Link groups of 4-manifolds

The notion of a Bing cell is introduced, and it is used to define invariants, link groups, of 4-manifolds. Bing cells combine some features of both surfaces and 4-dimensional handlebodies, and the link group \lambda(M) measures certain aspects of the handle structure of a 4-manifold M. This group is a quotient of the fundamental group, and examples of manifolds are given with \pi_1(M) not equal to \lambda(M). The main construction of the paper is a generalization of the Milnor group, which is used to formulate an obstruction to embeddability of Bing cells into 4-space. Applications to the A-B slice problem and to the structure of topological arbiters are discussed.Comment: 34 pages, 7 figures. v.3: minor phrasing change

Automorphisms of 3-manifolds and representations of 4-manifolds

In the first part of this thesis se prove that any (orientation- preserving) homeomorphism of a (orientable) connected sum of 3-manifolds can be written as a product g°f where f preserves factors and g is a composition of loop homeomorphisms and permutations of factors .The method yields results about the higher homotopy groups of the space of automorphisms of a general 3-manifold. In the second part we give a calculus of links to classify 4-manifolds similar to Kirby’s calculus for 3-manifolds. Using link pictures with certain identified links and corresponding allowable moves. We also consider a stable classification of 4-manifolds using such link pictures

A recipe for exotic 2-links in closed 4-manifolds whose components are topological unknots

We describe a construction procedure of infinite sets of $2$-links in closed simply connected 4-manifolds that are topologically isotopic, smoothly inequivalent and componentwise topologically unknotted. These 2-links are the first examples of such kind in the literature. The examples provided have surface and free groups as their 2-link groups. We also point out an exotic Brunnian behaviour of such families, which highlights the important role of linking in creating exotic phenomena.Comment: v2: The paper has been rewritten to include the case of 2-links. New title. An author has been adde

Torus action on quaternionic projective plane and related spaces

For an action of a compact torus $T$ on a smooth compact manifold~$X$ with isolated fixed points the number $\frac{1}{2}\dim X-\dim T$ is called the complexity of the action. In this paper we study certain examples of torus actions of complexity one and describe their orbit spaces. We prove that $\mathbb{H}P^2/T^3\cong S^5$ and $S^6/T^2\cong S^4$, for the homogeneous spaces $\mathbb{H}P^2=Sp(3)/(Sp(2)\times Sp(1))$ and $S^6=G_2/SU(3)$. Here the maximal tori of the corresponding Lie groups $Sp(3)$ and $G_2$ act on the homogeneous spaces by the left multiplication. Next we consider the quaternionic analogues of smooth toric surfaces: they give a class of 8-dimensional manifolds with the action of $T^3$, generalizing $\mathbb{H}P^2$. We prove that their orbit spaces are homeomorphic to $S^5$ as well. We link this result to Kuiper--Massey theorem and some of its generalizations.Comment: 22 pages, 6 figure