Filtered instanton Floer homology and the homology cobordism group

Abstract

For any s ∈ [−∞, 0] and oriented homology 3-sphere Y , we introduce a homology cobordism invariant rs(Y ) whose value is in (0, ∞]. The values {rs(Y )} are contained in the critical values of the SU (2)-Chern-Simons functional of Y , and we have a negative definite cobordism inequality and a connected sum formula. As applications, we have several new results on the homology cobordism group. As one of such results, we give infinitely many homology 3-spheres which cannot bound any definite 4-manifold. As another result, we show that if the 1-surgery of a knot has the Frøyshov invariant negative, then all positive 1/n-surgeries of the knot are linearly independent in the homology cobordism group. In another direction, we use {rs} to define a filtration on the homology cobordism group which is parametrized by [0, ∞]. Moreover, as a hyperbolic example, we compute an approximate value of rs for the 1/2-surgery along the mirror of the knot 52 in Rolfsen's knot table. Z 6.2. A pseudometric on Ker h 1

Homology cobordism, knot concordance, and Heegaard Floer homology

“…In the negative direction, Frøyshov (in unpublished work), F. Lin [Lin17], and Stoffregen [Sto17] showed that there are classes in Θ 3 Z that do not admit Seifert fibered representatives. Nozaki-Sato-Taniguchi [NST19] improved this result to show that there are classes that do not admit a Seifert fibered representative nor a representative that is surgery on a knot in S 3 . However, none of these results were sufficient to obstruct Θ 3 Z from being generated by Seifert fibered spaces.…”

An adjunction inequality for the Bauer-Furuta type invariants, with applications to sliceness and 4-manifold topology

“…Endo [11] proved that T contains Z ∞ (a certain class of preztel knots) as a subgroup by using Furuta's result [20]. There are many developments of studies of Z ∞subgroups in T : In Yang-Mills gauge theory side, there are several related studies of Z ∞ -subgroups in T [22,54,61]. In Heegaard Floer theory, several concordance invariants were constructed: the tau-invariant τ : C → Z [56], the nu + invariant ν + : C → Z ≥0 [28], and the upsilon invariant Υ : C → PL([0, 2], R) [58].…”

Equivariant aspects of singular instanton Floer homology

“…In this subsection, we did not attempt to systematically exploit the Chern-Simons filtration to study concordances, and we content ourselves with the definition of one homology concordance invariant. For example, we believe that by a slight modification of our axiomatization of enriched complexes one can define analogues of the invariants r s introduced in [NST19]. Another possible direction is to apply the construction of Subsection 4.7 in the context of enriched S-complexes.…”