Non-smoothable surfaces in the 4-sphere

TL;DR

This paper constructs examples of non-smoothable, non-orientable surfaces with knot group order 2 in the 4-sphere, addressing specific open questions in the K3 problem list.

Contribution

It provides explicit examples of non-smoothable surfaces in the 4-sphere, solving two open questions related to non-orientable surfaces and their knot groups.

Findings

Constructed non-smoothable surfaces in the 4-sphere.

Surfaces are non-orientable with knot group of order 2.

Answered two specific questions from the K3 problem list.

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.