Cobordism of disk knots

We study cobordisms and cobordisms rel boundary of PL locally-flat disk knots D ⁿ⁻² ↪ D ⁿ . Any two disk knots are cobordant if the cobordisms are not required to fix the boundary sphere knots, and any two even-dimensional disk knots with isotopic boundary knots are cobordant rel boundary. However, the cobordism rel boundary theory of odd-dimensional disk knots is more subtle. Generalizing results of J. Levine on the cobordism of sphere knots, we define disk knot Seifert matrices and show that two higher-dimensional disk knots with isotopic boundaries are cobordant rel boundary if and only if their disk knot Seifert matrices are algebraically cobordant. We also ask which algebraic cobordism classes can be realized given a fixed boundary knot and provide a complete classification when the boundary knot has no 2-torsion in its middle-dimensional Alexander module. In the course of this classification, we establish a close connection between the Blanchfield pairing of a disk knot and the Farber-Levine torsion pairing of its boundary knot (in fact, for disk knots satisfying certain connectivity assumptions, the disk knot Blanchfield pairing will determine the boundary Farber-Levine pairing). In addition, we study the dependence of disk knot Seifert matrices on choices of Seifert surface, demonstrating that all such Seifert matrices are rationally S-equivalent, but not necessarily integrally S-equivalent.     

Cobordism of algebraic knots via Seifert forms

Summary We prove that, for anyn strictly greater than 2, there exist nonisotopic algebraic spherical knots of dimension 2n–1 which are cobordant. We first consider plane curve singularities. In that case we determine the Witt-class of the associated rational Seifert form and we attach to such a singularity a finite abelian group which is an invariant of the integral monodromy. This allows us to gather information about cobordism and isotopy classes of the higher dimensional algebraic knots obtained after suspension, by means of the dictionary relating knots and Seifert forms.A recent paper of Szczepanski [SZ] seemed to give partial results about the cobordism of algebraic knots. However, we shall show that these results cannot be true.Oblatum 28-VIII-1991 & 15-V-1992     

Grope cobordism of classical knots

Motivated by the lower central series of a group, we define the notion of a grope cobordism between two knots in a 3-manifold. Just like an iterated group commutator, 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.The derived commutator series of a group also has a three-dimensional analogy, namely knots modulo symmetric grope cobordism. 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 finds the new von Neumann signatures of a knot.     

Homology cobordism and classical knot invariants

In this paper, we define and investigate -homology cobordism invariants of -homology 3-spheres which turn out to be related to classical invariants of knots. As an application, we show that many lens spaces have infinite order in the -homology cobordism group and we prove a lower bound for the slice genus of a knot on which integral surgery yields a given -homology sphere. We also give some new examples of 3-manifolds which cannot be obtained by integral surgery on a knot. Received: May 7, 2001     

L2‐signatures,homology localization,and amenable groups

Aimed at geometric applications, we prove the homology cobordism invariance of the L²‐Betti numbers and L²‐signature defects associated to the class of amenable groups lying in Strebel's class D(R), which includes some interesting infinite/finite non‐torsion‐free groups. This result includes the only prior known condition, that Γ is a poly‐torsion‐free abelian group (or a finite p‐group). We define a new commutator series that refines Harvey's torsion‐free derived series of groups, using the localizations of groups and rings of Bousfield, Vogel, and Cohn. The series, called the local derived series, has versions for homology with arbitrary coefficients and satisfies functoriality and an injectivity theorem. We combine these two new tools to give some applications to distinct homology cobordism types within the same simple homotopy type in higher dimensions, to concordance of knots in three manifolds, and to spherical space forms in dimension 3. © 2012 Wiley Periodicals, Inc.     

Abelian quotients of monoids of homology cylinders

A homology cylinder over a surface consists of a homology cobordism between two copies of the surface and markings of its boundary. The set of isomorphism classes of homology cylinders over a fixed surface has a natural monoid structure and it is known that this monoid can be seen as an enlargement of the mapping class group of the surface. We now focus on abelian quotients of this monoid. We show that both the monoid of all homology cylinders and that of irreducible homology cylinders are not finitely generated and moreover they have big abelian quotients. These properties contrast with the fact that the mapping class group is perfect in general. The proof is given by applying sutured Floer homology theory to homologically fibered knots studied in a previous paper.     

Equivariant cobordism of flag varieties and of symmetric varieties

We obtain an explicit presentation for the equivariant cobordism ring of a complete flag variety. An immediate corollary is a Borel presentation for the ordinary cobordism ring. Another application is an equivariant Schubert calculus in cobordism. We also describe the rational equivariant cobordism rings of wonderful symmetric varieties of minimal rank.     

