Homotopy invariance of Novikov-Shubin invariants and Betti numbers

We give short proofs of the Gromov-Shubin theorem on the homotopy invariance of the Novikov-Shubin invariants and of the Dodziuk theorem on the homotopy invariance of the Betti numbers of the universal covering of a closed manifold in this paper. We show that the homotopy invariance of these invariants is no more difficult to prove than the homotopy invariance of ordinary homology theory.

     

For a fixed Turaev shadow Jones-Vassiliev invariants depend polynomially on the gleams

We use Turaev's technique of shadows and gleams to parametrize the set of all knots in S ³ with the same Hopf projection. We show that the Vassiliev invariants arising from the Jones polynomial J _t (K) are polynomials in the gleams, i.e., for , the n-th order Vassiliev invariant u _n , defined by , is a polynomial of degree 2n in the gleams. Received: April 30, 1996     

Isotopy and invariants of Albert algebras

Let k be a field with characteristic different from 2 and 3. Let B be a central simple algebra of degree 3 over a quadratic extension K/k, which admits involutions of second kind. In this paper, we prove that if the Albert algebras and have same and invariants, then they are isotopic. We prove that for a given Albert algebra J, there exists an Albert algebra J' with , and . We conclude with a construction of Albert division algebras, which are pure second Tits' constructions. Received: December 9, 1997.     

Constructing modular separating invariants

We consider a finite dimensional modular representation V of a cyclic group of prime order p. We show that two points in V that are in different orbits can be separated by a homogeneous invariant polynomial that has degree one or p and that involves variables from at most two summands in the dual representation. Simultaneously, we describe an explicit construction for a separating set consisting of polynomials with these properties.     

The Alexander polynomial and finite type 3-manifold invariants

Using elementary counting methods, we calculate a universal perturbative invariant (also known as the LMO invariant) of a 3-manifold M, satisfying , in terms of the Alexander polynomial of M. We show that +1 surgery on a knot in the 3-sphere induces an injective map from finite type invariants of integral homology 3-spheres to finite type invariants of knots. We also show that weight systems of degree 2m on knots, obtained by applying finite type 3m invariants of integral homology 3-spheres, lie in the algebra of Alexander-Conway weight systems, thus answering the questions raised in [Ga]. Received: 27 April 1998 / in final form: 8 August 1999     

Milnor numbers, spanning trees, and the Alexander–Conway polynomial

We study relations between the Alexander–Conway polynomial _L and Milnor higher linking numbers of links from the point of view of finite-type (Vassiliev) invariants. We give a formula for the first non-vanishing coefficient of _L of an m-component link L all of whose Milnor numbers μ_i1…ip vanish for pn. We express this coefficient as a polynomial in Milnor numbers of L. Depending on whether the parity of n is odd or even, the terms in this polynomial correspond either to spanning trees in certain graphs or to decompositions of certain 3-graphs into pairs of spanning trees. Our results complement determinantal formulas of Traldi and Levine obtained by geometric methods.     

Invariance of Milnor numbers and topology of complex polynomials

We give a global version of Lê-Ramanujam μ-constant theorem for polynomials. Let , , be a family of polynomials of n complex variables with isolated singularities, whose coefficients are polynomials in t. We consider the case where some numerical invariants are constant (the affine Milnor number μ(t), the Milnor number at infinity λ(t), the number of critical values, the number of affine critical values, the number of critical values at infinity). Let n=2, we also suppose the degree of the is a constant, then the polynomials and are topologically equivalent. For we suppose that critical values at infinity depend continuously on t, then we prove that the geometric monodromy representations of the are all equivalent. Received: January 14, 2002     

Mutation and gauge theory I: Yang-Mills invariants

Mutation of 3-manifolds (cutting and regluing along a genus 2 surface using a central involution) is shown to preserve the instanton Floer homology of homology 3-spheres. A related operation on 4-manifolds is shown to preserve the Donaldson polynomial invariant. Received: November 20, 1998.     

On the Structures of the Milnor K-Groups of Some Complete Discrete Valuation Fields

For a complete discrete valuation field K, the unit group of K has a natural decreasing filtration with respect to the valuation, and the graded quotients of this filtration are written in terms of the residue field. The Milnor K-group of a field is a generalization of the unit group. The Milnor K-group of K has a natural decreasing filtration of the same kind. However, if K is of mixed characteristics and has an absolute ramification index greater than one, the structure is not yet known. The aim of this paper is to determine the structure of the Milnor K-group of some special K, which are of mixed characteristics (0,p), whose residue fields are allowed to be imperfect, and which are of absolute ramification index p(p–1).     

Khovanov homology and the slice genus

We use Lee’s work on the Khovanov homology to define a knot invariant s. We show that s(K) is a concordance invariant and that it provides a lower bound for the smooth slice genus of K. As a corollary, we give a purely combinatorial proof of the Milnor conjecture.     

