15
$\begingroup$

The Thom spectrum MO is a module over the ring spectrum π≤0S=HZ, where S is the sphere spectrum. In particular, MO is equivalent to the Eilenberg-MacLane spectrum Hπ*(MO). On the other hand, MU and MSO are not modules over π≤0S, because they have nontrivial k-invariants.

I wonder if the above result about MO can be generalized to other Thom spectra like MU and MSO by considering higher truncations π≤kS of the sphere spectrum. Such a result would be interesting because it is related to the question of representing Thom spectra as (weak) simplicial objects in the k-category of symmetric monoidal (k+1)-groups (k-groupoids with invertible objects), in the sense that interpreting symmetric monoidal (k+1)-groups as stable homotopy k-types and taking the homotopy colimit should give back the Thom spectrum under consideration. Such models are interesting because they are more strict then (say) Segal's Γ-spaces. A negative answer would necessarily preclude the existence of such models because a stable homotopy k-type is a module over π≤kS, and the homotopy colimit of a simplicial diagram of modules over π≤kS in the category of spectra is a again a module over π≤kS.

Is MU or MSO a module over the ring spectrum π≤kS for some k>0? Same question for KU and KO.

(Incidentally, the above result for MO implies that MO can be represented as a simplicial abelian group. I wonder if there is a geometric model for MO as a simplicial abelian group along the lines of Galatius-Madsen-Tillmann-Weiss theorem, i.e., n-simplices should be related to unoriented n-manifolds. Such a model would necessarily make explicit use of properties of unoriented manifolds as opposed to oriented or stably complex manifolds.)

$\endgroup$
1
  • 2
    $\begingroup$ I would be very surprised if this worked for $MU$. $\endgroup$ Mar 12, 2013 at 23:42

1 Answer 1

25
$\begingroup$

For any $k$ and any $0<n<\infty$, $K(n)\wedge \pi_{\leq k}S=0$. Indeed, this is true for any spectrum with finitely many homotopy groups, since $K(n)\wedge H\mathbb{Z}=0$ and any such spectrum has a finite filtration into Eilenberg-MacLane spectra. If a spectrum $M$ admits a $\pi_{\leq k}S$-module structure, then $M$ is a retract of $\pi_{\leq k}S\wedge M$, and so $K(n)\wedge M$ is a retract of $K(n)\wedge\pi_{\leq k}S\wedge M=0$ and hence $K(n)\wedge M=0$. But this is not true for $n=1$ and all of your examples (for $MSO$, you must work at an odd prime).

$\endgroup$
1
  • 1
    $\begingroup$ Very nice, thanks a lot. Can this argument be extended to the more general case of (not necessarily group-like) E_∞-spaces, as opposed to connective spectra? In other words, can MU, MSO, KU, or KO be represented as a homotopy colimit of k-truncated E_∞-spaces in the ∞-category of E_∞-spaces for some k? $\endgroup$ Mar 13, 2013 at 17:55

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.