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-$p$ homology $H _{*} X = H_{ *} ( X , {\bf Z} / p {\bf Z} )$ comes equipped with a natural right action of the Steenrod algebra $\mathcal{A}$ which is unstable: at the prime $2$, for example, this means that

\begin{equation*} 0 = \text{Sq} ^ { i } : H _ { n } X \rightarrow H _ { n - i } X , 2 i > n. \end{equation*}

Write $\mathcal{U}_{*}$ for the category of all unstable right modules over $\mathcal{A}$. This category has enough projective objects; indeed, there is an object $G ( n )$, $n \geq 0$, of $\mathcal{U}_{*}$ and a natural isomorphism

\begin{equation*} \operatorname{Hom}_{\cal U_*}( G ( n ) , M ) \cong M _ { n }, \end{equation*}

where $M _ { n }$ is the vector spaces of elements of degree $n$ in $M$. The module $G ( n )$ can be explicitly calculated. For example, if $p = 2$ and $x _ { n } \in G ( n )_{n}$ is the universal class, then the evaluation mapping $\mathcal A \rightarrow G ( n )$ sending $\theta$ to $x _ { n } \theta$ defines an isomorphism

\begin{equation*} \Sigma ^ { n } \mathcal{A} / \{ Sq ^ { i } : 2 i > n \} \mathcal{A} \cong G ( n ). \end{equation*}

These are the dual Brown–Gitler modules.

This pleasant bit of algebra can be only partly reproduced in algebraic topology. For example, for general $n$ there is no space whose (reduced) homology is $G ( n )$; specifically, if $p = 2$, the module $G ( 8 )$ 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 $p$-complete spectrum $T ( n )$ so that $H_{*} T ( n ) \cong G ( n )$ and for all pointed CW-complexes $Z$, the mapping

\begin{equation*} [ T ( n ) , \Sigma ^ { \infty } Z ] \rightarrow \overline { H } _ { n } Z \end{equation*}

sending $f$ to $f * ( x _ { n } )$ is surjective. Here, $\sum ^ { \infty } Z$ is the suspension spectrum of $Z$, the symbol $[ \cdot , \cdot ]$ denotes stable homotopy classes of mappings, and $\overline { H }$ is reduced homology. The spectra $T ( n )$ are the dual Brown–Gitler spectra. The Brown–Gitler spectra themselves can be obtained by the formula

\begin{equation*} B ( n ) = \Sigma ^ { n } D T ( n ), \end{equation*}

where $D$ denotes the Spanier–Whitehead duality functor. The suspension factor is a normalization introduced to put the bottom cohomology class of $B ( n )$ in degree $0$. An easy calculation shows that $B ( 2 n ) \simeq B ( 2 n + 1 )$ for all prime numbers and all $n \geq 0$.

For a general spectrum $X$ and $n \not \equiv \pm 1$ modulo $2 p$, the group $[ T ( n ) , X ]$ is naturally isomorphic to the group $D _ { n } H_{*} \Omega ^ { \infty } X$ of homogeneous elements of degree $n$ in the Cartier–Dieudonné module $D _{*} H _{*} \Omega ^ { \infty } X$ of the Abelian Hopf algebra $H_{ *} \Omega ^ { \infty } X$. In fact, one way to construct the Brown–Gitler spectra is to note that the functor

\begin{equation*} X \mapsto D _ { 2n } H *\Omega X \end{equation*}

is the degree-$2 n$ group of an extraordinary homology theory; then $B ( 2 n )$ is the $p$-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 $n$-manifolds" Comment. Math. Helv. , 54 : 3 (1979) pp. 405–430
[a3]	G. Carlsson, "G.B. Segal's Burnside ring conjecture for $( \mathbf{Z} / 2 ) ^ { k }$" 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 $p$" Invent. Math. , 108 : 1 (1992) pp. 163–227
[a8]	J. Lannes, "Sur les espaces fonctionnels dont la source est le classifiant d'un $p$-groupe abélien élémentaire" IHES Publ. Math. , 75 (1992) pp. 135–244
[a9]	M. Mahowald, "A new infinite family in $\square _ { 2 } \pi _ { * } ^ { s }$" 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