182 research outputs found
Contact homology of good toric contact manifolds
In this paper we show that any good toric contact manifold has well defined
cylindrical contact homology and describe how it can be combinatorially
computed from the associated moment cone. As an application we compute the
cylindrical contact homology of a particularly nice family of examples that
appear in the work of Gauntlett-Martelli-Sparks-Waldram on Sasaki-Einstein
metrics. We show in particular that these give rise to a new infinite family of
non-equivalent contact structures on in the unique homotopy
class of almost contact structures with vanishing first Chern class.Comment: 30 pages. Version 2: minor corrections, improved exposition and
expanded subsection 6.2 (see Remark 1.5). Version 3: minor corrections,
clarified assumptions in section 4, added references, to appear in Compositio
Mathematic
Cascades and perturbed Morse-Bott functions
Let be a Morse-Bott function on a finite
dimensional closed smooth manifold . Choosing an appropriate Riemannian
metric on and Morse-Smale functions on the
critical submanifolds , one can construct a Morse chain complex whose
boundary operator is defined by counting cascades \cite{FraTheA}. Similar data,
which also includes a parameter that scales the Morse-Smale
functions , can be used to define an explicit perturbation of the
Morse-Bott function to a Morse-Smale function \cite{AusMor} \cite{BanDyn}. In this paper we show that the
Morse-Smale-Witten chain complex of is the same as the Morse chain
complex defined using cascades for any sufficiently small. That
is, the two chain complexes have the same generators, and their boundary
operators are the same (up to a choice of sign). Thus, the Morse Homology
Theorem implies that the homology of the cascade chain complex of is isomorphic to the singular homology
.Comment: 34 pages, 2 figure
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