Characterizations of spectra with $\mathcal {U}$-injective cohomology which satisfy the Brown-Gitler property

Abstract:

We work in the stable homotopy category of $p$–complete connective spectra having mod $p$ homology of finite type. $H^*(X)$ means cohomology with $\mathbf {Z}/p$ coefficients, and is a left module over the Steenrod algebra $\mathcal {A}$. A spectrum $Z$ is called spacelike if it is a wedge summand of a suspension spectrum, and a spectrum $X$ satisfies the Brown–Gitler property if the natural map $[X,Z] \rightarrow \operatorname {Hom}_{\mathcal {A}}(H^*(Z),H^*(X))$ is onto, for all spacelike $Z$. It is known that there exist spectra $T(n)$ satisfying the Brown–Gitler property, and with $H^*(T(n))$ isomorphic to the injective envelope of $H^*(S^n)$ in the category $\mathcal {U}$ of unstable $\mathcal {A}$–modules. Call a spectrum $X$ standard if it is a wedge of spectra of the form $L \wedge T(n)$, where $L$ is a stable wedge summand of the classifying space of some elementary abelian $p$–group. Such spectra have $\mathcal {U}$–injective cohomology, and all $\mathcal {U}$–injectives appear in this way. Working directly with the two properties of $T(n)$ stated above, we clarify and extend earlier work by many people on Brown–Gitler spectra. Our main theorem is that, if $X$ is a spectrum with $\mathcal {U}$–injective cohomology, the following conditions are equivalent: (A) there exist a spectrum $Y$ whose cohomology is a reduced $\mathcal {U}$–injective and a map $X \rightarrow Y$ that is epic in cohomology, (B) there exist a spacelike spectrum $Z$ and a map $X \rightarrow Z$ that is epic in cohomology, (C) $\epsilon :\Sigma ^{\infty }\Omega ^{\infty }X \rightarrow X$ is monic in cohomology, (D) $X$ satisfies the Brown–Gitler property, (E) $X$ is spacelike, (F) $X$ is standard. ($M \in \mathcal {U}$ is reduced if it has no nontrivial submodule which is a suspension.) As an application, we prove that the Snaith summands of $\Omega ^2S^3$ are Brown–Gitler spectra–a new result for the most interesting summands at odd primes. Another application combines the theorem with the second author’s work on the Whitehead conjecture. Of independent interest, enroute to proving that (B) implies (C), we prove that the homology suspension has the following property: if an $n$–connected space $X$ admits a map to an $n$–fold suspension that is monic in mod $p$ homology, then $\epsilon : \Sigma ^n\Omega ^n X \rightarrow X$ is onto in mod $p$ homology.