104 research outputs found
Traces in monoidal categories
The main result of this paper is the construction of a trace and a trace
pairing for endomorphisms satisfying suitable conditions in a monoidal
category. This construction is a common generalization of the trace for
endomorphisms of dualizable ob jects in a balanced monoidal category and the
trace of nuclear operators on a locally convex topological vector space with
the approximation property
Pulling Apart 2-spheres in 4-manifolds
An obstruction theory for representing homotopy classes of surfaces in
4-manifolds by immersions with pairwise disjoint images is developed, using the
theory of non-repeating Whitney towers. The accompanying higher-order
intersection invariants provide a geometric generalization of Milnor's
link-homotopy invariants, and can give the complete obstruction to pulling
apart 2-spheres in certain families of 4-manifolds. It is also shown that in an
arbitrary simply connected 4-manifold any number of parallel copies of an
immersed surface with vanishing self-intersection number can be pulled apart,
and that this is not always possible in the non-simply connected setting. The
order 1 intersection invariant is shown to be the complete obstruction to
pulling apart 2-spheres in any 4-manifold after taking connected sums with
finitely many copies of S^2\times S^2; and the order 2 intersection
indeterminacies for quadruples of immersed 2-spheres in a simply connected
4-manifold are shown to lead to interesting number theoretic questions.Comment: Revised to conform with the published version in Documenta
Mathematic
All two dimensional links are null homotopic
We show that any number of disjointly embedded 2-spheres in 4-space can be
pulled apart by a link homotopy, ie, by a motion in which the 2-spheres stay
disjoint but are allowed to self-intersect.Comment: 18 pages. Published copy, also available at
http://www.maths.warwick.ac.uk/gt/GTVol3/paper10.abs.htm
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
Grope cobordism of classical knots
We explain the notion of a grope cobordism between two knots in a 3-manifold.
Each grope cobordism has a type that can be described by a rooted unitrivalent
tree. By filtering these trees in different ways, we show how the
Goussarov-Habiro approach to finite type invariants of knots is closely related
to our notion of grope cobordism. Thus our results can be viewed as a geometric
interpretation of finite type invariants.
An interesting refinement we study are knots modulo symmetric grope cobordism
in 3-space. On one hand this theory maps onto the usual Vassiliev theory and on
the other hand it maps onto the Cochran-Orr-Teichner filtration of the knot
concordance group, via symmetric grope cobordism in 4-space. In particular, the
graded theory contains information on finite type invariants (with degree h
terms mapping to Vassiliev degree 2^h), Blanchfield forms or S-equivalence at
h=2, Casson-Gordon invariants at h=3, and for h=4 one has the new von Neumann
signatures of a knot.Comment: Final version. To appear in Topology. See
http://www.math.cornell.edu/~jconant/pagethree.html for a PDF file with
better figure qualit
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