Twisted Blanchfield pairings and twisted signatures I: Algebraic background

Abstract

This is the first paper in a series of three devoted to studying twisted linking forms of knots and three-manifolds. Its function is to provide the algebraic foundations for the next two papers by describing how to define and calculate signature invariants associated to a linking form M ˆM Ñ Fptq{Frt ˘1 s for F " R, C, where M is a torsion Frt ˘1s-module. Along the way, we classify such linking forms up to isometry and Witt equivalence and study whether they can be represented by matrices.Definition (see Definition 5.1). The signature jump of a linking form pM, λq over Crt ˘1s at ξ P S 1 is defined as the following integer:(1.3) δσ pM,λq pξq " ´ÿ n odd ǫ"˘1 ǫPpn, ǫ, ξ, Cq.The signature jump of a real linking form is defined analogously. Before using signature jumps to construct the signature function, we describe how they can be used to classify linking forms up to Witt equivalence. Recall that the Witt group W pFptq, Frt ˘1sq of linking forms over Frt ˘1s consists of the monoid of non-singular linking forms modulo the submonoid of metabolic linking forms. Two linking forms over Frt ˘1s are Witt equivalent if they agree in W pFptq, Frt ˘1sq. This Witt group is well understood [25,27,41] and its description can be summarized by noting thatProposition (Proposition 5.11). Let ξ P S 1 . If a non-singular linking form pM, λq over Frt ˘1s is representable by a matrix Aptq, then the following equation holds: lim θÑ0 `sign Ape iθ ξq ´lim θÑ0 ´sign Ape iθ ξq " 2δσ pM,λq pξq. Non-singular linking forms over Rrt ˘1s are always representable. Moreover, if multiplication by t˘1 is an isomorphism, the form is representable by a diagonalizable matrix [7, Proposition 4.1].The situation over Crt ˘1s is substantially different. There are forms that are non-representable, see Subsection 3.4. Furthermore, forms that cannot be represented by a diagonal matrix, occur much more often over the complex numbers. Despite these surprises, using signature jumps, we are able to prove the following characterisation of representable linking forms. It is stated as Corollary 5.15, but most of the technical work is done in Subsection 3.5.Proposition (Corollary 5.15). Over Crt ˘1s, metabolic forms are representable, representability is invariant under Witt equivalence and, given a non-singular linking form pM, λq, the following are equivalent:(1) pM, λq is representable;(2) pM, λq is Witt equivalent to a representable one;(3) the total signature jump Σ ξPS 1 δσ pM,λq pξq vanishes.Remark 1.1. The question of representability up to Witt equivalence can be understood via the long exact sequence in L-theory [19,36,39,40]: it asks whether the map W pFptqq Ñ W pFptq, Frt ˘1sq is surjective. While this sequence is well understood for Frt ˘1s (and more generally for Dedekind domains), to the best of our knowledge, the explicit equivalences for Corollary 5.15 are new. In particular, the question of representability up to isometry (instead of Witt equivalence) does not appear to be addressed in the literature.Using signature jumps, we now describe the signat...