morse theory for M and prove, in particular, that M has the homotopy type of a space obtained from a manifold fibered over a circle, by attaching cells of dimension n. We compute the Novikov homology of M for a large class of homomorphisms of the fundamental group of M to R.