An infinite-rank summand of the homology cobordism group

Abstract

We show that the three-dimensional homology cobordism group admits an infinite-rank summand. It was previously known that the homology cobordism group contains a Z ∞ -subgroup [Fur90, FS90] and a Z-summand [Frø02]. Our proof proceeds by introducing an algebraic variant of the involutive Heegaard Floer package of Hendricks-Manolescu and Hendricks-Manolescu-Zemke. This is inspired by an analogous argument in the setting of knot concordance due to the second author.

On the quotient of the homology cobordism group by Seifert spaces

“…The local equivalence class of this iota-complex is an invariant of the homology cobordism class of Y , and the set of iota-complexes modulo local equivalence forms a group under tensor product. For technical reasons, it is often convenient to consider a slightly weaker notion of equivalence, called almost local equivalence, and the associated group Î of almost iota-complexes modulo almost local-equivalence, as in [DHST18]. There is a group homomorphism ĥ : Θ 3 Z → Î induced by sending [Y ] to the almost local equivalence class of its iota-complex.…”

“…(2) A computation of the almost local equivalence class of linear combinations of C(n − 1), for n ≥ 2, following the strategy of [DHST18, Section 8.1]. (3) A comparison of the results from step (2) with the computation of ĥ(Θ SF ) in [DHST18,Theorem 8.1].…”

More concordance homomorphisms from knot Floer homology

“…Lastly, we point out that the techniques in this paper are the knot Floer analogues of the techniques used in [DHST18] to study the three-dimensional homology cobordism group.…”

Linear independence of rationally slice knots

“…We refer to the pair (CFK F[U ,V] (K), ι K ) as the ι K -complex of a knot K. Moreover, Zemke [Zem19b] (see also [Zem19a, Theorem 1.5]) showed that up to an algebraic equivalence called local equivalence, the ι K -complex of a knot is a concordance invariant. We use a slightly coarser equivalence relation called almost local equivalence, which is motivated by [DHST18] (see also [DHST19]), to show the following. Note that the following theorem implies Theorem 1.1 immediately since the (p, −1)-cable of a rationally slice knot is rationally slice.…”

“…It can sometimes be cumbersome to work with ι K -complexes. Let (U, V) denote the ideal generated by U and V. Motivated by [DHST18] (see also [DHST19]), we make the following more relaxed definition:…”