Criteria for homotopic maps to be so along monotone homotopies

The state spaces of machines admit the structure of time. A homotopy theory respecting this additional structure can detect machine behavior unseen by classical homotopy theory. In an attempt to bootstrap classical tools into the world of abstract spacetime, we identify criteria for classically homotopic, monotone maps of pospaces to future homotope, or homotope along homotopies monotone in both coordinates, to a common map. We show that consequently, a hypercontinuous lattice equipped with its Lawson topology is future contractible, or contractible along a future homotopy, if its underlying space has connected CW type.Comment: 7 pages, 5 figures, partially presented at GETCO 2006. title change; strengthened Cor. 3.3. -> Prop. 3.7, Prop. 3.2 -> Lem. 3.2; corrected def of category of continuous lattices in sec. 2; added 5 figures, 8 eg's, Def. 3.4, Lemmas 2.8, 3.5, refs [1],[4],[5]; rewording throughout; conclusion and abstract rewritte

Symplectic homology and the Eilenberg-Steenrod axioms

We give a definition of symplectic homology for pairs of filled Liouville cobordisms, and show that it satisfies analogues of the Eilenberg-Steenrod axioms except for the dimension axiom. The resulting long exact sequence of a pair generalizes various earlier long exact sequences such as the handle attaching sequence, the Legendrian duality sequence, and the exact sequence relating symplectic homology and Rabinowitz Floer homology. New consequences of this framework include a Mayer-Vietoris exact sequence for symplectic homology, invariance of Rabinowitz Floer homology under subcritical handle attachment, and a new product on Rabinowitz Floer homology unifying the pair-of-pants product on symplectic homology with a secondary coproduct on positive symplectic homology. In the appendix, joint with Peter Albers, we discuss obstructions to the existence of certain Liouville cobordisms.Comment: v3: corrected Lemma 7.11. Various other minor modifications and reformatting. Final version to be published in Algebraic and Geometric Topolog

Lagrangian cobordisms and Lagrangian surgery

Lagrangian $k$-surgery modifies an immersed Lagrangian submanifold by topological $k$-surgery while removing a self-intersection. Associated to a $k$-surgery is a Lagrangian surgery trace cobordism. We prove that every Lagrangian cobordism is exactly homotopic to a concatenation of suspension cobordisms and Lagrangian surgery traces. This exact homotopy can be chosen with as small Hofer norm as desired. Furthermore, we show that each Lagrangian surgery trace bounds a holomorphic teardrop pairing the Morse cochain associated with the handle attachment to the Floer cochain generated by the self-intersection. We give a sample computation for how these decompositions can be used to algorithmically construct bounding cochains for Lagrangian submanifolds. In an appendix, we describe a 2-ended embedded monotone Lagrangian cobordism which is not the suspension of a Hamiltonian isotopy following a suggestion of Abouzaid and Auroux.Comment: 59 pages, 32 figures. Version 3: Version accepted to Commentarii Mathematici Helvetici. Incorporated helpful suggestions from refere

Unobstructed Lagrangian cobordism groups of surfaces

We study Lagrangian cobordism groups of closed symplectic surfaces of genus $g \geq 2$ whose relations are given by unobstructed, immersed Lagrangian cobordisms. Building upon work of Abouzaid and Perrier, we compute these cobordism groups and show that they are isomorphic to the Grothendieck group of the derived Fukaya category of the surface.Comment: 60 pages, 15 figure

HOMFLYPT Skein Theory, String Topology and 2-Categories

We show that relations in Homflypt type skein theory of an oriented $3$-manifold $M$ are induced from a $2$-groupoid defined from the fundamental $2$-groupoid of a space of singular links in $M$. The module relations are defined by homomorphisms related to string topology. They appear from a representation of the groupoid into free modules on a set of model objects. The construction on the fundamental $2$-groupoid is defined by the singularity stratification and relates Vassiliev and skein theory. Several explicit properties are discussed, and some implications for skein modules are derived.Comment: 55 pages, 1 figur