93 research outputs found
Regular homotopy and total curvature
We consider properties of the total absolute geodesic curvature functional on
circle immersions into a Riemann surface. In particular, we study its behavior
under regular homotopies, its infima in regular homotopy classes, and the
homotopy types of spaces of its local minima.
We consider properties of the total curvature functional on the space of
2-sphere immersions into 3-space. We show that the infimum over all sphere
eversions of the maximum of the total curvature during an eversion is at most
8\pi and we establish a non-injectivity result for local minima.Comment: This is the version published by Algebraic & Geometric Topology on 23
March 2006. arXiv admin note: this version concatenates two articles
published in Algebraic & Geometric Topolog
Duality between Lagrangian and Legendrian invariants
Consider a pair , of a Weinstein manifold with an exact Lagrangian
submanifold , with ideal contact boundary , where is a
contact manifold and is a Legendrian submanifold. We
introduce the Chekanov-Eliashberg DG-algebra, , with
coefficients in chains of the based loop space of and study its
relation to the Floer cohomology of . Using the augmentation
induced by , can be expressed as the Adams cobar
construction applied to a Legendrian coalgebra, .
We define a twisting cochain:via holomorphic curve counts, where
denotes the bar construction and the graded linear dual. We show under
simply-connectedness assumptions that the corresponding Koszul complex is
acyclic which then implies that and are Koszul
dual. In particular, induces a quasi-isomorphism between
and the cobar of the Floer homology of , .
We use the duality result to show that under certain connectivity and locally
finiteness assumptions, is quasi-isomorphic to for any Lagrangian filling of . Our constructions have
interpretations in terms of wrapped Floer cohomology after versions of
Lagrangian handle attachments. In particular, we outline a proof that
is quasi-isomorphic to the wrapped Floer cohomology of a
fiber disk in the Weinstein domain obtained by attaching
to along (or, in the
terminology of arXiv:1604.02540 the wrapped Floer cohomology of in with
wrapping stopped by ). Along the way, we give a definition of wrapped
Floer cohomology without Hamiltonian perturbations.Comment: 126 pages, 20 figures. Substantial overall revision based on
referee's comments. The main results remain the same but the exposition has
been improve
- β¦