Kent E. Orr

Dehn surgery equivalence relations on 3-manifolds

Suppose M is an oriented 3-manifold. A Dehn surgery on M (defined below) is a process by which M is altered by deleting a tubular neighbourhood of an embedded circle and replacing it again via some diffeomorphism of the boundary torus. It was shown by Lickorish [ Li ] and Wallace [ Wa ] that any closed oriented connected 3-manifold can be obtained from any other such manifold by a finite sequence of Dehn surgeries. Thus under this equivalence relation all closed oriented 3-manifolds are equivalent. We shall investigate this same question for more restricted classes of surgeries. In particular we shall insist that our Dehn surgeries preserve the integral (or rational) homology groups. Specifically, if M 0 and M 1 have isomorphic integral (respectively rational) homology groups, is there a sequence of Dehn surgeries, each of which preserves integral (respectively rational) homology, that transforms M 0 to M 1 ? What is the situation if we further restrict the Dehn surgeries to preserve more of the fundamental group? Is there a difference if we require ‘integral’ surgeries? We also show that these Dehn surgery relations are strongly connected to the following questions concerning another point of view towards understanding 3-manifolds. Is there a Heegard splitting of M 0 , M 0 = H 1 ∪ f H 2 ( H i are handlebodies of genus g and f is a homeomorphism of their common boundary surface), and a homeomorphism g of ∂ H 1 such that M 1 has a Heegard splitting using g ∘ f as the identification? Since there are many natural subgroups of the mapping class group, such as the Torelli subgroup and the ‘Johnson subgroup’, one can ask the same question where g is restricted to lie in one of these subgroups. This is related to work of Morita on Cassons invariant for homology 3-spheres [ Mo1 ]. Even under these restrictions it has been known for some time that any homology 3-sphere is related to S 3 . This fact has been used to define, calculate and understand invariants of homology 3-spheres (such as Cassons invariant) by choosing such a ‘path to S 3 ’ in the ‘space’ of 3-manifolds.

CLUSTER COMPLEXES VIA SEMI-INVARIANTS

We define and study virtual representation spaces for vectors having both positive and negative dimensions at the vertices of a quiver without oriented cycles. We consider the natural semi-invariants on these spaces which we call virtual semi-invariants and prove that they satisfy the three basic theorems: the first fundamental theorem, the saturation theorem and the canonical decomposition theorem. In the special case of Dynkin quivers with n vertices, this gives the fundamental interrelationship between supports of the semi-invariants and the tilting triangulation of the ( n −1)-sphere.

Not all links are concordant to boundary links

A link is a smooth, oriented submanifold L = {Kx, . . . , Km} of S which is the ordered disjoint union of m manifolds each piecewise-linearly homeomorphic to the «-sphere (if m = 1, L is called a knot). Knots and links play an essential role in the classification of manifolds and, in this regard, perhaps the most important equivalence relation on links is that of link concordance. LQ and L{ are concordant if there is a smooth, oriented submanifold C = {Cx, . . . , Cm} of S x [0,1] which meets the boundary transversely in dC, is piecewise-linearly homeomorphic to L0 x [0, 1] and meets S n+2 x {/} in L. for / = 0, 1. The particular situation which led to the introduction of this equivalence relation and which indicates its importance is as follows. If S is an immersed 2-disk or 2-sphere in a 4-manifold X, x0 is a singular value and B is a small 4-ball neighborhood of x 0 , then S n B is a link in *S. If L were concordant to a link whose components bound disjoint 2-disks in S (the latter is called a trivial link) then the singularity at x0 could be removed. Thus the fundamental problem is to classify (for fixed m, n) the set of concordance classes. In the mid-1960s M. Kervaire and J. Levine gave an algebraic classification of the high-dimensional (n > 1) knot concordance groups [L2]. For even n these are the trivial group and for odd n they are infinitely generated. In a sequence of papers S. Cappell

Stability of lower central series of compact 3-Manifold groups

Abstract The length of a group G is the least ordinal α such that Gα = Gα + 1 where Gα is the αth term of the transfinite lower central series. We begin by establishing connections between lower central series length and the Parafree Conjecture, four-dimensional topological surgery, and link concordance. We prove that the length of all surface groups and most Fuchsian groups is at most ω. We show that the length of the group a Seifert fibration over a base of non-positive even Euler characteristic is at most ω. Our major result is the existence of closed hyperbolic 3-manifolds with length at least 2ω. We observe that any closed orientable 3-manifold group has the same lower central series quotients as a hyperbolic one.

HOMOLOGY BOUNDARY LINKS AND BLANCHFIELD FORMS: CONCORDANCE CLASSIFICATION AND NEW TANGLE-THEORETIC CONSTRUCTIONS

THIS work forms part of our on-going effort to classify the set of concordance classes of links.RecallthatalinkL= (K,,. . . ,K,}inS “+’ is a locally flat piecewise-linear, oriented submanifold of S”+2 of which each component Ki is homeomorphic to S”. The exterior E(L) of a link L is the closure of the complement of a small regular neighborhood N(L) of L. A longitude of a component Ki is a parallel of Ki lying on the boundary of the tubular neighborhood (untwisted if n = 1). A meridian pi is a path from a basepoint to JN(L) which traverses a fiber of 8N(L) and returns. A Seifert Surface for Ki is a connected, compact, oriented, (n + 1)-manifold K E E(L) such that aK is a longitude of Ki. Links Lo, L1 are concordant (or cobordant) if there is a smooth, oriented submanifold C = {C,, . . . , C,} of S n+2 x [0, l] which meets the boundary transversely in X, is piecewise-linearly homeomorphic to L,, x [0, 11, and meets Sn+2 x {i} in Li for i = 0, 1. In the mid-603 M. Kervaire and J. Levine gave an algebraic classification of knot concordance groups (m = 1) in high dimensions (n > 1) [l]. For even n these are trivial and for odd n they are infinitely generated, being isomorphic to certain Witt groups obtained from information garnered from the Seifert surface. Extending Levine’s knot cobordism classification to links is difficult for several reasons. Firstly, if m > 1, the natural operation of connected-sum is not well-defined on concordance classes so there is no obvious group structure. Secondly, the Seifert surfaces for different components of a link may intersect, obstructing at least the naive generalization of the Seifert form information. However, the techniques do extend well to the class of boundary links. A boundary link is one which admits a collection of m disjoint Seifert surfaces, one for each component. In fact, S. Cappell and J. Shaneson classified boundary links modulo boundary link cobordism in 1980 using their homology surgery groups, followed later by Ki Ko and W. Mio who accomplished this via Seifert surfaces [2-41. A boundary link cobordism is a cobordism C between Lo and L1 for which there exist disjointly embedded 2n-manifolds IV= {IV,, . . . , ZVm) in E(C) such that IVn(S”+2 x {i}) is a system of Seifert surfaces for the boundary link Li, i = 0, 1, and such that

Hidden torsion, 3-manifolds, and homology cobordism

This paper continues our exploration of homology cobordism of 3-manifolds using our recent results on Cheeger-Gromov �-invariants associated to amenable representations. We introduce a new type of torsion in 3-manifold groups we call hidden torsion, and an algebraic approximation we call local hidden torsion. We construct infinitely many hyperbolic 3-manifolds which have local hidden torsion in the transfinite lower central subgroup. By realizing Cheeger- Gromov invariants over amenable groups, we show that our hyperbolic 3-manifolds are not pair- wise homology cobordant, yet remain indistinguishable by any prior known homology cobordism invariants.