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     

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.     

Almost Alternating Diagrams and Fibered Links in S3

Let R be a Seifert surface obtained by applying Seifert's algorithmto a connected diagram D for a link L. In this paper, lettingD be almost alternating, we give a practical algorithm to determinewhether L is a fibered link and R is a fiber surface. We furthershow that L is a fibered link and R is a fiber surface for Lif and only if R is a Hopf plumbing, that is, a successive plumbingof a finite number of Hopf bands. It has been known for sometime that this is true if D is alternating, and we show thatit is not always true if D is 2-almost alternating. In the appendix,we partially answer C. Adams's open question concerning almostalternating diagrams. 2000 Mathematical Subject Classification:57M25.     

Cobordism of algebraic knots

Summary The isolated singularities of complex hypersurfaces are studied by considering the topology of the highly connected submanifolds of spheres determined by the singularity. By introducing the notion of the link of a perturbation of the singularity and using techniques of surgery theory, we are able to describe which invariants associated to a singularity can be used to determine the cobordism type of the singularity.It is shown that the cobordism type is determined by the set of weakly distinguished bases. This result is used to draw a distinction between the classical case of two variables and the higher dimensional problem. That is, we show that the result of Le which states that the cobordism and topological classifications of singularities coincide in the classical dimension does not hold for singularities of functions of more than three variables. Examples of topologically distinct but cobordant singularities are obtained using results of Ebeling.     

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 invariance of the family index

We give a K‐theory proof of the invariance under cobordism of the family index. We consider elliptic pseudodifferential families on a continuous fibre bundle with smooth fibres $M\hookrightarrow \mbox{$\cal M$}\rightarrow B$, and define a notion of cobordant families using K¹‐groups on fibrations with boundary. We show that the index of two such families is the same using properties of the push‐forward map in K‐theory to reduce it to families on $B\times \mathbb {R}^n$.     

Rokhlin's inequalities for real algebraic curves and¶an extremal property

Rokhlin's inequalities for real algebraic curves and an extremal property for one of the inequalities are derived via a link cobordism approach. Received: 18 February 1997 / Revised version: 23 September 1998     

Représentations du groupe des tresses généralisées dans des groupes de Lie

Let Γ be a Lie group.Then any automorphism of the free group of rank n induces a diffeomorphism of Γ ⁿ . We use this remark and a result of P. Vogel to construct linear representations of a certain automorphism group of the algebraic closure of the free group. This automorphism group is closely related to the string link cobordism group.

Received:     

Equivariant Cobordism of Torus Orbifolds

Torus orbifolds are topological generalizations of symplectic toric orbifolds.The authours give a construction of smooth orbifolds with torus actions whose boundary is a disjoint union of torus orbifolds using a toric topological method. As a result, they show that any orientable locally standard torus orbifold is equivariantly cobordant to some copies of orbifold complex projective spaces. They also discuss some further equivariant cobordism results including the cases when torus orbifolds are actually torus manifolds.     

Torsion algebraic cycles and étale cobordism

We prove that the classical integral cycle class map from algebraic cycles to étale cohomology factors through a quotient of ?-adic étale cobordism over an algebraically closed field of positive characteristic. This shows that there is a strong topological obstruction for cohomology classes to be algebraic and that examples of Atiyah, Hirzebruch and Totaro also work in positive characteristic.     

Oriented Cohomology Theories of Algebraic Varieties

This article contains proofs of the results announced by Panin and Smirnov (http://www. math.uiuc.edu/k-theory/0459/2000) in the part concerning general properties of oriented cohomology theories of algebraic varieties. It is constructed one-to-one correspondences between orientations, Chern structures and Thom structures on a given ring cohomology theory. The theory is illustrated by motivic cohomology, algebraic K-theory, algebraic cobordism theory and by other examples.     

