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

TL;DR

This paper establishes conditions under which homotopic smooth embeddings of surfaces in 5-manifolds are isotopic, introducing a new invariant based on homotopy groups to classify these embeddings.

Contribution

It generalizes previous results by identifying new criteria for isotopy and constructs a novel invariant for classifying surface embeddings in 5-manifolds.

Findings

Homotopic embeddings with a common algebraic dual 3-sphere are isotopic.

Embeddings in simply connected 5-manifolds are isotopic if homotopic.

A new invariant based on homotopy groups classifies embeddings within a homotopy class.

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.