An obstruction to slicing knots using the eta invariant
AbstractWe establish a connection between the η invariant of Atiyah, Patodi and Singer ([1, 2]) and the condition that a knot K [subset or is implied by] S^{3} 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 S^{3} − K into a closed manifold M and, for given unitary representations of π_{1}(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 π_{1}(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 π_{1}(M) and is broken into two parts. In 2·1, homomorphisms from π_{1}(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 π_{1}(M) allow η to be used as an invariant are developed in Section 3 and [script P]_{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 [script P]_{k}. (Received August 11 1998)(Revised May 25 1999) Footnotes^{1} I would like to thank Peter Teichner for pointing out an error in a previous draft. ^{2} This paper was written using Paul Taylor's commutative diagrams package for T_{E}X. |