12,953 research outputs found
Combinatorial Heegaard Floer homology and nice Heegaard diagrams
We consider a stabilized version of hat Heegaard Floer homology of a
3-manifold Y (i.e. the U=0 variant of Heegaard Floer homology for closed
3-manifolds). We give a combinatorial algorithm for constructing this
invariant, starting from a Heegaard decomposition for Y, and give a
combinatorial proof of its invariance properties
Contact surgeries and the transverse invariant in knot Floer homology
We study naturality properties of the transverse invariant in knot Floer
homology under contact (+1)-surgery. This can be used as a calculational tool
for the transverse invariant. As a consequence, we show that the
Eliashberg-Chekanov twist knots E_n are not transversely simple for n odd and
n>3.Comment: Corrected naturality discussion
Universal manifold pairings and positivity
Gluing two manifolds M_1 and M_2 with a common boundary S yields a closed
manifold M. Extending to formal linear combinations x=Sum_i(a_i M_i) yields a
sesquilinear pairing p= with values in (formal linear combinations of)
closed manifolds. Topological quantum field theory (TQFT) represents this
universal pairing p onto a finite dimensional quotient pairing q with values in
C which in physically motivated cases is positive definite. To see if such a
"unitary" TQFT can potentially detect any nontrivial x, we ask if is
non-zero whenever x is non-zero. If this is the case, we call the pairing p
positive. The question arises for each dimension d=0,1,2,.... We find p(d)
positive for d=0,1, and 2 and not positive for d=4. We conjecture that p(3) is
also positive. Similar questions may be phrased for (manifold, submanifold)
pairs and manifolds with other additional structure. The results in dimension 4
imply that unitary TQFTs cannot distinguish homotopy equivalent simply
connected 4-manifolds, nor can they distinguish smoothly s-cobordant
4-manifolds. This may illuminate the difficulties that have been met by several
authors in their attempts to formulate unitary TQFTs for d=3+1. There is a
further physical implication of this paper. Whereas 3-dimensional Chern-Simons
theory appears to be well-encoded within 2-dimensional quantum physics, eg in
the fractional quantum Hall effect, Donaldson-Seiberg-Witten theory cannot be
captured by a 3-dimensional quantum system. The positivity of the physical
Hilbert spaces means they cannot see null vectors of the universal pairing;
such vectors must map to zero.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol9/paper53.abs.htm
Floer homology and splicing knot complements
We obtain a formula for the Heegaard Floer homology (hat theory) of the
three-manifold obtained by splicing the complements of the knots
, , in terms of the knot Floer homology of and
. We also present a few applications. If denotes the rank of the
Heegaard Floer group for the knot obtained by
-surgery over we show that the rank of
is bounded below by
We also show that if splicing the complement of a knot with the
trefoil complements gives a homology sphere -space then is trivial and
is a homology sphere -space.Comment: Some errors in version 2 of the paper are corrected, and the
exposition is slightly improve
Homotopy versus isotopy: spheres with duals in 4-manifolds
David Gabai recently proved a smooth 4-dimensional "Light Bulb Theorem" in
the absence of 2-torsion in the fundamental group. We extend his result to
4-manifolds with arbitrary fundamental group by showing that an invariant of
Mike Freedman and Frank Quinn gives the complete obstruction to "homotopy
implies isotopy" for embedded 2-spheres which have a common geometric dual. The
invariant takes values in an Z/2Z-vector space generated by elements of order 2
in the fundamental group and has applications to unknotting numbers and
pseudo-isotopy classes of self-diffeomorphisms. Our methods also give an
alternative approach to Gabai's theorem using various maneuvers with Whitney
disks and a fundamental isotopy between surgeries along dual circles in an
orientable surface.Comment: Included into section 2 of this version is a proof that the operation
of `sliding a Whitney disk over itself' preserves the isotopy class of the
resulting Whitney move in the current setting. Some expository clarifications
have also been added. Main results and proofs are unchanged from the previous
version. 39 pages, 25 figure
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