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 $(X,L)$, of a Weinstein manifold $X$ with an exact Lagrangian submanifold $L$, with ideal contact boundary $(Y,\Lambda)$, where $Y$ is a contact manifold and $\Lambda\subset Y$ is a Legendrian submanifold. We introduce the Chekanov-Eliashberg DG-algebra, $CE^{\ast}(\Lambda)$, with coefficients in chains of the based loop space of $\Lambda$ and study its relation to the Floer cohomology $CF^{\ast}(L)$ of $L$. Using the augmentation induced by $L$, $CE^{\ast}(\Lambda)$ can be expressed as the Adams cobar construction $\Omega$ applied to a Legendrian coalgebra, $LC_{\ast}(\Lambda)$. We define a twisting cochain:$$\mathfrak{t} \colon LC_{\ast}(\Lambda) \to \mathrm{B} (CF^*(L))^\#$$via holomorphic curve counts, where $\mathrm{B}$ 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 $CE^*(\Lambda)$ and $CF^{\ast}(L)$ are Koszul dual. In particular, $\mathfrak{t}$ induces a quasi-isomorphism between $CE^*(\Lambda)$ and the cobar of the Floer homology of $L$, $\Omega CF_*(L)$. We use the duality result to show that under certain connectivity and locally finiteness assumptions, $CE^*(\Lambda)$ is quasi-isomorphic to $C_{-*}(\Omega L)$ for any Lagrangian filling $L$ of $\Lambda$. Our constructions have interpretations in terms of wrapped Floer cohomology after versions of Lagrangian handle attachments. In particular, we outline a proof that $CE^{\ast}(\Lambda)$ is quasi-isomorphic to the wrapped Floer cohomology of a fiber disk $C$ in the Weinstein domain obtained by attaching $T^{\ast}(\Lambda\times[0,\infty))$ to $X$ along $\Lambda$ (or, in the terminology of arXiv:1604.02540 the wrapped Floer cohomology of $C$ in $X$ with wrapping stopped by $\Lambda$). 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