View source for Brown-Gitler spectra

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Spectra introduced by E.H. Brown Jr. and S. Gitler [[#References|[a1]]] to study higher-order obstructions to immersions of manifolds (cf. also [[Immersion|Immersion]]; [[Spectrum of spaces|Spectrum of spaces]]). They immediately found wide applicability in a variety of areas of [[Homotopy|homotopy]] theory, most notably in the stable homotopy groups of spheres ([[#References|[a9]]] and [[#References|[a4]]]), in studying homotopy classes of mappings out of various classifying spaces ([[#References|[a3]]], [[#References|[a10]]] and [[#References|[a8]]]), and, as might be expected, in studying the immersion conjecture for manifolds ([[#References|[a2]]] and [[#References|[a5]]]).

The modulo-$p$ [[Homology|homology]] $H _{*} X = H_{ *} ( X , {\bf Z} / p {\bf Z} )$ comes equipped with a natural right action of the [[Steenrod algebra|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 &gt; n. \end{equation*}

Write $\mathcal{U}_{*}$ for the [[Category|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 &gt; 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|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|co-algebra]] over the Steenrod algebra. However, after stabilizing, this objection does not apply and the following result from [[#References|[a1]]], [[#References|[a4]]], [[#References|[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|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 [[#References|[a6]]]. This can be greatly, but not completely, destabilized. See [[#References|[a7]]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  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</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  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</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  G. Carlsson,   "G.B. Segal's Burnside ring conjecture for $( \mathbf{Z} / 2 ) ^ { k }$"  ''Topology'' , '''22'''  (1983)  pp. 83–103</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  R.L. Cohen,   "Odd primary infinite families in stable homotopy theory"  ''Memoirs Amer. Math. Soc.'' , '''30''' :  242  (1981)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  R.L. Cohen,   "The immersion conjecture for differentiable manifolds"  ''Ann. of Math. (2)'' , '''122''' :  2  (1985)  pp. 237–328</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  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</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  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</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  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</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  M. Mahowald,   "A new infinite family in $\square _ { 2 } \pi _ { * } ^ { s }$"  ''Topology'' , '''16''' :  3  (1977)  pp. 249–256</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  H. Miller,   "The Sullivan conjecture on maps from classifying spaces"  ''Ann. of Math. (2)'' , '''120''' :  1  (1984)  pp. 39–87</td></tr></table>