558 research outputs found
Embedded contact homology and open book decompositions
This is the first of a series of papers devoted to proving the equivalence of
Heegaard Floer homology and embedded contact homology (abbreviated ECH). In
this paper we prove that, given a closed, oriented, contact -manifold, there
is an equivalence between ECH of the closed -manifold and a version of ECH,
defined on the complement of the binding of an adapted open book decomposition.Comment: First revision: A mistake in our use of Morse-Bott techniques in ECH
has been fixed and results on sutured ECH have been added. Second revision:
bibliography updated. Third revision: exposition improved and details added
following referees' suggestions. Fourth revision: we added an appendix
sketching the proof of the Morse-Bott gluing statements used in the articl
A survey of Heegaard Floer homology
This work has two goals. The first is to provide a conceptual introduction to
Heegaard Floer homology, the second is to survey the current state of the
field, without aiming for completeness. After reviewing the structure of
Heegaard Floer homology, we list some of its most important applications. Many
of these are purely topological results, not referring to Heegaard Floer
homology itself. Then, we briefly outline the construction of Lagrangian
intersection Floer homology. We construct the Heegaard Floer chain complex as a
special case of the above, and try to motivate the role of the various
seemingly ad hoc features such as admissibility, the choice of basepoint, and
Spin^c-structures. We also discuss the proof of invariance of the homology up
to isomorphism under all the choices made, and how to define Heegaard Floer
homology using this in a functorial way (naturality). Next, we explain why
Heegaard Floer homology is computable, and how it lends itself to the various
combinatorial descriptions. The last chapter gives an overview of the
definition and applications of sutured Floer homology, which includes sketches
of some of the key proofs. Throughout, we have tried to collect some of the
important open conjectures in the area. For example, a positive answer to two
of these would give a new proof of the Poincar\'e conjecture.Comment: 38 pages, 1 figure, a few minor correction
Optimal topological simplification of discrete functions on surfaces
We solve the problem of minimizing the number of critical points among all
functions on a surface within a prescribed distance {\delta} from a given input
function. The result is achieved by establishing a connection between discrete
Morse theory and persistent homology. Our method completely removes homological
noise with persistence less than 2{\delta}, constructively proving the
tightness of a lower bound on the number of critical points given by the
stability theorem of persistent homology in dimension two for any input
function. We also show that an optimal solution can be computed in linear time
after persistence pairs have been computed.Comment: 27 pages, 8 figure
Convex cocompactness and stability in mapping class groups
We introduce a strong notion of quasiconvexity in finitely generated groups,
which we call stability. Stability agrees with quasiconvexity in hyperbolic
groups and is preserved under quasi-isometry for finitely generated groups. We
show that the stable subgroups of mapping class groups are precisely the convex
cocompact subgroups. This generalizes a well-known result of Behrstock and is
related to questions asked by Farb-Mosher and Farb.Comment: 15 pages, 1 figur
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