An extension of Casson's invariant by Kevin Walker

By Kevin Walker

This publication describes an invariant, l, of orientated rational homology 3-spheres that is a generalization of labor of Andrew Casson within the integer homology sphere case. allow R(X) denote the gap of conjugacy sessions of representations of p(X) into SU(2). allow (W, W, F) be a Heegaard splitting of a rational homology sphere M. Then l(M) is asserted to be an accurately outlined intersection variety of R(W) and R(W) within R(F). The definition of this intersection quantity is a fragile activity, because the areas concerned have singularities. A formulation describing how l transforms below Dehn surgical procedure is proved. The formulation comprises Alexander polynomials and Dedekind sums, and will be used to provide a slightly hassle-free facts of the lifestyles of l. it's also proven that after M is a Z-homology sphere, l(M) determines the Rochlin invariant of M

Show description

Read or Download An extension of Casson's invariant PDF

Best topology books

Modern Geometry: Introduction to Homology Theory Pt. 3: Methods and Applications

During the last fifteen years, the geometrical and topological equipment of the speculation of manifolds have assumed a valuable function within the such a lot complicated parts of natural and utilized arithmetic in addition to theoretical physics. the 3 volumes of "Modern Geometry - equipment and functions" comprise a concrete exposition of those equipment including their major purposes in arithmetic and physics.

Borel Liftings of Borel Sets: Some Decidable and Undecidable Statements

One of many goals of this paintings is to enquire a few common homes of Borel units that are undecidable in $ZFC$. The authors' start line is the subsequent straightforward, notwithstanding non-trivial outcome: contemplate $X \subset 2omega\times2omega$, set $Y=\pi(X)$, the place $\pi$ denotes the canonical projection of $2omega\times2omega$ onto the 1st issue, and consider that $(\star)$ : ""Any compact subset of $Y$ is the projection of a few compact subset of $X$"".

Extra resources for An extension of Casson's invariant

Sample text

Therefore, every abelian variety admits a polarization. 7. The theory of symmetric homomorphisms for abelian varieties is analogous to the theory of symmetric bilinear forms for real vector spaces. Let B(−, −) be a symmetric bilinear form on a real vector space V . The symmetric homomorphism λL associated to a line bundle L is analogous to the symmetric bilinear form B(x, y) = Q(x + y) − Q(x) − Q(y) 22 4. POLARIZATIONS associated to a quadratic form Q. 3 is an analog to the fact that every form B arises this way, with 1 Q(x) = B(x, x).

The functor X classifies OB -linear polarized abelian varieties up to isomorphism (with certain restrictions on the slopes of the p-divisible group and the Weil pairings). We shall now introduce a different functor X which classifies OB,(p) -linear polarized abelian varieties with rational level structure up to isogeny. The functor X will be shown to be equivalent to X . 4. EQUIVALENCE OF THE MODULI PROBLEMS 39 Assume that S is a locally noetherian connected scheme on which p is locally nilpotent.

Since the image η(Lp ) is compact, the index [η(Lp ) : N · T p (A)] must be finite. Therefore η(Lp ) is a virtual subgroup κ(η) ∈ VSubp (A). Automorphisms of Lp do not alter the image η(Lp ), so the virtual subgroup η(Lp ) depends only on the K0p -orbit of η. 1. A rational level structure η : V p − level structure if and only if the virtual subgroup κ(η) is T p (A). 4. The Tate representation Let k be a field, and let be distinct from the characteristic of k. If A is an abelian variety over k, the -adic Tate module of A is defined by T (A) = T (A ⊗k k).

Download PDF sample

Rated 4.74 of 5 – based on 50 votes