On the isotopy classes of embeddings of surfaces in 5-manifolds

Abstract

Let f, g be two homotopic smooth embeddings of a closed surface in a closed oriented 5-dimensional manifold. We show that if f admits a common algebraic dual 3-sphere, or if the fundamental group of the ambient space is trivial, then f and g must be isotopic. This generalizes a result of Kosanovic, Schneiderman, and Teichner. The proof is based on the construction of an invariant that classifies the isotopy classes of smooth embeddings of surfaces in ambient 5-dimensional manifolds within a homotopy class, which may be of independent interest. The invariant is defined in terms of the homotopy groups of the 5-dimensional manifold.