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First lets fix a prime $p$ (I really care about $p=2$ but would be happy to know about other primes as well). When localized at a prime the spectrum $ku$ (Complex connective K-theory) splits as a wedge of suspensions of a spectrum $l$. In their paper on the Cooperation algebra of the Adams Summand, Baker-Richter state that the complex orientation $MU \to l$ factors through the complex orientation of $BP$ and that it can be arranged so that the image of $x_n \in \pi_{2n}MU$ and $v_i \in \pi_{2p^i-2}BP$ go to zero in $\pi_*l$. Let $BP(n)$ denote the "truncation" of $BP$ so that $\pi_*BP(n) \cong \pi_*BP/(v_{n+1},v_{n+2},...)$. (This may not be the notation in the literature, but I am pretty sure I have seen people mention these objects).

My question is can we extend the complex orientation of $l$ over some of these other spectra? Does the structure of $l$ and $l_*BP$ as a module over $BP(n)$ change a lot when going from $BP(n)$ to $BP(n+1)$?

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ku splits as a wedge of (p-1) summands of l, so if you're mostly interested in p=2, well, ... – Tilman Mar 24 '11 at 22:05
I understand that. My question is about whether or not we can factor the complex orientation through "truncations" of $BP$. – Sean Tilson Mar 25 '11 at 2:22
does it change the question if I ask about $ku$ instead? – Sean Tilson Mar 25 '11 at 2:23
up vote 4 down vote accepted

Yes, the complex orientation can be factored through these truncations of BP. Either classical methods (the Baas-Sullivan theory of manifolds with singularity - see Baas' "On bordism theory of manifolds with singularities") or more modern methods (see e.g. Strickland's "Products on MU-modules") produce truncated Brown-Peterson $BP\langle n\rangle$ as a tower of "quotients" $$ MU \to \cdots \to BP \to \cdots \to BP\langle 2\rangle \to \ell \to H\mathbb{Z} \to H\mathbb{Z}/p $$ and this produces a sequence of compatible complex orientations on these, provided of course that you've produced compatible multiplicative structures on all of the $BP\langle n\rangle$.

The problem doesn't really change if you use $ku$. Also, note that $ku$ and $\ell$ have nicer and more natural multiplicative structures and orientations than any version of $BP\langle n\rangle$ is known to in general.

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Thanks Tyler, what about the module structures? can I naively truncate the formulas in Ravenel or Wilson? Also, it occurred to me that it might just be easier to think about $l_*MU$ instead of $l_*BP$ for my particular issue. Can one get a similar result for truncations of $MU$? Thanks for the reference! – Sean Tilson Mar 26 '11 at 1:02
Sean: Yes, there is a similar result about truncations of MU. So far as module structures, the left module structure of $\ell_*, \ell_* BP$ are all restrictions to the module structures over $BP_*, BP\langle n\rangle_*$, or any other factorization obtained from the quotient map on homotopy groups. I'm not sure which formulas you're after (the right unit, maybe?) but you can get it by modding out. – Tyler Lawson Mar 27 '11 at 13:57
One small warning about ku that I didn't say is that the standard orientation of ku isn't the one that is compatible with $BP\langle 1\rangle$; it comes from a map $MU_* \to ku_*$ that is a quotient by a different regular sequence. – Tyler Lawson Mar 27 '11 at 14:00

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