L²‐signatures, homology localization, and amenable groups

Abstract

Aimed at geometric applications, we prove the homology cobordism invariance of the L2‐Betti numbers and L2‐signature defects associated to the class of amenable groups lying in Strebel's class D(R), which includes some interesting infinite/finite non‐torsion‐free groups. This result includes the only prior known condition, that Γ is a poly‐torsion‐free abelian group (or a finite p‐group). We define a new commutator series that refines Harvey's torsion‐free derived series of groups, using the localizations of groups and rings of Bousfield, Vogel, and Cohn. The series, called the local derived series, has versions for homology with arbitrary coefficients and satisfies functoriality and an injectivity theorem. We combine these two new tools to give some applications to distinct homology cobordism types within the same simple homotopy type in higher dimensions, to concordance of knots in three manifolds, and to spherical space forms in dimension 3. © 2012 Wiley Periodicals, Inc.

Nonconcordant links with homology cobordant zero-framed surgery manifolds

“…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 .…”

“…A more detailed discussion is given in Section 5. For the purpose of distinguishing exteriors, we use the amenable Cheeger-Gromov ρ-invariant technology for bordered 3-manifolds (particularly for link exteriors) developed in [Cha 2014], generalising applications of ρ-invariants to concordance and homology cobordism in [Cochran et al 2003;2009;Cha and Orr 2012].…”

“…We state a special case which will be sufficient for our purpose. For a closed 3-manifold M and a homomorphism φ : π 1 (M) → G, denote the von Neumann-Cheeger-Gromov ρ-invariant by ρ (2) (M, φ) ∈ ‫.ޒ‬ See, e.g., [Cochran et al 2003, Section 5] as well as [Chang and Weinberger 2003;Harvey 2008;Cha 2008;Cha and Orr 2012] for definitions and useful properties of ρ (2) (M, φ). Precise references for the properties that we need will be recalled as we go along.…”

Unknotted gropes, Whitney towers, and doubly slicing knots

“…(1) For any integer m ≥ 3, there exists a family of slice knots To show that the examples and their linear combinations are not grope/Whitney slice, we will first relate the grope/Whitney sliceness to the (m, n)-solvability stuided in [Kim06] in Section 6.1, and then use amenable ρ (2) -invariant techniques developed in [CO12,Cha14a,Cha14b] in Sections 6.2, 6.3, and 6.4.…”

“…In Section 6.1, we prove that a knot is (m, n)-solvable whenever it is height (m + 2, n + 2) Whitney slice, that is, W m+2,n+2 ⊂ F m,n (see Theorem 6.5). Then, in Sections 6.2-6.4, we use the amenable signature theorem of [CO12,Cha14a] and combine it with the ideas of [Kim06] to extract obstructions to being (m, n)-solvable from the von Neumann-Cheeger-Gromov ρ (2) -invariants of the zero surgery manifolds. A corollary of (the above outlined proof of) Theorem C is that its analog holds for the solvable bi-filtration {F m,n } (in place of both {G m,n } and {W m,n }) as well.…”