Homotopy spheres i 505 remark s smale 25 and j stallings 27 c zeeman 1331 have proved that every homotopy n sphere n 34 is actually homeomorphic to the standard sphere sn furthermore smale has proved 26 that two homotopy n spheres n 3 4 are h cobordant if and only if they are diffeomorphic. In the mathematical field of algebraic topology the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other they are examples of topological invariants which reflect in algebraic terms the structure of spheres viewed as topological spaces forgetting about their precise geometry unlike homology groups which are also topological invariants the homotopy groups are surprisingly complex and difficult to compute the hopf fibration is a nontrivial mapp. Homotopy spheres are s parallelizable which homotopy spheres bound parallelizable manifolds spherical modifications framed spherical modifications the groups bp 2k a cohomology operation references. The calculation of the homotopy groups of the spheres pi i s n was considered in its time especially in the 1950s as one of the central problems in topology topologists hoped that these groups could be successfully calculated completely and that they would help to solve other classification problems in homotopy
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