Stable étale realization and étale cobordism

We show that there is a stable homotopy theory of profinite spaces and use it for two main applications. On the one hand we construct an étale topological realization of the stable A¹-homotopy theory of smooth schemes over a base field of arbitrary characteristic in analogy to the complex realization functor for fields of characteristic zero.On the other hand we get a natural setting for étale cohomology theories. In particular, we define and discuss an étale topological cobordism theory for schemes. It is equipped with an Atiyah-Hirzebruch spectral sequence starting from étale cohomology. Finally, we construct maps from algebraic to étale cobordism and discuss algebraic cobordism with finite coefficients over an algebraically closed field after inverting a Bott element.     

Rigidity for orientable functors

In the following paper we introduce the notion of orientable functor (orientable cohomology theory) on the category of projective smooth schemes and define a family of transfer maps. Applying this technique, we prove that with finite coefficients orientable cohomology of a projective variety is invariant with respect to the base-change given by an extension of algebraically closed fields. This statement generalizes the classical result of Suslin, concerning algebraic K-theory of algebraically closed fields. Besides K-theory, we treat such examples of orientable functors as etale cohomology, motivic cohomology, algebraic cobordism. We also demonstrate a method to endow algebraic cobordism with multiplicative structure and Chern classes.     

The Rost invariant has trivial kernel for quasi-split groups of low rank

For G a simple simply connected algebraic group defined over a field F, Rost has shown that there exists a canonical map . This includes the Arason invariant for quadratic forms and Rost's mod 3 invariant for exceptional Jordan algebras as special cases. We show that R _G has trivial kernel if G is quasi-split of type E ₆ or E ₇. A case-by-case analysis shows that it has trivial kernel whenever G is quasi-split of low rank. Received: November 1, 2000     

Cobordism Classes Fibering with Fiber RP（9）

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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.     

Equivariant pretheories and invariants of torsors

In the present paper we introduce and study the notion of an equivariant pretheory (basic examples are equivariant Chow groups of Edidin and Graham, Thomason??s equivariant K-theory and equivariant algebraic cobordism). Using the language of equivariant pretheories we generalize the theorem of Karpenko and Merkurjev on G-torsors and rational cycles. As an application, to every G-torsor E and a G-equivariant pretheory we associate a ring which serves as an invariant of E. In the case of Chow groups this ring encodes the information about the motivic J-invariant of E, in the case of Grothendieck??s K ₀ indexes of the respective Tits algebras and in the case of algebraic cobordism ?? it gives a quotient of the cobordism ring of G.     

Degenerate fibers of three-dimensional manifolds fibered into rational surfaces

It is proved that the components of degenerate fibers of three-dimensional algebraic manifolds fibered into rational surfaces are rational or irrational ruled suriaces. An example is constructed of a three-dimensional algebraic manifold, fibered into rational surfaces, whose degenerate fiber contains an irrational ruled surface which cannot be eliminated by birational transformations that do not alter the common fiber.Translated from Matematicheskie Zametki, Vol. 7, No. 2, pp. 191–202, February, 1970.In conclusion the author expresses his gratitude to his scientific adviser Yu. I Martin for his assistance.     

Base independent algebraic cobordism

The purpose of this article is to show that the bivariant algebraic A-cobordism groups considered previously by the author are independent of the chosen base ring A. This result is proven by analyzing the bivariant ideal generated by the so called snc relations, and, while the alternative characterization we obtain for this ideal is interesting by itself because of its simplicity, perhaps more importantly it allows us to easily extend the definition of bivariant algebraic cobordism to divisorial Noetherian derived schemes of finite Krull dimension. As an interesting corollary, we define the corresponding homology theory called algebraic bordism. We also generalize projective bundle formula, the theory of Chern classes, the Conner–Floyd theorem and the Grothendieck–Riemann–Roch theorem to this setting. The general definitions of bivariant cobordism are based on the careful study of ample line bundles and quasi-projective morphisms of Noetherian derived schemes, also undertaken in this work.