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

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).