I would like to understand the cell structure of integrally oriented Thom spectra. A Thom spectrum over a space $X$ is something you can build from a stable spherical bundle, which is classified by a map $\varphi: X \to BGL_1 \mathbb{S}$. When $R$ is an $A_\infty$ ring spectrum, it's possible to deloop $GL_1 R$ once to get $BGL_1 R$, and the spherical bundle $\varphi$ is said to be $R$-orientable when the map $$X \xrightarrow{\varphi} BGL_1 \mathbb{S} \xrightarrow{BGL_1 \eta} BGL_1 R$$ is null. The point of an orientation (i.e., a choice of lift of $\varphi$ to the fiber of $BGL_1 \eta$) is that it buys you a Thom isomorphism $$R \wedge T(\varphi) = R \wedge \Sigma^\infty_+ X.$$

$\newcommand{\Z}{\mathbb{Z}}\renewcommand{\S}{\mathbb{S}}\renewcommand{\phi}{\varphi}\newcommand{\sm}{\wedge}\newcommand{\Susp}{\Sigma}\newcommand{\Loops}{\Omega}$I would like to specify that my bundle is $H\Z$-oriented --- I think that'll be important for a couple reasons below. What it means homotopically for $\phi$ to be $H\Z$-orientable is evident by looking at the homotopy groups of $BGL_1 R$, which are given by the following formula: $$\pi_n GL_1 R = \begin{cases}(\pi_0 R)^\times & \hbox{when $n = 0$,} \\ \pi_n R & \hbox{otherwise.}\end{cases}$$ In the case of $BGL_1 \S$, these groups are complicated because $\pi_n \S$ is complicated, but at least we know $\pi_1 BGL_1 \S = \Z/2$. The spectrum $H\Z$ is easier, where we have the complete calculation $BGL_1 H\Z = \Susp H\Z/2$, and the map $BGL_1 \eta$ is an isomorphism in degree $1$. Altogether this means...

- ... that an $H\Z$-oriented spherical fibration has a unique lift to the fiber, and
- ... that the fiber has known homotopy type as well. It is given as a connected cover $$\operatorname{fib}(BGL_1 \S \to BGL_1 H\Z) = \Loops^\infty \S^1[2, \infty).$$

Now with all the pieces in place, let me ask the baby version of my question first: suppose that $X$ takes the form $X = S^n$, $n \ge 2$. Then the spherical bundle $\phi$ is actually selecting an element $\omega \in \pi_{n-1} \S$ in the stable homotopy groups of spheres.

Can the Thom spectrum $T(\phi)$ be identified with $M_0(\omega)$, the cone on $\omega$ with bottom cell in dimension $0$?

This is true in the paltry number of cases that I know classically: the projective spaces $\mathbb{R}\mathrm{P}^1$ and $\mathbb{C}\mathrm{P}^1$ can be identified with the spheres $S^1$ and $S^2$ respectively, and they carry reduced tautological line bundles whose Thom spectra can be identified with $\Susp^{-1} \Susp^\infty \mathbb{R}\mathrm{P}^2$ and $\Susp^{-2} \Susp^\infty \mathbb{C}\mathrm{P}^2$, which themselves are the Moore spectra $M_0(2\iota)$ and $M_0(\eta)$. (Of course, $\mathbb{R}\mathrm{P}^1$ doesn't carry an $H\mathbb{Z}$-orientable bundle, but that's not super important to me that this example is captured by an answer --- it's just motivation.) The proofs of these that I know (from Atiyah's *Thom Complexes*) don't go in a way that lend themselves to generalization along the lines of the highlighted question, though.

Now the more serious question: I'm much more interested to know what happens in the case where $X$ is a cell complex with more than one cell. Toward that end:

What information about the attaching maps of the cells of $\Susp^\infty X$ and about the action of the map $\phi$ is necessary to describe the attaching maps of the cells of $T(\phi)$?

Here's a guess at a way of phrasing a response to this question: one possible way of encoding the attaching maps of $\Susp^\infty_+ X$ is to claim perfect knowledge of the differentials in the Atiyah-Hirzebruch spectral sequence $$H_*(X; \pi_* \S) \Rightarrow \pi_* \Susp^\infty_+ X.$$ So my question could be interpreted as: what do I need to know about the map $\phi$ to use it to twist the differentials in that Atiyah-Hirzebruch spectral sequence to study the spectral sequence $$H_*(X; \pi_* \S) \cong H_*(T(\phi); \pi_* \S) \Rightarrow \pi_* T(\phi)$$ instead? (Note: the isomorphism in this last line is where I finally use the $H\Z$-orientable hypothesis. I think it's required to begin comparing cell structures using these methods.)