Surfaces in 4-manifolds and extendible mapping classes

We study smooth proper embeddings of compact orientable surfaces in compact orientable $4$-manifolds and elements in the mapping class group of that surface which are induced by diffeomorphisms of the ambient $4$-manifolds. We call such mapping classes extendible. An embedding for which all mapping classes are extendible is called flexible. We show that for most of the surfaces there exists no flexible embedding in a $4$-manifold with homology type of a $4$-ball or of a $4$-sphere. As an application of our method, we address a question of Etnyre and Lekili and show that there exists no simple open book decomposition of $S^5$ with a spin page where all $3$-dimensional open books admit open book embeddings. We also provide many constructions and criteria for extendible and non-extendible mapping classes, and discuss a connection between extendibility and sliceness of links in a homology $4$-ball with $S^3$ boundary. Finally, we give a new generating set of the group of extendible mapping classes for the trivial embedding of a closed genus $g$ surface in $S^4$, consisting of $3g$ generators. This improves a previous result of Hirose giving a generating set of size $6g-1$.