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 ,
Double L-theory
We develop new algebraic methods refining the Witt group of linking forms and Ranicki's torsion algebraic L-groups into double Witt groups and double L-groups. At each prime ideal of the underlying ring, our double Witt groups capture infinitely many more integral signatures of the linking form than the single Witt groups. The double L-groups are an algebraic theory of `double cobordism', refining L-theory analogously. We exhibit an exact sequence relating the double L-groups to classical projective L-theory via a double homology surgery obstruction group. 
The algebraic techniques are applied to high-dimensional knot theory to define new invariants for the study of doubly-slice knots. In particular we prove a homomorphism from the n-dimensional double concordance group to a double L-group, which factors the construction of the Blanchfield form. Some results of Stoltzfus in this area are reproved and we show that every Seifert matrix for a doubly-slice knot is hyperbolic.
Net Spaces in Categorical Topology a
In the paper we introduce the notion of S‐nets, S a nonempty construct, as a generalization of the usual nets. S‐net spaces and continuous maps between them are then defined in a natural way. For S‐net spaces we introduce a number of axioms and study the categories obtained. We find some topological and Cartesian closed categories of S‐net spaces and describe their relations to certain known topological categories.