An obstruction to slicing knots using the eta invariant

Abstract

We establish a connection between the η invariant of Atiyah, Patodi and Singer ([1, 2]) and the condition that a knot K ⊂ S3 be slice. We produce a new family of metabelian obstructions to slicing K such as those first developed by Casson and Gordon in [4] in the mid 1970s. Surgery is used to turn the knot complement S3 − K into a closed manifold M and, for given unitary representations of π1(M), η can be defined. Levine has recently shown in [11] that η acts as an homology cobordism invariant for a certain subvariety of the representation space of π1(N), where N is zero-framed surgery on a knot concordance. We demonstrate a large family of such representations, show they are extensions of similar representations on the boundary of N and prove that for slice knots, the value of η defined by these representations must vanish.The paper is organized as follows; Section 1 consists of background material on η and Levine's work on how it is used as a concordance invariant [11]. Section 2 deals with unitary representations of π1(M) and is broken into two parts. In 2·1, homomorphisms from π1(M) to a metabelian group Γ are developed using the Blanchfield pairing. Unitary representations of Γ are then considered in 2·2. Conditions ensuring that such two stage representations of π1(M) allow η to be used as an invariant are developed in Section 3 and [Pscr ]k, the family of such representations, is defined. Section 4 contains the main result of the paper, Theorem 4·3. Lastly, in Section 5, we demonstrate the construction of representations in [Pscr ]k.