Thom spectrum
A spectrum of spaces, equivalent to the spectrum associated to a certain structure series (cf. -structure).
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. [1], [2]). Analogously,
is a Thom spectrum, but using sphere bundles, [3].
References
[1] | M. Mahowold, "A new infinite family in ![]() |
[2] | S. Priddy, "![]() |
[3] | M. Mahowold, "Ring spectra which are Thom complexes" Duke Math. J. , 46 : 3 (1979) pp. 549–559 |
Thom spectrum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Thom_spectrum&oldid=18456