Link concordance, homology cobordism, and Hirzebruch-type defects from iterated $p$-covers

Abstract

Abstract. We obtain new invariants of topological link concordance and homology cobordism of 3-manifolds from Hirzebruch-type intersection form defects of towers of iterated p-covers. Our invariants can extract geometric information from an arbitrary depth of the derived series of the fundamental group, and can detect torsion which is invisible via signature invariants. Applications illustrating these features include the following: (1) There are infinitely many homology equivalent rational 3-spheres which are indistinguishable via multisignatures, η-invariants, and L 2 -signatures but have distinct homology cobordism types. (2) There is an infinite family of 2-torsion (amphichiral) knots with non-slice iterated Bing doubles; as a special case, we give the first proof of the conjecture that the Bing double of the figure eight knot is not slice. (3) There exist infinitely many torsion elements at any depth of the Cochran-Orr-Teichner filtration of link concordance.

Twisted torsion invariants and link concordance

“…It is therefore an interesting question to determine which methods and invariants detect the non-sliceness of Bing doubles. For interesting results, we refer to Cimasoni [Ci06], Harvey [Ha08], Cha [Ch10,Ch09], and Cha-Livingston-Ruberman [CLR08]. In particular in [CLR08] it was proved that if K is not algebraically slice, B(K) is not slice.…”

“…In what follows we show that an appropriate twisted torsion invariant of the Bing double B(4 1 ) is not a norm, and consequently B(4 1 ) is not slice. (The fact that B(4 1 ) is not slice had first been shown in [Ch10] …”

“…In Section 6 we show how to compute twisted torsion of boundary links using a boundary link Seifert matrix and we discuss the behavior of our invariants for links which are boundary slice. Finally in Section 7 we prove a formula for the twisted invariants of satellite links and we use this formula to reprove the fact (first proved by the first author [Ch10]) that the Bing double of the Figure 8 knot is not slice.…”

“…We refer to the work of the first author [Ch10], the first author and Kim [CK08], Cochran, Harvey and Leidy [CHL08] and van Cott [vCo09] for interesting results on iterated Bing doubling.…”

Nonconcordant links with homology cobordant zero-framed surgery manifolds

“…which restricts to a homeomorphism on the boundary preserving longitudes and meridians (see, e.g., [Cha 2010, proof of Proposition 4.8; Cha and Orr 2013, Lemma 5.3]). As in [Cha et al 2012, Lemma 2.1], by [Stallings 1965] it follows that f induces (2) The curve η is nonzero in π 1 (L I )/π 1 (L I ) m .…”