A slicing obstruction from the $\frac {10}{8}$ theorem

Abstract

Abstract. From Furuta's 10 8 theorem, we derive a smooth slicing obstruction for knots in S 3 using a spin 4-manifold whose boundary is 0-surgery on a knot. We show that this obstruction is able to detect torsion elements in the smooth concordance group and find topologically slice knots which are not smoothly slice.

Relative genus bounds in indefinite four-manifolds

“…Proof. The proof is analogous to [11]. A neighborhood of the slice disk Σ in X • together with the removed B4 gives an embedding of X 0 (K), the trace of the 0-surgery on K, inside X.…”

“…We now turn to case (b). Assume by contradiction that the inequality (11) does not hold. By blowing up appropriately we can repeatedly reduce the quantity c…”

“…In a different direction, Furuta's celebrated 10/8-theorem [16] gives constraints on the intersection forms of smooth spin 4-manifolds. Donald and Vafaee [11] used the 10/8-theorem to derive a new sliceness obstruction (in the four-ball). Their result was strengthened by Truong in [65], by applying a refinement of Furuta's theorem (called the 10/8 + 4 theorem) due to Hopkins-Lin-Shi-Xu [27].…”