Non-smoothable surfaces in the 4-sphere

Abstract

We construct examples of non-smoothable surfaces in the $4$-sphere, thereby answering Question 4.32 on the K3 problem list. These surfaces are non-orientable and have knot group of order $2$, thus simultaneously answering Question 4.29(a) on the K3 problem list.

v1: 17 pages. v2: Reorganisation of the introduction. Weakened Theorem 1.2 to account for an oversight in the proof; see the new Proposition 5.4. The advertised main theorem remains unchanged. 22 pages. 2 Figures