Homotopy of Brown-Gitler spectra

Question

Let $A^\vee = \mathbb{F}_2[\bar\xi_1, \bar\xi_2, ...]$ be the mod-2 dual Steenrod algebra. One can define a weight filtration on $A^\vee$ by setting $wt(\bar\xi_i)=2^i$ and $wt(xy)=wt(x)wt(y)$. There are subcomodules $B_0(k) \subseteq A^\vee$ for each $k \geq 0$ called the Brown-Gitler comodules defined as the span of monomials of weight $\leq k$. These comodules arise as the mod-2 homology of the Brown-Gitler spectra, denoted here as $H\mathbb{Z}_k$; i.e., we have that $H_*(H\mathbb{Z}_k; \mathbb{F}_2) = B_0(k)$.

What are the homotopy groups $\pi_*H\mathbb{Z}_k$? I know that the $H\mathbb{Z}_k$ are finite-cell complexes, but running the Adams-Novikov doesn't seem to give me any substantial information. Any information or direction would be appreciated.

Answer 1

This is not an answer but perhaps gives interesting context. The Brown-Gitler spectra are (up to suspension) wedge summands in the spectrum $\Sigma^\infty_+\Omega^2S^3$. Up to homotopy there is a unique nontrivial double loop map $\alpha\colon\Omega^2S^3\to BO$, which can be regarded as giving a virtual vector bundle over $\Omega^2S^3$. It is known (I think due to Mahowald) that the Thom spectrum $(\Omega^2S^3)^\alpha$ is just the mod $2$ Eilenberg-MacLane spectrum $H$. This means that we have a Thom isomorphism $H_*(\Omega^2S^3)_+\simeq H_*H$. If this was an isomorphism of $H_*H$-comodules then we would have the same Adams Spectral Sequence $E_2$ term for $\Sigma^\infty_+\Omega^2S^3$ and $H$, which would give $\pi_*^S(\Omega^2S^3)_+=\pi^S_0(\Omega^2S^3)_+=\mathbb{Z}/2$. However, the Thom isomorphism does not in fact preserve comodule structures so this does not work. Nonetheless, there is a very nice picture of how things fit together, explained in a paper of Steve Mitchell (Power series methods in unoriented cobordism). There is a standard interpretation of $\operatorname{spec}(H_*H)$ as $\operatorname{Aut}_1(G_a)$, the scheme of automorphisms of the additive formal group of the form $f(x)=\sum_{i\geq 0}a_ix^{2^i}$ with $a_0=1$. We can also consider the scheme $\operatorname{End}_0(G_a)$ of endomorphisms with $a_0=0$, and this can be identified with $\operatorname{spec}(H_*(\Omega^2S^3)_+))$. The Steenrod coalgebra structures of $H_*H$ and $H_*(\Omega^2S^3)_+$ correspond to he actions of $\operatorname{Aut}_1(G_a)$ on $\operatorname{Aut}_1(G_a)$ and $\operatorname{End}_0(G_a)$ by right composition.The Thom isomorphism corresponds to the isomorphism $\operatorname{End}_0(G_a)\simeq\operatorname{Aut}_1(G_a)$ given by adding $x$. One can probably illuminate some features of the Adams Spectral Sequence for $\pi^S_*(\Omega^2S^3_+)$ from this point of view. However, as Nick Kuhn said, you cannot expect to calculate this completely. Indeed, the sphere spectrum $S^0$ (which can be regarded as the zeroth Brown-Gitler spectrum) is evidently a retract of $\Sigma^\infty_+\Omega^2S^3$, and we certainly cannot calculate $\pi^S_*(S^0)$. The calculations should get easier for higher Brown-Gitler spectra, but only by a finite ratio.