Is $D(n)$ a Thom spectrum?

Fix $p=2$ and let $D(n)$ be a spectrum which filters the Eilenberg-Moore spectrum $H\mathbb{Z}/2$, i.e. $\mathrm{colim}\ D(n)=H\mathbb{Z}/2$. This spectrum can be considered as the cofibre of $Sp^{2^{n-1}}S^0\to Sp^{2^n}S^0$ with $Sp^kS^0$ being the $k$-th symmetric power of the sphere spectrum.

We know $D(n)$ maps to a Thom spectrum, $H\mathbb{Z}/2$, and I think we can identify it with stable summand of a Thom spectrum. I wonder whether or not if it is known to be a Thom spectrum itself?!

I will be happy if, at the prime $2$, we can find a map $X\to \Omega^2S^3$ so that the Thom spectrum of the composition $X\to\Omega^2S^3\to BO$ is $D(n)$ or some (de-)suspension of it. Here, the map $\Omega^2S^3\to BO$ is the extension of $S^1\to BO$, representing a generator of $\pi_1BO$, to a double loop map, using the infinite loop structure on $BO$ provided by the Bott periodicity; it is known due to Mahowald that the Thom spectrum of $\Omega^2S^3\to BO$ is $H\mathbb{Z}/2$.

• I don't understand your definition of $D(n)$. If you define it as the cofibre then it doesn't have a map to $HZ/2$. – Oscar Randal-Williams Sep 27 '15 at 17:56
• @OscarRandal-Williams As far as I know, the definition is standard. I presume the easiest way to see the existence of such a map is the cohomology; $H^*(D(n);\mathbb{Z}/2)$ is monogenic over one generator of dimension $0$, with basis elements corresponding to Steenrod operations of length no more than $n$. I don't know of a more geometric argument to construct such a map $D(n)\to H\mathbb{Z}/2$ and I need to think about it; it might be somehow an answer to my question. A reference for above is Section 4 of Mitchell-Priddy's paper on Splitting by Steinberg idempotent (Proposition 4.3). – user51223 Sep 27 '15 at 19:35
• Isn't that the cohomology of the symmetric power, not of the cofibre of the map between symmetric powers? – Oscar Randal-Williams Sep 27 '15 at 19:49
• The cohomology of symmetric power is a little bit different; its basis has been computed by Nakaoka, and consists of such operation with length no more than $n$ and not belonging to the ideal geneated by Bockstein, I guess. It is recorded in Theorem 4.1 of Mitchell-Priddy. I think one motivation for defining $D(n)$ was to generalise Dold-Thom on $Sp^\infty S^0=H\mathbb{Z}$. – user51223 Sep 27 '15 at 19:55
• Just so I don't sound like an idiot in the first substantial comment... are the most-locally-scoped definitions available in, say, this paper? – Jesse C. McKeown Sep 27 '15 at 21:12