# Thom isomorphism's effect on module structure of n-oriented spectra

This question is specifically related to the spectra $X(n)$ used in Devinatz, Hopkins and Smith's proof of the nilpotence conjectures, but any general answer in terms of the Thom isomorphism would also be appreciated.

The spectra $X(n)$ are Thom spectra coming from the maps $\Omega SU(n)\to\Omega SU\simeq BU\to BF$. We say that a spectrum $E$ has a complex orientation of degree $n$, or is $n$-oriented, if there is a class $x\in \tilde{E}^2(\mathbb{C}P^n)$ whose restriction to $\tilde{E}^2(\mathbb{C}P^1)\cong\pi_0(E)$ is 1 (similarly to complex orientations). There is a one-to-one correspondence between $n$-orientations of $E$ and ring-spectrum maps $X(n)\to E$, again just like in the complex oriented case. We know also that there is a Thom isomorphism for such spectra: $t:E\wedge X(n)\overset{\simeq}\to E\wedge \Omega SU(n)_+$. It's not hard to see that $X(n+1)$ is $n$-oriented, with the orientation coming from the Thomification of the map $\Omega SU(n)\to\Omega SU(n+1)$. Moreover, $X(n+1)$ is an $X(n)$-algebra. My question is whether or not the Thom isomorphism respects the module structure $X(n)\wedge X(n+1)\to X(n+1)$. That is, is there an obvious map $X(n+1)\wedge\Omega SU(n)_+\to X(n+1)$ and is its precomposition with the Thom isomorphism the same as the given module action?

• Jon, as the map $X(n) \to X(n+1)$ is a map of ring spectra, this seems to me like it likely boils down to verifying compatibility of the structure of a ring spectrum on $X(n)$ which the Thom isomorphism. Yes? Jun 13, 2013 at 3:15
• Yeah, pretty much. I'm currently looking at some oldish papers of Mark Mahowald on Thom spectra which are ring spectra, and specifically Thom spectra coming from loop spaces (hence $A_\infty$-ring spectra) to see how he does this kind of thing. I think I'm close to understanding it (though I've thought that about plenty of things before, and been miserably wrong). Jun 13, 2013 at 3:29
• Aha, yes. Okay. I was overcomplicating things I think. The $X(n)$-module structure on $X(n+1)$ is not complicated, and one can take advantage of the Thom isomorphism on $X(n+1)\wedge X(n+1)$ as well. I'll try to make sure what I've written down makes sense tomorrow. If so, I'll post it, though, of course, I'd love to see what others have to say. Jun 13, 2013 at 3:50

Recall the "shear" map for an $H$-space $X$ given by $\sigma: X\times X\to X\times X$, $\sigma(x,y)=(x,x^{-1}y)$. This is a homotopy equivalence, with homotopy inverse given by $(x,y)\mapsto (x,xy)$, I believe. However, notice that if we compose with the multiplication map, we don't get the same maps, i.e. $(x,y)\mapsto (x,x^{-1}y)\mapsto y$ rather than $xy$. Now let's say we've got some map $f:X\to BF$. Then we've got a few maps floating around, $ff:X\times X\overset{\mu}\to X\overset{f}\to BF$ whose Thom spectrum is $Th(f)\wedge Th(f)$ and $f0:X\times X\overset{\sigma}\to X\times X\overset{\mu}\to X\overset{f}\to BF$ whose associated Thom spectrum is $Th(f)\wedge \Sigma^\infty_+ X$. But since $\sigma$ is an equivalence, we must have that its Thomification is an equivalence, hence $Th(f)\wedge Th(f)\overset{\sim}{\underset{\sigma}\to}Th(f)\wedge \Sigma^\infty_+X$ is also an equivalence (inducing the Thom isomorphism). But notice that the map $\mu\circ \sigma$ also Thomifies to a map $Th(f)\wedge\Sigma^\infty_+\to Th(f)$. And basically by drawing out the commutative diagram, you'll see that this "action" of $\Sigma^\infty_+ X$ on $Th(f)$ is the same thing as going backwards along the Thom isomorphism and then applying the multiplication.
The general case for a spectrum which is $X(n)$-oriented or whatever, follows from the fact that $\Omega SU(n)\to \Omega SU(n+1)$ is an inclusion (and so we can factor the map $\Omega SU(n)\to BF$ through $\Omega SU(n)\to\Omega SU(n+1)\to BF$).