On diffeomorphisms over non-orientable surfaces standardly embedded in the 4-sphere

TL;DR

This paper characterizes when a diffeomorphism of a non-orientable surface embedded in the 4-sphere can be extended to the ambient space, linking extendability to preservation of a specific quadratic form.

Contribution

It establishes a precise criterion for extendability of surface diffeomorphisms based on quadratic form preservation, connecting surface topology with 4-sphere embeddings.

Findings

Extendability of diffeomorphisms is equivalent to preserving the Guillou-Marin quadratic form.

Provides a topological criterion for diffeomorphism extension over non-orientable surfaces.

Links algebraic invariants with geometric embedding properties.

Abstract

For a non-orientable closed surface standardly embedded in the 4-sphere, a diffeomorphism over this surface is extendable if and only if this diffeomorphism preserves the Guillou-Marin quadratic form of this embedded surface.