We’ll discuss two interesting examples that reveal difficulties that can arise when considering cxdimension-two embeddings of a manifold into another manifold:

1. A topological embedding of the torus into the 4-sphere that is not homotopic to any PL map
2. A continuous map of the torus into a (nonimpact) 4-manifold that is a homotopy equivalence yet is not homotopic to any embedding.