 two maps formula_54 are homotopic if and only if formula_55.In other words, degree is an isomorphism between formula_56 and formula_57.Moreover, the Hopf theorem states that for any formula_1-dimensional closed oriented manifold "M", two maps formula_59 are homotopic if and only if formula_60

A self-map formula_61 of the "n"-sphere is extendable to a map formula_62 from the "n"-ball to the "n"-sphere if and only if formula_63.