Blanchfield and Seifert algebra in high dimensional knot theory
Novikov [12] initiated the study of the algebraic properties of quadratic forms over polynomial extensions by a far-reaching analogue of the Pontrjagin-Thom transversality construction of a Seifert surface of a knot and the infinite cyclic cover of the knot exterior. In this paper the analogy is applied to explain the relationship between the Seifert forms over a ring with involution A and Blanchfield forms over the Laurent polynomial extension A[z, z]. The rings A and A[z, z] correspond to the two ways of associating algebraic invariants to an n-knot k : S ⊂ S with A = Z : (i) The Z[z, z]-module invariants of the canonical infinite cyclic cover M = pR of the exterior of k M = cl.(S\k(S)×D) ⊂ S with k(S) × D ⊂ S a regular neighbourhood of k(S) in S, p : M → S a map inducing an isomorphism p : H(S) ∼= H(M), and ∂M = S × S. (ii) The Z-module invariants of a codimension 1 submanifold N ⊂ S with boundary ∂N = k(S) ⊂ S ,
Non-Commutative Localization in Algebra and Topology: Invariants of boundary link cobordism II. The Blanchfield-Duval form
We use the Blanchfield-Duval form to define complete invariants for the cobordism group C_{2q-1}(F_\mu) of (2q-1)-dimensional \mu-component boundary links (for q\geq2). 
The author solved the same problem in math.AT/0110249 via Seifert forms. Although Seifert forms are convenient in explicit computations, the Blanchfield-Duval form is more intrinsic and appears naturally in homology surgery theory. 
The free cover of the complement of a link is constructed by pasting together infinitely many copies of the complement of a \mu-component Seifert surface. We prove that the algebraic analogue of this construction, a functor denoted B, identifies the author's earlier invariants with those defined here. We show that B is equivalent to a universal localization of categories and describe the structure of the modules sent to zero. Taking coefficients in a semi-simple Artinian ring, we deduce that the Witt group of Seifert forms is isomorphic to the Witt group of Blanchfield-Duval forms.
A simple proof that the concordance group of algebraically slice knots is infinitely generated
A simple proof of the result stated in the title is obtained by making the Casson-Gordon invariant additive. A. J. Casson and C. McA. Gordon proved in [1], [2] that there are algebraically slice knots which are not slice knots. In other words, the concordance group &, of algebraically slice knots is nontrivial. A natural question: Is dT infinitely generated? The author learned from Wu-chung Hsiang that Casson had obtained the affirmative answer to this question for some time; however, Casson has not published his proof to date. The purpose of this short note is to present a simple proof, which would possibly be different from Casson's. The main point in our argument is the observation that we can make the Casson-Gordon invariant additive by slightly generalizing its definition, thereby making it possible to detect linear independence in 9,. We shall use the language and notation of [2]. 1. Additivity of Casson-Gordon invariant for 3-manifolds. In [2], Casson and Gordon defined, for a closed oriented 3-manifold M and an epimorphism 4: H,(M) Z,,,, the invariant ar(M, 4), 0 M is the m-fold cyclic covering induced by 4. Pick up an m-fold cyclic branched covering of 4-manifolds W-> W, branched over a surface F c int W, such that a( W -> W) = (M -> M). (The existence of such (W, F) follows from Lemma 2.2 of [2].) Then define 2[F]2r(m r) ar(M, ) sign W er( W) -2 m For our purpose, we have to deal with arbitrary homomorphism 4. For simplicity, we shall restrict ourselves to the case m = p, a prime, so that a homomorphism H,(M) Zm is either epimorphic or trivial. DEFINITION. Let 4: H1(M) -> Zp be a homomorphism, where M is an oriented closed 3-manifold, p a prime. Define ar(M, )) as above if 4 is epimorphic, and define ar(M, 4) = 0 if 4 is trivial, for 0 < r <p. This invariant is additive in the following sense. Let M', M" be two closed Received by the editors June 30, 1980. 1980 Mathematics Subject Classification Primary 55A25; Secondary 57D90. 'This work was done in Fall, 1979, while the author was visiting the Institute for Advanced Study, Princeton, New Jersey, and partially supported by an NSF grant. The author is grateful to Professor W. C. Hsiang for helpful conversations. O 1981 American Mathematical Society 0002-9939/8 1 /0000-0442/$02.00