The Poincar\'e-Hopf Theorem for relative braid classes

TL;DR

This paper extends braid Floer homology to derive a Poincaré-Hopf type theorem, linking the Euler-Floer characteristic to the existence of closed integral curves and periodic points for vector fields and diffeomorphisms on the 2-disc.

Contribution

It introduces a Poincaré-Hopf type theorem based on the Euler-Floer characteristic, applicable to proper relative braid classes on the 2-disc and computable via finite cube complexes.

Findings

Euler-Floer characteristic forces closed integral curves.

The method applies to arbitrary vector fields and diffeomorphisms.

Computable via finite cube complexes.

Abstract

Braid Floer homology is an invariant of proper relative braid classes. Closed integral curves of 1-periodic Hamiltonian vector fields on the 2-disc may be regarded as braids. If the Braid Floer homology of associated proper relative braid classes is non-trivial, then additional closed integral curves of the Hamiltonian equations are forced via a Morse type theory. In this article we show that certain information contained in the braid Floer homology - the Euler-Floer characteristic - also forces closed integral curves and periodic points of arbitrary vector fields and diffeomorphisms and leads to a Poincar\'e-Hopf type Theorem. The Euler-Floer characteristic for any proper relative braid class can be computed via a finite cube complex that serves as a model for the given braid class. The results in this paper are restricted to the 2-disc, but can be extend to two-dimensional surfaces (with or without boundary).