Algebraic cobordism revisited

We define a cobordism theory in algebraic geometry based on normal crossing degenerations with double point singularities. The main result is the equivalence of double point cobordism to the theory of algebraic cobordism previously defined by Levine and Morel. Double point cobordism provides a simple, geometric presentation of algebraic cobordism theory. As a corollary, the Lazard ring given by products of projective spaces rationally generates all nonsingular projective varieties modulo double point degenerations. Double point degenerations arise naturally in relative Donaldson–Thomas theory. We use double point cobordism to prove all the degree 0 conjectures in Donaldson–Thomas theory: absolute, relative, and equivariant.     

Cobordism for Hamiltonian loop group actions and flat connections on the punctured two-sphere

We develop a theory of symplectic cobordism and a Duistermaat-Heckman principle for Hamiltonian loop group actions. As an application, we construct a symplectic cobordism between moduli spaces of flat connections on the three holed sphere and disjoint unions of toric varieties. This cobordism yields formulas for the mixed Pontrjagin numbers of the moduli spaces, equivalent to Witten's formulas in the case of symplectic volumes. Received June 15, 1998     

Some Remarks on Real and Algebraic Cobordism

We investigate connections between Real cobordism, algebraic cobordism, quadratic forms, the Rost Motive, Morava K(n)-theories and analogues of homotopy classes of Hopf invariant 1.     

On Fredholm index, transversal approximations and Quillen's geometric complex cobordism of Hilbert manifolds with some applications to flag varieties of loop groups

In [Contemp. Math. 258 (2000) 1-19], by using Fredholm index we developed a version of Quillen's geometric cobordism theory for infinite dimensional Hilbert manifolds. This cobordism theory has a graded group structure under topological union operation and has push-forward maps for complex orientable Fredholm maps. In this work, by using Quinn's Transversality Theorem [Proc. Sympos. Pure. Math. 15 (1970) 213-222], it will be shown that this cobordism theory has a graded ring structure under transversal intersection operation and has pull-back maps for smooth maps. It will be shown that the Thom isomorphism in this theory will be satisfied for finite dimensional vector bundles over separable Hilbert manifolds and the projection formula for Gysin maps will be proved. After we discuss the relation between this theory and classical cobordism, we describe some applications to the complex cobordism of flag varieties of loop groups and we do some calculations.     

单形和3维立方体乘积上小覆盖的等变协边类的个数

本文研究了小覆盖的等变协边分类. 利用示性函数和Stong同态确定了单形和3维立方体乘积上小覆盖的等变协边类的个数, 推广了现有文献中的相关结果.     

A theory of cobordism for non-spherical links

We define an equivalence relation, called algebraic cobordism, on the set of bilinear forms over the integers. When , we prove that two 2n - 1 dimensional, simple fibered links are cobordant if and only if they have algebraically cobordant Seifert forms. As an algebraic link is a simple fibered link, our criterion for cobordism allows us to study isolated singularities of complex hypersurfaces up to cobordism. Received: August 24, 1995     

An invariant of tangle cobordisms

We construct a new invariant of tangle cobordisms. The invariant of a tangle is a complex of bimodules over certain rings, well-defined up to chain homotopy equivalence. The invariant of a tangle cobordism is a homomorphism between complexes of bimodules assigned to boundaries of the cobordism.

     

具有非常余维数不动点集的可换对合

给定一组正整数l1相似文献   

棱柱上的小覆盖的等变配边分类

多面体上的小覆盖的等变配边类是由它的切表示集所决定的.本文通过将棱柱上的小覆盖的切表示集约化到一种素形式,来确定其等变配边分类.     

Manifolds of positive scalar curvature and conformal cobordism theory

For a supergoup , we study closed -manifolds with positive conformal classes. We use the relative Yamabe invariant from [2] to define the conformal cobordism relation on the category of such manifolds. We prove that the corresponding conformal cobordism groups are isomorphic to the cobordism groups defined by Stolz in [19]. As a corollary, we show that the conformal concordance relation on positive conformal classes coincides with the standard concordance relation on positive scalar curvature metrics. Our main technical tools come from analysis and conformal geometry. Received: 22 August 2000 / Published online: 5 September 2002     

单形和3维立方体乘积上小覆盖的等变协边类的个数

本文研究了小覆盖的等变协边分类.利用示性函数和Stong同态确定了单形和3维立方体乘积上小覆盖的等变协边类的个数,推广了现有文献中的相关结果.     

具有常余维数不动点集的可换对合

设（Z2）k广作用于光滑闭流形Mn，其不动点集具有常维数n-r，J是具有上述性质的未定向的n维上协边类[Mn]构成的集合.本文通过构造上协边环MO*的一组生成元决定了J的群结构