Finite type invariants and fatgraphs

We define an invariant ∇G(M) of pairs M,G, where M is a 3-manifold obtained by surgery on some framed link in the cylinder Σ×I, Σ is a connected surface with at least one boundary component, and G is a fatgraph spine of Σ. In effect, ∇G is the composition with the ιn maps of Le-Murakami-Ohtsuki of the link invariant of Andersen-Mattes-Reshetikhin computed relative to choices determined by the fatgraph G; this provides a basic connection between 2d geometry and 3d quantum topology. For each fixed G, this invariant is shown to be universal for homology cylinders, i.e., ∇G establishes an isomorphism from an appropriate vector space of homology cylinders to a certain algebra of Jacobi diagrams. Via composition for any pair of fatgraph spines G,G^′ of Σ, we derive a representation of the Ptolemy groupoid, i.e., the combinatorial model for the fundamental path groupoid of Teichmüller space, as a group of automorphisms of this algebra. The space comes equipped with a geometrically natural product induced by stacking cylinders on top of one another and furthermore supports related operations which arise by gluing a homology handlebody to one end of a cylinder or to another homology handlebody. We compute how ∇G interacts with all three operations explicitly in terms of natural products on Jacobi diagrams and certain diagrammatic constants. Our main result gives an explicit extension of the LMO invariant of 3-manifolds to the Ptolemy groupoid in terms of these operations, and this groupoid extension nearly fits the paradigm of a TQFT. We finally re-derive the Morita-Penner cocycle representing the first Johnson homomorphism using a variant/generalization of ∇G.     

Motivic Milnor fibre of cyclic <Emphasis Type="Italic">L</Emphasis><Subscript>∞</Subscript>-algebras

We define the motivic Milnor fiber of cyclic L _∞-algebras of dimension three using the method of Denef and Loeser of motivic integration. It is proved by Nicaise and Sebag that the topological Euler characteristic of the motivic Milnor fiber is equal to the Euler characteristic of the étale cohomology of the analytic Milnor fiber. We prove that the value of Behrend function on the germ moduli space determined by a cyclic L _∞-algebra L is equal to the Euler characteristic of the analytic Milnor fiber. Thus we prove that the Behrend function depends only on the formal neighborhood of the moduli space.     

Stable mappings and logarithmic relative symplectic forms

Let D be the image of a stable, weighted homogeneous map , with , and which is not a trivial deformation of a lower-dimensional map. By proving a variant of the Buchsbaum-Eisenbud structure theorem for grade 3 Gorenstein quotients, we show the existence of a form which restrictst to a non-degenerate holomorphic 2-form on the Milnor fibres of D; experiments with the computer algebra programme Macaulay suggest this restriction is closed, and is thus a holomorphic symplectic form. Received January 23, 1997; in final form July 15, 1998     

SAGBI bases in rings of multiplicative invariants

Let k be a field and G be a finite subgroup of . We show that the ring of multiplicative invariants has a finite SAGBI basis if and only if G is generated by reflections. Received: March 5, 2002     

Milnor invariants and Massey products

In this paper Massey products in the cohomology of the complementary space of a link in a three-dimensional homology sphere are calculated. It is proved that these products are determined by the Milnor invariants of the link and determine them. This generalizes the known connection between the linking coefficients and the cup-product in the cohomology of the complementary space. The existence of a connection between Massey products and Milnor invariants was stated as a conjecture by Stallings.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 66, pp. 189–203, 1976.     

Essentially finite generation of valuation rings in terms of classical invariants

The main goal of this paper is to study some properties of an extension of valuations from classical invariants. More specifically, we consider a valued field $(K,\nu )$ and an extension ω of ν to a finite extension L of K. Then we study when the valuation ring of ω is essentially finitely generated over the valuation ring of ν. We present a necessary condition in terms of classic invariants of the extension by Hagen Knaf and show that in some particular cases, this condition is also sufficient. We also study when the corresponding extension of graded algebras is finitely generated. For this problem we present an equivalent condition (which is weaker than the one for the finite generation of the valuation rings).     

Infinitesimal Torelli theorem for surfaces of general type with certain invariants

We prove the infinitesimal Torelli theorem for general minimal complex surfaces X's with the first Chern number 3, geometric genus 1, and irregularity 0 which have non-trivial 3-torsion divisors. We also show that the coarse moduli space for surfaces with the invariants as above is a 14-dimensional unirational variety.     

Homotopy, Δ-equivalence and concordance for knots in the complement of a trivial link

Link-homotopy and self Δ-equivalence are equivalence relations on links. It was shown by J. Milnor (resp. the last author) that Milnor invariants determine whether or not a link is link-homotopic (resp. self Δ-equivalent) to a trivial link. We study link-homotopy and self Δ-equivalence on a certain component of a link with fixing the other components, in other words, homotopy and Δ-equivalence of knots in the complement of a certain link. We show that Milnor invariants determine whether a knot in the complement of a trivial link is null-homotopic, and give a sufficient condition for such a knot to be Δ-equivalent to the trivial knot. We also give a sufficient condition for knots in the complements of the trivial knot to be equivalent up to Δ-equivalence and concordance.     

Picard groups of multiplicative invariants

Let S = kA denote the group algebra of a finitely generated free abelian group A over the field k and let G be a finite subgroup of GL(A). Then G acts on S by means of the unique extension of the natural GL(A)-action on A. We determine the Picard group Pic R of the algebra of invariants R = S ^G . As an application, we produce new polycyclic group algebras with nontrivial torsion in K ₀ . Received: April 25, 1996     

On the Haefliger-Hirsch-Wu invariants for embeddings and immersions

We prove beyond the metastable dimension the PL cases of the classical theorems due to Haefliger, Harris, Hirsch and Weber on the deleted product criteria for embeddings and immersions. The isotopy and regular homotopy versions of the above theorems are also improved. We show by examples that they cannot be improved further. These results have many interesting corollaries, e.g.? 1) Any closed homologically 2-connected smooth 7-manifold smoothly embeds in .? 2) If and then the set of PL embeddings up to PL isotopy is in 1-1 correspondence with?. Received: July 6, 2000