1,705 research outputs found
Stable concordance of knots in 3-manifolds
Knots and links in 3-manifolds are studied by applying intersection
invariants to singular concordances. The resulting link invariants generalize
the Arf invariant, the mod 2 Sato-Levine invariants, and Milnor's triple
linking numbers. Besides fitting into a general theory of Whitney towers, these
invariants provide obstructions to the existence of a singular concordance
which can be homotoped to an embedding after stabilization by connected sums
with . Results include classifications of stably slice links in
orientable 3-manifolds, stable knot concordance in products of an orientable
surface with the circle, and stable link concordance for many links of
null-homotopic knots in orientable 3-manifolds.Comment: 59 pages, 28 figure
Concordance groups of links
We define a notion of concordance based on Euler characteristic, and show that it gives rise to a concordance group of links in the 3-sphere, which has the concordance group of knots as a direct summand with infinitely generated complement. We consider variants of this using oriented and nonoriented surfaces as well as smooth and locally flat embeddings
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
Whitney tower concordance of classical links
This paper computes Whitney tower filtrations of classical links. Whitney
towers consist of iterated stages of Whitney disks and allow a tree-valued
intersection theory, showing that the associated graded quotients of the
filtration are finitely generated abelian groups. Twisted Whitney towers are
studied and a new quadratic refinement of the intersection theory is
introduced, measuring Whitney disk framing obstructions. It is shown that the
filtrations are completely classified by Milnor invariants together with new
higher-order Sato-Levine and higher-order Arf invariants, which are
obstructions to framing a twisted Whitney tower in the 4-ball bounded by a link
in the 3-sphere. Applications include computation of the grope filtration, and
new geometric characterizations of Milnor's link invariants.Comment: Only change is the addition of this comment: This paper subsumes the
entire preprint "Geometric Filtrations of Classical Link Concordance"
(arXiv:1101.3477v2 [math.GT]) and the first six sections of the preprint
"Universal Quadratic Forms and Untwisting Whitney Towers" (arXiv:1101.3480v2
[math.GT]
Intersection forms of toric hyperkaehler varieties
This note proves combinatorially that the intersection pairing on the middle
dimensional compactly supported cohomology of a smooth toric hyperkaehler
variety is always definite, providing a large number of non-trivial L^2
harmonic forms for toric hyperkaehler metrics on these varieties. This is
motivated by a result of Hitchin about the definiteness of the pairing of L^2
harmonic forms on complete hyperkaehler manifolds of linear growth.Comment: Latex, 7 page
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