Let be a structure series, and let be the bundle over induced by the mapping . Let be the Thom space of . The mapping induces a mapping , where is suspension and ( is the one-dimensional trivial bundle). One obtains a spectrum of spaces , associated with the structure series , and a Thom spectrum is any spectrum that is (homotopy) equivalent to a spectrum of the form . It represents -cobordism theory. Thus, the series of classical Lie groups , , , and lead to the Thom spectra , , , and .
Let be Artin's braid group on strings (cf. Braid theory). The homomorphism , where is the symmetric group, yields a mapping such that a structure series arises ( is canonically imbedded in ). The corresponding Thom spectrum is equivalent to the Eilenberg–MacLane spectrum , so that is a Thom spectrum (cf. , ). Analogously, is a Thom spectrum, but using sphere bundles, .
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Thom spectrum. Yu.B. Rudyak (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Thom_spectrum&oldid=18456