Triple torsion, triple cup products, and embedding obstructions for rational homology 3-spheres

Abstract

Freedman and Krushkal introduced a triple torsion linking form for rational homology $3$-spheres and used it to obstruct locally flat embeddings in $S^4$. For every odd prime $p$, we identify their triple torsion form, computed with parameter $t=p$ on rational homology $3$-spheres whose first homology has exponent $p$, with the mod-$p$ triple cup product under torsion-linking duality. For algebraically split $\pm p$-framed surgery links, this gives a signed formula in terms of Milnor's integral length-three invariants $\bar\mu_{ijk}$, with the framing-sign factor dictated by torsion-linking duality. We then use Borromean band-sums to realize arbitrary mod-$p$ triple cup tensors on rational homology $3$-spheres with $H_1\cong(\mathbb Z/p)^6$ and fixed hyperbolic ordinary torsion linking form. Finally, using the classical spinor/Klein model for the split six-dimensional quadratic space, we classify the tensors with no dual null Hantzsche pair. This produces, for every odd prime $p$, a rational homology $3$-sphere with hyperbolic ordinary torsion linking form but with no locally flat embedding in $S^4$, and indeed no locally flat embedding in any integer homology $4$-sphere.