Difference between revisions of "Brown-Gitler spectra"
(Importing text file) |
Ulf Rehmann (talk | contribs) m (moved Brown–Gitler spectra to Brown-Gitler spectra: ascii title) |
(No difference)
|
Revision as of 18:51, 24 March 2012
Spectra introduced by E.H. Brown Jr. and S. Gitler [a1] to study higher-order obstructions to immersions of manifolds (cf. also Immersion; Spectrum of spaces). They immediately found wide applicability in a variety of areas of homotopy theory, most notably in the stable homotopy groups of spheres ([a9] and [a4]), in studying homotopy classes of mappings out of various classifying spaces ([a3], [a10] and [a8]), and, as might be expected, in studying the immersion conjecture for manifolds ([a2] and [a5]).
The modulo- homology
comes equipped with a natural right action of the Steenrod algebra
which is unstable: at the prime
, for example, this means that
![]() |
Write for the category of all unstable right modules over
. This category has enough projective objects; indeed, there is an object
,
, of
and a natural isomorphism
![]() |
where is the vector spaces of elements of degree
in
. The module
can be explicitly calculated. For example, if
and
is the universal class, then the evaluation mapping
sending
to
defines an isomorphism
![]() |
These are the dual Brown–Gitler modules.
This pleasant bit of algebra can be only partly reproduced in algebraic topology. For example, for general there is no space whose (reduced) homology is
; specifically, if
, the module
cannot support the structure of an unstable co-algebra over the Steenrod algebra. However, after stabilizing, this objection does not apply and the following result from [a1], [a4], [a6] holds: There is a unique
-complete spectrum
so that
and for all pointed CW-complexes
, the mapping
![]() |
sending to
is surjective. Here,
is the suspension spectrum of
, the symbol
denotes stable homotopy classes of mappings, and
is reduced homology. The spectra
are the dual Brown–Gitler spectra. The Brown–Gitler spectra themselves can be obtained by the formula
![]() |
where denotes the Spanier–Whitehead duality functor. The suspension factor is a normalization introduced to put the bottom cohomology class of
in degree
. An easy calculation shows that
for all prime numbers and all
.
For a general spectrum and
modulo
, the group
is naturally isomorphic to the group
of homogeneous elements of degree
in the Cartier–Dieudonné module
of the Abelian Hopf algebra
. In fact, one way to construct the Brown–Gitler spectra is to note that the functor
![]() |
is the degree- group of an extraordinary homology theory; then
is the
-completion of the representing spectrum. See [a6]. This can be greatly, but not completely, destabilized. See [a7].
References
[a1] | E.H. Brown Jr., S. Gitler, "A spectrum whose cohomology is a certain cyclic module over the Steenrod algebra" Topology , 12 (1973) pp. 283–295 |
[a2] | E.H. Brown Jr., F.P. Peterson, "A universal space for normal bundles of ![]() |
[a3] | G. Carlsson, "G.B. Segal's Burnside ring conjecture for ![]() |
[a4] | R.L. Cohen, "Odd primary infinite families in stable homotopy theory" Memoirs Amer. Math. Soc. , 30 : 242 (1981) |
[a5] | R.L. Cohen, "The immersion conjecture for differentiable manifolds" Ann. of Math. (2) , 122 : 2 (1985) pp. 237–328 |
[a6] | P. Goerss, J. Lannes, F. Morel, "Hopf algebras, Witt vectors, and Brown–Gitler spectra" , Algebraic Topology (Oaxtepec, 1991) , Contemp. Math. , 146 , Amer. Math. Soc. (1993) pp. 111–128 |
[a7] | P. Goerss, J. Lannes, F. Morel, "Vecteurs de Witt non-commutatifs et représentabilité de l'homologie modulo ![]() |
[a8] | J. Lannes, "Sur les espaces fonctionnels dont la source est le classifiant d'un ![]() |
[a9] | M. Mahowald, "A new infinite family in ![]() |
[a10] | H. Miller, "The Sullivan conjecture on maps from classifying spaces" Ann. of Math. (2) , 120 : 1 (1984) pp. 39–87 |
Brown-Gitler spectra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Brown-Gitler_spectra&oldid=17469