[Paper Review] An inequality for the h-invariant in instanton Floer theory

Kim A. Frøyshov|ArXiv.org|Nov 4, 2001

Geometric and Algebraic Topology12 references4 citations

TL;DR

This paper establishes a sharp inequality in instanton Floer homology relating the h-invariant of a homology sphere boundary to the genus of an embedded surface with self-intersection +1 in a 4-manifold with b₂⁺ = 1. Using moduli space techniques and projectively flat U(2) connections, it proves h(Y) + ⌈g/2⌉ ≥ e(ℒ), where e(ℒ) is a lattice invariant derived from the orthogonal complement of the surface class, generalizing Donaldson's diagonalization theorem and enabling bounds on h-invariant changes under ±1 surgery on knots.

ABSTRACT

In math.DG/9903083 (henceforth referred to as EA) we defined an integer invariant $h(Y)$ for oriented integral homology 3-spheres $Y$ which only depends on the rational homology cobordism class of $Y$ and is additive under connected sums. In this paper we establish lower bounds for $h(Y)$ when $Y$ is the boundary of a smooth, compact, oriented 4-manifold with $b_2^+=1$. As applications, we give an upper bound for how much $h$ changes under -1 surgery on knots in terms of the slice genus of the knot, and compute $h$ for a family of Brieskorn spheres. This paper contains, in revised form, most of the material from v1 of EA that was left out in the final version of that paper. In particular, Theorem 1 of the present paper is virtually the same as Theorem 1 of v1 of EA. The proof is also essentially the same, but the exposition has been improved, with more details.

Motivation & Objective

• To establish a new inequality involving the h-invariant in instanton Floer homology for 4-manifolds with homology sphere boundary and b₂⁺ = 1.
• To relate the h-invariant to the genus of an embedded surface of self-intersection +1 and the structure of the orthogonal complement lattice in cohomology.
• To generalize Donaldson's diagonalization theorem via Floer-theoretic invariants and lattice invariants e(ℒ).
• To derive bounds on how the h-invariant changes under ±1 surgery on knots, using rational homology cobordisms.

Proposed method

• Uses instanton Floer theory to define the h-invariant as a surjective homomorphism from the homology cobordism group to ℤ.
• Applies moduli space techniques for SO(3) connections on 4-manifolds with boundary, focusing on parametrized families and reducible connections.
• Introduces the lattice invariant e(ℒ) for unimodular negative definite lattices ℒ, defined via a sum over extremal vectors and characteristic classes.
• Employs projectively flat U(2) connections and their relation to flat SU(2) connections over 3-manifolds to compute Floer chain complexes.
• Analyzes the structure of the moduli space of flat SO(3) connections on 3-tori, showing exactly two non-degenerate flat connections differing by index 4.
• Uses the family of metrics and cutting-down techniques to control boundary and end behavior of the moduli space.

Experimental results

Research questions

• RQ1 How does the h-invariant of a homology sphere boundary relate to the genus and self-intersection of an embedded surface in a 4-manifold with b₂⁺ = 1?
• RQ2 What is the precise role of the lattice invariant e(ℒ) in bounding the h-invariant in terms of surface genus?
• RQ3 Can the h-invariant be bounded under ±1 surgery on knots, and what is the sharpness of such bounds?
• RQ4 How do the invariants h(Y) and e(ℒ) behave under connected sums and cobordisms?
• RQ5 What is the structure of the moduli space of flat SO(3) connections on the 3-torus, and how does it relate to Floer homology?

Key findings

• The inequality h(Y) + ⌈g/2⌉ ≥ e(ℒ) holds for any smooth, compact, oriented 4-manifold X with boundary a homology sphere Y and b₂⁺(X) = 1, where g is the genus of an embedded surface Σ with Σ·Σ = 1.
• The lattice invariant e(ℒ) vanishes if and only if ℒ is diagonal, and e(−kE₈ ⊕ ℒ) = k when ℒ is diagonal and k ≥ 0.
• For the Brieskorn sphere Σ(2,2k−1,4k−3), the h-invariant is exactly ⌊k/2⌋, computed via the inequality and matching upper and lower bounds.
• The h-invariant changes by at most ⌈g/2⌉ under −1 surgery on a knot of slice genus g, giving 0 ≤ h(Y_{γ,−1}) − h(Y) ≤ ⌈g/2⌉.
• The moduli space of flat SO(3) connections on the 3-torus has exactly two non-degenerate points, differing by index 4, which is consistent with the known degree 4 involution on Floer homology.
• The construction via projectively flat U(2) connections allows explicit computation of the Floer chain complex for non-trivial SO(3) bundles over T³.