On non-orientable surfaces in 4-manifolds

TL;DR

This paper extends key theorems in 4-manifold topology to include non-orientable surfaces, providing conditions for representing such surfaces as connected sums involving unknotted projective planes.

Contribution

It introduces new conditions under which non-orientable surfaces in 4-manifolds can be represented, extending the Gabai light bulb theorem and related results.

Findings

Extended the Gabai 4D light bulb theorem to non-orientable surfaces.

Established conditions for representing non-orientable surfaces as connected sums.

Generalized the 'one is enough' theorem to non-orientable cases.

Abstract

We find conditions under which a non-orientable closed surface S embedded into an orientable closed 4-manifold X can be represented by a connected sum of an embedded closed surface in X and an unknotted projective plane in a 4-sphere. This allows us to extend the Gabai 4-dimensional light bulb theorem and the Auckly-Kim-Melvin-Ruberman-Schwartz "one is enough" theorem to the case of non-orientable surfaces.