Brown-Gitler spectra

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 -manifolds" Comment. Math. Helv. , 54 : 3 (1979) pp. 405–430
[a3]	G. Carlsson, "G.B. Segal's Burnside ring conjecture for " Topology , 22 (1983) pp. 83–103
[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 " Invent. Math. , 108 : 1 (1992) pp. 163–227
[a8]	J. Lannes, "Sur les espaces fonctionnels dont la source est le classifiant d'un -groupe abélien élémentaire" IHES Publ. Math. , 75 (1992) pp. 135–244
[a9]	M. Mahowald, "A new infinite family in " Topology , 16 : 3 (1977) pp. 249–256
[a10]	H. Miller, "The Sullivan conjecture on maps from classifying spaces" Ann. of Math. (2) , 120 : 1 (1984) pp. 39–87