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IX Framed Manifolds

Elsevier B.V.
DOI: 10.1016/s0079-8169(08)62210-2
  • Mathematics


Publisher Summary This chapter discusses a construction that associates to every map from a manifold Mk+n to a sphere Sn a submanifold Vk of M and a framing of its normal bundle. The chapter discusses some general results concerning the problem of completing a frame field to a framing. Construction of the set Ωk(Mn+k) of framed cobordism classes of framed k-dimensional submanifolds of a manifold Mn+k is presented, which can be given a group structure. The chapter presents as an example the calculation of the group Ω0(Mn) for Mn a closed, compact, connected, and orientable manifold. The link with homotopy theory is presented, where it is proved that Ωk(Mn+k) corresponds bijectively to the set [Mn+k,Sn] of homotopy classes of maps Mn+k →Sn. Using this correspondence, a few standard operations of homotopy theory are interpreted as operations on framed submanifolds. The chapter also examines π-manifolds—that is, manifolds that have trivial normal bundle when imbedded in a Euclidean space of high dimension.

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