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A. K. Bousfield’s \(H\mathbb {Z}\)-localization of groups inverts homologically two-connected homomorphisms of groups. J. P. Levine’s algebraic closure of groups inverts homomorphisms between finitely generated and finitely presented groups which are homologically two-connected and for which the image normally generates. We resolve an old problem concerning Bousfield \(H\mathbb {Z}\)-localization of groups, and answer two questions of Levine regarding algebraic closure of groups. In particular, we show that the kernel of the natural homomorphism from a group \(G\) to its Bousfield \(H\mathbb {Z}\)-localization is not always a \(G\)-perfect subgroup. In the case of algebraic closure of groups, we prove the analogous result that this kernel is not always a union of invisible subgroups.

This paper continues our exploration of homology cobordism of 3-manifolds using our recent results on Cheeger-Gromov rho-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 pairwise homology cobordant, yet remain indistinguishable by any prior known homology cobordism invariants. Additionally we give an answer to a question about transfinite lower central series of homology cobordant 3-manifold groups, asked by T. D. Cochran and M. H. Freedman., 24 pages; a new theorem answering a question of Cochran and Freedman (Kirby List 3.78) added; referee's comments incorporated; to appear in J. Topology

We relate certain abelian invariants of a knot, namely the Alexander polynomial, the Blanchfield form, and the Arf invariant, to intersection data of a Whitney tower in the 4-ball bounded by the knot. We also give a new 3-dimensional algorithm for computing these invariants., Comment: 34 pages, 14 figures; to appear in Mathematische Zeitschrift

We construct many examples of nonslice knots in 3-space that cannot be distinguished from slice knots by previously known invariants. Using Whitney towers in place of embedded disks, we define a geometric filtration of the 3-dimensional topological knot concordance group. The bottom part of the filtration exhibits all classical concordance invariants, including the CassonGordon invariants. As a first step, we construct an infinite sequence of new obstructions that vanish on slice knots. These take values in the L-theory of skew fields associated to certain universal groups. Finally, we use the dimension theory of von Neumann algebras to define an L 2 -signature and use this to detect the first unknown step in our obstruction theory.

We give a geometric slice-like characterization for the vanishing of Milnor's link invariants by proving the k-slice conjecture. This conjecture states that a linkhas vanishing Milnor -invariants of length )2k if and only ifbounds disjoint surfaces in a four disk in such a way that the fundamental group of the complement admits free nilpotent quotients of class k. In the course of our proof, we compute the dimension )3 homology groups of "nitely generated free nilpotent Lie rings and groups. We develop a new algorithm for constructing a weighted chain resolution for a nilpotent group with torsion free lower central series quotients, and with the property that its associated graded complex is the Koszul complex of the associated graded Lie ring. This give a new derivation of the May spectral sequence relating the group homology of the nilpotent group to the Lie ring homology of its associated graded Lie ring. Finally, we de"ne -invariants of `picturesa and use these to describe a generating set of cocycles in the cohomology of the free nilpotent groups. Some sample computations follow. 2001 Elsevier Science Ltd. All rights reserved.

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.

If G and H are finitely generated, residually nilpotent metabelian groups, H is termed para-G if there is a homomorphism of G into H which induces an isomorphism between the corresponding terms of their lower central quotient groups. We prove that this is an equivalence relation. It is a much coarser relation than isomorphism, our ultimate concern. It turns out that many of the groups in a given equivalence class share various properties including finite presentability. There are examples, such as the lamplighter group, where an equivalence class consists of a single isomorphism class and others where this is not the case. We give several examples where we solve the Isomorphism Problem. We prove also that the sequence of torsion-free ranks of the lower central quotients of a finitely generated metabelian group is computable. In a future paper we plan on proving that there is an algorithm to compute the numerator and denominator of the rational Poincar\'e series of a finitely generated metabelian group and will carry out this computation in a number of examples, which may shed a tiny bit of light on the Isomorphism Problem. Our proofs use localization, class field theory and some constructive commutative algebra., Comment: 24 pages. Substantially rewritten. Some new results

