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We prove that for each positive integer n, the V n -equivalence classes of ribbon knot types form a subgroup R n , of index two, of the free abelian group V n constructed by the author and Stanford. As a corollary, any non-ribbon know whose Arf invariant is trivial cannot be distinguished from ribbon knots by finitely many independent Vassiliev invariants. Furthermore, except...
For a finite group G we calculate the E-Tate cohomology t(E) *G and the E-homology E * (BG + ) as functors of the augmented commutative ring E * (BG + ) when E * ( ) is a complex oriented, ν n -periodic cohomology theory with one-dimensional graded coefficient ring E * .
This paper defines the Normal Euler Number and the Normal Euler Class for polyhedral surfaces in 4-space by means of singularities of projections into hyperplanes. There exist polyhedral analogues of nearly all of Whitney's theorems on Normal Euler Classes of surfaces smoothly immersed in 4-space. However, although the Normal Euler Number of a smooth embedding of the real projective plane in 4-space...
We study phantom maps and homology theories in a stable homotopy category L via a certain Abelian category A. We express the group P(X, Y) of phantom maps X → Y as an Ext group in A, and give conditions on X or Y which guarantee that it vanishes. We also determine P(X, HB). We show that any composite of two phantom maps is zero, and use this to reduce Margolis's axiomatisation conjecture to an...
The present paper is a continuation of [11,6] devoted to the study of finite type invariants of integral homology 3-spheres. We introduce the notion of manifold weight systems, and show that type m invariants of integral homology 3-spheres are determined (modulo invariants of type m - 1) by their associated manifold weight systems. In particular, we deduce a vanishing theorem for finite type invariants...
Let N g be the moduli space of stable holomorphic vector bundles of rank 2 and fixed determinant of odd degree over a smooth complex projective curve of genus g. This paper gives a complete and very simple description of the rational cohomology ring H * (N g ). A structural formula is proved for H * (N g ), which was originally conjectured by Mumford. It...
We provide the natural extension, from the dynamical point of view, of the Poincare-Hopf theorem to noncompact manifolds. On the other hand, given a compact set K being an attractor for a flow generated by a C 1 tangent vector field X on an n-manifold, we prove that the Euler characteristic of its region of attraction A, χ(A), is defined and satisfies Ind A (X) = (-1) n χ(A)...
It is shown that an infinite loop space with no odd torsion in its integral homology also has no odd torsion in its homotopy. Combined with known results of Steve Wilson, this gives a complete classification; all such spaces are products of the Wilson spaces, which are the building blocks of the spaces in the omega spectrum for BP.
We give new proofs of the convexity and connectedness properties of the moment map using the technique of symplectic cutting and extend these results to the case of orbifolds.
We show that, in some cases, a Euclidean cone structure on a closed 3-manifold can be deformed into hyperbolic or spherical cone structures by moving the singular angle. We describe other deformations on the complement of the singular set by using generalized Dehn surgery parameters. In order to do that, we study the relationship between algebraic deformations of the holonomy representation and...
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