Seifert and Blanchfield complexes

Abstract

Seifert complexes are abstractions of the duality properties of a Seifert surface of an n-knot, generalizing the form of Seifert [262]. Blanchfield complexes are abstractions of the duality properties of the infinite cyclic cover of the exterior of an n-knot, generalizing the form of Blanchfield [23]. These types of algebraic Poincaré complexes were already introduced in Ranicki [237]. The object of this chapter is to recall the definitions, and to make precise the relationship between the Blanchfield and Seifert complexes, using the L-theory of the Fredholm localizations

$$\Omega _ + ^{ - 1}A\left( s \right),\Omega _ + ^{ - 1}A\left( {s,{s^{ - 1}},{{\left( {1 - s} \right)}^{ - 1}}} \right) = I{I^{ - 1}}A\left[ {z,{z^{ - 1}}} \right]\left( {\bar s = 1 - s,s = {{\left( {1 - z} \right)}^{ - 1}},\bar z = {z^{ - 1}}} \right)$$

for any ring with involution A. In particular, the cobordism groups of Seifert complexes over A are identified with the cobordism groups of Blanchfield complexes over A[z, z ⁻¹].