Kawauchi defined a group structure on the set of homology S1×S2's under an equivalence relation called H˜-cobordism. This group receives a homomorphism from the knot concordance group, given by the ...operation of zero-surgery. We apply knot concordance invariants derived from knot Floer homology to study the kernel of the zero-surgery homomorphism. As a consequence, we show that the kernel contains a Z∞-subgroup generated by topologically slice knots in the smooth category.
It has been a central open problem in Heegaard Floer theory whether cobordisms of links induce homomorphisms on the associated link Floer homology groups. We provide an affirmative answer by ...introducing a natural notion of cobordism between sutured manifolds, and showing that such a cobordism induces a map on sutured Floer homology. This map is a common generalization of the hat version of the closed 3-manifold cobordism map in Heegaard Floer theory, and the contact gluing map defined by Honda, Kazez, and Matić. We show that sutured Floer homology, together with the above cobordism maps, forms a type of TQFT in the sense of Atiyah. Applied to the sutured manifold cobordism complementary to a decorated link cobordism, our theory gives rise to the desired map on link Floer homology. Hence, link Floer homology is a categorification of the multi-variable Alexander polynomial. We outline an alternative definition of the contact gluing map using only the contact element and handle maps. Finally, we show that a Weinstein sutured manifold cobordism preserves the contact element.
Smooth subvarieties of Jacobians Benoist, Olivier; Debarre, Olivier
Épijournal de géométrie algébrique,
06/2023, Letnik:
Special volume in honour of...
Journal Article
Recenzirano
Odprti dostop
We give new examples of algebraic integral cohomology classes on smooth
projective complex varieties that are not integral linear combinations of
classes of smooth subvarieties. Some of our examples ...have dimension 6, the
lowest possible. The classes that we consider are minimal cohomology classes on
Jacobians of very general curves. Our main tool is complex cobordism.
We construct a geometric cycle model for a Hodge filtered extension of complex cobordism for every smooth manifold with a filtration on its de Rham complex with complex coefficients. Using a ...refinement of the Pontryagin–Thom construction, we construct an explicit isomorphism between our geometric model and the abstract model of Hodge filtered complex cobordism of Hopkins–Quick for every complex manifold with the Hodge filtration.
Ribbon homology cobordisms Daemi, Aliakbar; Lidman, Tye; Vela-Vick, David Shea ...
Advances in mathematics (New York. 1965),
10/2022, Letnik:
408
Journal Article
Recenzirano
Odprti dostop
We study 4-dimensional homology cobordisms without 3-handles, showing that they interact nicely with Thurston geometries, character varieties, and instanton and Heegaard Floer homologies. Using ...these, we derive obstructions to such cobordisms. As one example of these obstructions, we generalize other recent results on the behavior of knot Floer homology under ribbon concordances. Finally, we provide topological applications, including to Dehn surgery problems.
Let X be an algebraic variety over k such that X = X ⊗k k is cellular. We study torsion elements in the Chow ring CH∗(X) which correspond to viy in the algebraic cobordism Ω∗( X ) where 0 ≠y ∈CH∗( X ...)/p and vi is the generator of BP∗with |vi| = −2(pi − 1). In particular, we try to compute CH∗(X) from Ω∗( X ) when X are twisted complete flag varieties.
Legendrian knots and exact Lagrangian cobordisms Ekholm, Tobias; Honda, Ko; Kálmán, Tamás
Journal of the European Mathematical Society : JEMS,
01/2016, Letnik:
18, Številka:
11
Journal Article
Recenzirano
Odprti dostop
We introduce constructions of exact Lagrangian cobordisms with cylindrical Legendrian ends and study their invariants which arise from Symplectic Field Theory. A pair $(X,L)$ consisting of an exact ...symplectic manifold $X$ and an exact Lagrangian cobordism $L\subset X$ which agrees with cylinders over Legendrian links $\Lambda_+$ and $\Lambda_-$ at the positive and negative ends induces a differential graded algebra (DGA) map from the Legendrian contact homology DGA of $\Lambda_+$ to that of $\Lambda_-$. We give a gradient flow tree description of the DGA maps for certain pairs $(X,L)$, which in turn yields a purely combinatorial description of the cobordism map for elementary cobordisms, i.e., cobordisms that correspond to certain local modifications of Legendrian knots. As an application, we find exact Lagrangian surfaces that fill a fixed Legendrian link and are not isotopic through exact Lagrangian surfaces.
We study homological mirror symmetry for toric varieties, exploring the relationship between various Fukaya-Seidel categories which have been employed for constructing the mirror to a toric variety. ...In particular, we realize tropical Lagrangian sections as objects of a partially wrapped category and construct a Lagrangian correspondence mirror to the inclusion of a toric divisor. As a corollary, we prove that tropical sections generate the Fukaya-Seidel category, completing a Floer-theoretic proof of homological mirror symmetry for projective toric varieties. In the course of the proof, we develop techniques for constructing Lagrangian cobordisms and Lagrangian correspondences in Liouville domains, which may be of independent interest.
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