Linking-form splitting and four-dimensional genera of knots

Abstract

By Gilmer's splitting lemma, the linking form of a rational homology sphere that bounds a 4-manifold is the orthogonal sum of a metabolic summand and a summand presented by a form of the rank of the intersection form. The lemma underlies Gilmer's slice genus theorem and the rank criteria of Murakami-Yasuhara and Gilmer-Livingston for nonorientable surfaces and the 4-dimensional clasp number. We show that the lemma and these results, as stated, are false. For each prime $p \equiv 3 \pmod 8$ we construct knots bounding smooth Möbius bands, and knots bounding smooth punctured tori with clasp number one, whose linking forms have no such summand of rank at most two. We then determine what remains true: the lemma holds up to Witt equivalence and for cube-free orders, and its conclusions about summands hold away from exceptional primes and for orientable surfaces of genus at least two. Finally, we find the optimal Witt-class bound for the topological nonorientable 4-genus.