Mod 2 instanton homology and 4-manifolds with boundary

TL;DR

This paper introduces a new homomorphism from the 3-dimensional homology cobordism group to integers using instanton homology, revealing distinctions from existing invariants and providing bounds related to 4-manifolds with boundary.

Contribution

It constructs a novel homomorphism $q_2$ from the homology cobordism group using $Z/2$ instanton homology, which is independent of known invariants like $h$ and $d$.

Findings

$q_2$ is not a rational linear combination of $h$ and $d$.

$q_2(Y) \\ge 0$ for certain negative definite 4-manifolds.

Strict inequality holds if the intersection form is non-standard.

Abstract

Using instanton homology with coefficients in $Z/2$ we construct a homomorphism $q_2$ from the homology cobordism group in dimension 3 to the integers which is not a rational linear combination of the instanton $h$--invariant and the Heegaard Floer correction term $d$. If an oriented homology $3$--sphere $Y$ bounds a smooth, compact, negative definite $4$--manifold without $2$--torsion in its homology then $q_2(Y)\ge0$, with strict inequality if the intersection form is non-standard.