Ribbon homology cobordisms

Abstract

We study 4-dimensional homology cobordisms without 3-handles via Heegaard and instanton Floer homologies, character varieties, and Thurston geometries. We provide obstructions to such cobordisms arising from each of these theories, and illustrate some topological applications.

Ribbon concordance of knots is a partial order

“…In the conclusion we point out that this theorem potentially generalizes to homology ribbon cobordism in the sense of [5] and we consider the possibility of answering some other questions from [8, Section 6].…”

“…Both of these results follow from a result of Gerstenhaber-Rothaus [6, Theorem 1(ii)] which allows one to extend a representation ρ : π 1 (X) → S O(N) to a representation ρ : π 1 (Y) → S O(N) which restricts to ρ using the fact that Y has a handle decomposition with k 1-handles and k 2-handles added to a collar neighborhood of X, and so that the 2-handles homologically cancel the 1-handles to obtain a homology cobordism (this is called a ribbon homology cobordism in [5]). Note that the map R N (Y) → R N (X) may be with respect to different coordinates, since the generators of π 1 (X) may be regarded as a subset of the generators of π 1 (Y), and hence this polynomial map is a projection onto the subspace corresponding to the generators of π 1 (X).…”

Ribbon Homology Cobordisms and Link Floer Homology

“…In particular, if knot Floer homology obstructs K 0 and K 1 from being ribbon concordant in S 3 × I with respect to the usual Morse function, they cannot be ribbon concordant through any ribbon Z-homology cobordism W : S 3 → S 3 . This is perhaps unsurprising, as any such ribbon homology cobordism is necessarily a homotopy S 3 × I by [DLVVW,Theorem 1.14] .…”

Dehn surgery and non-separating two-spheres

“…In this article, we study a more general question: when can surgery on a knot in a three-manifold (other than S 3 ) produce an S 2 × S 1 summand? In previous work of Daemi, the second author, Vela-Vick, and Wong [DLVVW19], some constraints were given on the geography problem. Here, we answer both the botany and geography problems in several different settings.…”