We define and study virtual representation spaces 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., 34 pages

We provide new information about the structure of the abelian group of topological concordance classes of knots in $S^3$. One consequence is that there is a subgroup of infinite rank consisting entirely of knots with vanishing Casson-Gordon invariants but whose non-triviality is detected by $L^{(2)}$ signatures., 26 pages, 3 figures

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 M0 and M1 have isomorphic integral (respectively rational) homology groups, is there a sequence of Dehn surgeries, each of which preserves integral (respectively rational) homology, that transforms M0 to M1? 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 M0, M0 = H1 ∪fH2 (Hi are handlebodies of genus g and f is a homeomorphism of their common boundary surface), and a homeomorphism g of ∂H1 such that M1 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 Casson's 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 S3. This fact has been used to define, calculate and understand invariants of homology 3-spheres (such as Casson's invariant) by choosing such a ‘path to S3’ in the ‘space’ of 3-manifolds.

Knot and link theory studies how one manifold embeds in another. Given a manifold embedding, one can alter that embedding in a neighborhood of a point by removing this neighborhood and replacing it with an embedded disk pair. In this way traditional knot theory, the study of embeddings of spheres in spheres, impacts the general manifold embedding problem. In dimension one, the manifold embedding problem is knot and link theory. This article attempts a rapid survey of the role of surgery in the development of knot and link theory. Surgery is one of the most powerful tools in dealing with the question “To what extent are manifolds (or manifold embeddings) uniquely determined by their homotopy type?” As we shall see, roughly speaking, knots and links are determined by their homotopy type (more precisely, Poincare embedding type) in codimension ≥ 3 and are much more complicated in codimension two. We proceed, largely, from an historical perspective, presenting most of seminal early results in the language and techniques in which they were first discovered. These results in knot theory are among the most significant early applications of surgery theory and contributed to its development. We will emphasize knotted and linked spheres, providing only a brief discussion of more general codimension two embedding questions. In particular, the theory of codimension two embedding, from the standpoint of classifying within a Poincare embedding type, deserves a long overdue survey paper. The present paper will not fill this void in the literature. Cappell and Shaneson give an excellent introduction to this subject in [CS78]. By no means is this survey comprehensive, and we apologize in advance for the omission of many areas where considerable and important work has been done. For example, we will omit the extensive subject of equivariant knot theory. We will also not include any discussion of the techniques of Dehn surgery that have proven so valuable in the study of three manifolds and classical knots. Furthermore, we will not touch on the related subject of immersion theory, and barely mention singularity theory. We urge the reader to consult one of the many excellent surveys which have covered the early (before 1977) development of codimension two knot theory in more depth. The articles by Cameron Gordon [Gor77] and Kervaire-Weber [KW77] on, respectively, low-dimensional and high-dimensional knot theory are excellent. A detailed discussion of surgery and embedding theory can be found in Ranicki’s book, [Ran81]. On the other hand, we are not aware of any previously existing comprehensive survey of recent developments in link theory.

Let $ {\cal Z}(M) $ be the 3-manifold invariant of Le, Murakami and Ohtsuki. We show that $ {\cal Z}(M) = 1 + o(n) $ , where $ o(n) $ denotes terms of degree $ \geq n $ , if M is a homology 3-sphere obtained from $ S^3 $ by surgery on an n-component Brunnian link whose Milnor $ \overline\mu $ -invariants of length $ \leq 2n $ vanish.¶We prove a realization theorem which is a partial converse to the above theorem.¶Using the Milnor filtration on links, we define a new bifiltration on the $ \Bbb Q $ vector space with basis the set of oriented diffeomorphism classes of homology 3-spheres. This includes the Milnor level 2 filtration defined by Ohtsuki. We show that the Milnor level 2 and level 3 filtrations coincide after reindexing.

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