 Homology obstruction to carrying surface curves onto one another

A homotopy equivalence of a closed surface cannot carry a simple closed curve of nonzero integral homology class onto a contractible simple closed curve. Any map into the latter factors through a circle whose inclusion induces zero on first homology; the former class would be killed by an isomorphism. Such a carrying map can nevertheless exist: project a handle generator onto a circle and compose with the contractible embedded circle.