= 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.
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