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 ,
Blanchfield and Seifert algebra in high-dimensional boundary link theory I: Algebraic K-theory
The classification of high-dimensional mu-component boundary links motivates decomposition theorems for the algebraic K-groups of the group ring A[F_mu] and the noncommutative Cohn localization Sigma^{-1}A[F_mu], for any mu>0 and an arbitrary ring A, with F_mu the free group on mu generators and Sigma the set of matrices over A[F_mu] which become invertible over A under the augmentation A[F_mu] to A. Blanchfield A[F_mu]-modules and Seifert A-modules are abstract algebraic analogues of the exteriors and Seifert surfaces of boundary links. Algebraic transversality for A[F_mu]-module chain complexes is used to establish a long exact sequence relating the algebraic K-groups of the Blanchfield and Seifert modules, and to obtain the decompositions of K_*(A[F_mu]) and K_*(Sigma^{-1}A[F_mu]) subject to a stable flatness condition on Sigma^{-1}A[F_mu] for the higher K-groups.
An Interpretation of G. Whitehead's Generalization of H. Hopf's Invariant
In the present paper', the generalized H. Hopf's invariant H: 7rdwn++(Sn+l) w7rd +n+i(Sqn+ ), due to G. Whitehead [10], is given a new definition which has some similarity with the original H. Hopf's definition [5]. The invariant H(f ) appears as depending in particular on the position in Sd+n+l of the inverse images M, Md byf: Sdln+l -_ Sn+1 of two regular values q, q' in Sn+1. Arnold Shapiro has defined the linking coefficient of two spheres Sp, Sq imbedded (without common point) in Em+i for p + q > m.2 In ? 5, the notion of linking coefficient is extended to (p + q m)-connected 7rmanifolds Mp, Mq in Em+i. It is an element of the stable homotopy group 7rW+N(SN), where r p + q m (7r-manifold = manifold which can be imbedded in some euclidean space with a trivial normal bundle). In the definition of H given in ? 3, M, and M' are 7r-manifolds but need not be (d n)-connected. As a consequence, H will in general also depend on the fields of normal vectors over M and M'. Therefore, it cannot be considered strictly as a linking coefficient which should be uniquely determined by the position in space of the two manifolds. It is an open question whether the method can be used to define the linking coefficient of (non-necessarily (p + q m)-connected) w-manifolds Mp, Mq in Em+i by going over to the quotient of 7Wp+qm,+N(SN) by some suitable subgroup. As an application of the new definition of H, it is proved that any regular imbedding (without self-intersection) of the d-sphere into euclidean (d + n)-space induces over Sd the trivial normal bundle provided that 2n > d + 1. A partial result in this direction was announced in [6].