The Magnus representation and higher-order Alexander invariants for homology cobordisms of surfaces

同调球的Alexander技巧

The Alexander trick for homology spheres

本文证明在维度至少为6时，固定边界的紧致可缩流形同胚群可缩，并由此推导出单侧$h$-配边的强唯一性。

Alexander invariants of ribbon tangles and planar algebras

Ribbon tangles are proper embeddings of tori and cylinders in the $4$-ball~$B^4$, "bounding" $3$-manifolds with only ribbon disks as singularities. We construct an Alexander invariant $\mathsf{A}$ of ribbon tangles equipped with a representation of the fundamental group of their exterior in a free abelian group $G$. This invariant induces a functor in a certain category $\mathsf{R}ib_G$ of tangles, which restricts to the exterior powers of Burau-Gassner representation for ribbon braids, that are analogous to usual braids in this context. We define a circuit algebra $\mathsf{C}ob_G$ over the op…

Factorization formulas and computations of higher-order Alexander invariants for homologically fibered knots

Homologically fibered knots are knots whose exteriors satisfy the same homological conditions as fibered knots. In our previous paper, we observed that for such a knot, higher-order Alexander invariants defined by Cochran, Harvey and Friedl are generally factorized into the part of the Magnus matrix and that of a certain Reidemeister torsion, both of which are known as invariants of homology cylinders over a surface. In this paper, we study more details of the invariants and give some concrete calculations by restricting to the case of the invariants associated with metabelian quotients of the…

Morita's trace maps on the group of homology cobordisms

Morita introduced in 2008 a 1-cocycle on the group of homology cobordisms of surfaces with values in an infinite-dimensional vector space. His 1-cocycle contains all the "traces" of Johnson homomorphisms which he introduced fifteen years earlier in his study of the mapping class group. In this paper, we propose a new version of Morita's 1-cocycle based on a simple and explicit construction. Our 1-cocycle is proved to satisfy several fundamental properties, including a connection with the Magnus representation and the LMO homomorphism. As an application, we show that the rational abelianization…

Cobordisms of sutured manifolds and the functoriality of link Floer homology

It has been a central open problem in Heegaard Floer theory whether cobordisms of links induce homomorphisms on the associated link Floer homology groups. We provide an affirmative answer by introducing a natural notion of cobordism between sutured manifolds, and showing that such a cobordism induces a map on sutured Floer homology. This map is a common generalization of the hat version of the closed 3-manifold cobordism map in Heegaard Floer theory, and the contact gluing map defined by Honda, Kazez, and Matić. We show that sutured Floer homology, together with the above cobordism maps, forms…

Computing cobordism maps in link Floer homology and the reduced Khovanov TQFT

We study the maps induced on link Floer homology by elementary decorated link cobordisms. We compute these for births, deaths, stabilizations, and destabilizations, and show that saddle cobordisms can be computed in terms of maps in a decorated skein exact triangle that extends the oriented skein exact triangle in knot Floer homology. In particular, we completely determine the Alexander and Maslov grading shifts. As a corollary, we compute the maps induced by elementary cobordisms between unlinks. We show that these give rise to a $(1+1)$-dimensional TQFT that coincides with the reduced Khovan…

提升三分支链环的Milnor不变量

Lifting Milnor Invariants for 3-Component Links

本文定义了三分支链环L的整数值不变量序列γ^k(L)，证明其在协边和弱协边下不变，并提升了某些Milnor不变量，通过引入Kojima-Yamasaki η-不变量的三分支类比h(L)来建立该结果，应用包括当指定分支的Alexander多项式平凡时的弱协边分类，以及刻画在B^4中边界连续嵌入圆盘且补空间基本群为ℤ的纽结。

Arithmetic topology in Ihara theory

Ihara initiated to study a certain Galois representation which may be seen as an arithmetic analogue of the Artin representation of a pure braid group. We pursue the analogies in Ihara theory further, following after some issues and their inter-relations in the theory of braids and links such as Milnor invariants, Johnson homomorphisms, Magnus-Gassner cocycles and Alexander invariants, and study relations with arithmetic in Ihara theory.

Resolutions of p-Modular TQFT's and Representations of Symmetric Groups

We construct families of TQFT's over the finite field Z/pZ starting from an integral TQFT obtained by Frohman and Nicas. These TQFT's are likely to describe the constant order contributions of the cyclotomic integer expansions of the Reshetikhin Turaev Ohtsuki theories. Their modular structure is intimately related to the p-modular representation theory of the symmetric groups S_n. We construct resolutions of simple p-modular TQFT's and S_n-modules over Z/pZ using powers of Lefschetz operators. These yield expansions of the irreducible p-modular S_n-characters in terms of the ordianry ones as …