TY - JOUR

T1 - The Poincaré-Hopf theorem for relative braid classes

AU - Munaò, Simone

AU - Van der Vorst, Robert

PY - 2017/12/30

Y1 - 2017/12/30

N2 - This paper studies the existence and multiplicity of closed integral curves of smooth vector fields on the closed 2-disc D2 in the complement of a suitable geometric braid y—called a skeleton. The strands in a skeleton y must be trajectories of the vector field and the 1-periodic orbits to be counted must also be representatives of a suitable relative braid class x in the complement of y. From Leray–Schauder degree theory a signed count of the closed trajectories representing x in the complement of y is independent of the vector field. It therefore suffices to compute, or estimate the signed count of periodic orbits for a particularly convenient vector field. We show that the signed count equals the Euler-Floer characteristic of Braid Floer homology, (cf. van den Berg et al. in J Differ Equ 259(5):1663–1721, 2015). The latter can be computed via a finite cube complex which serves as a model for the given braid class.

AB - This paper studies the existence and multiplicity of closed integral curves of smooth vector fields on the closed 2-disc D2 in the complement of a suitable geometric braid y—called a skeleton. The strands in a skeleton y must be trajectories of the vector field and the 1-periodic orbits to be counted must also be representatives of a suitable relative braid class x in the complement of y. From Leray–Schauder degree theory a signed count of the closed trajectories representing x in the complement of y is independent of the vector field. It therefore suffices to compute, or estimate the signed count of periodic orbits for a particularly convenient vector field. We show that the signed count equals the Euler-Floer characteristic of Braid Floer homology, (cf. van den Berg et al. in J Differ Equ 259(5):1663–1721, 2015). The latter can be computed via a finite cube complex which serves as a model for the given braid class.

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U2 - 10.1007/s00209-016-1841-4

DO - 10.1007/s00209-016-1841-4

M3 - Article

AN - SCOPUS:85007413616

VL - 287

SP - 679

EP - 703

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

IS - 1-2

ER -