coming back to this comparison between $ku$ and $\mathbb{S}[\mathbb{C}P^\infty]$... alas, I'm confused about basepoints. We have the inclusion ${1}\times BU(1)\to \mathbb{Z} \times BU$, "inclusion of line bundles into virtual vector bundles". But the latter space is an infinite loop space, therefore we get an infinite loop map $Q\mathbb{C}P^\infty \to \mathbb{Z} \times BU$. I'm confused about getting from here to an actual map $\mathbb{S}[\mathbb{C}P^\infty]\to ku$, because of basepoints
in the notation $\mathbb{S}[\mathbb{C}P^\infty]$ there is an "added disjoint basepoint" which I don't see where it enters
(and the spectrum map wouldn't really come from $\mathbb{S}[\mathbb{C}P^\infty]=\Sigma^\infty_+(\mathbb{C}P^\infty)$ but rather from an $\Omega$-spectrum replacement of that)
@BrunoStonek You are adjoining it wrong. The map $\mathbb{CP}^∞→BU×\mathbb{Z}$ is not a map of pointed spaces, so the adjunction goes from $Q(\mathbb{CP}^∞_+)→BU×\mathbb{Z}$ or, more naturally $S[\mathbb{CP}^∞]=\Sigma^∞_+\mathbb{CP}^∞→ku$
(also, you generally do not need to fibrantly replace objects when they are the source of the maps. Although I was doing it implicitly anyway...)
@DenisNardin yeah, I'm aware of this, it's just the presence of $Q$ that led me to say that... Now, on the point business, I'm still confused. why don't you get an extra point in $BU\times \mathbb{Z}$, too?
Why would you? $BU×\mathbb{Z}$ is a pointed space (pointed at the 0 virtual space). You can add a basepoint when you map from a space to a pointed space
(said in a more formal way: $(-)_+$ is the left adjoint of the forgetful functor $\mathrm{Top}_*→\mathrm{Top}$)
Are there small DGAs known that are quasiisomorphic to cochains on BG with coefficients in some ring R ? In the special case $G=(Z/p)^n$ with $F_p$ coefficients that DGA seems to be formal for p=2 and it cannot be formal for p>2. Are there (for p>2) some smaller (than all cochains/ than the cobar resolution) DGAs known that are quasiisomorphic to the singular cochain DGA ?
What's the original source or a good modern reference of the fact that K/p splits into sum of suspensions of Morava K(1)'s and Adams operations on them? I knew those facts, but never seen a clearly written stuff on them, and I am not able to produce a proof myself... I guess they were in one of Adams J(X) paper, but checking all of them could be quite painful
it's not going to be interesting for them (it would be more suitable for elementary school kids) but this also springs to mind mathoverflow.net/a/99237/6249
Hah. Yeah. I remember my first algebraic topology class with Jack Morava. He was saying how we could (and probably should) teach homotopy theory to 8 year olds.
@HenrikRüping I believe that you can do the following.
Suppose R is F_3 and G is cyclic of order three. Take a DGA which is a noncommutative polynomial algebra on two generators x and y in (cohomological) degree 1, and the differential satisfies dx = 0, dy = x^2. Then x represents the generator of H^1(G; F_3) and xy + yx (representing the Massey product <x,x,x>) represents the generator of H^2(G; F_3).
For G = C_p and R = F_p something similar works: take a polynomial algebra in (p-1) noncommuting variables that I'll call x, y_2, y_3, ... y_{p-1}, with d(y_2) = x^2, d(y_3) = x y_2 + y_2 x representing <x,x,x>, and generally d(y_i) a representative for the i-fold bracket <x,x,...,x>.
@DenisNardin I did, and he mentioned that on K(1) the Z_p^* action induced by action on multiplicative FGL coincide with Adams operation, without proof, and I did not find anything about the splitting of K/p, maybe I missed something there....
@MingcongZeng I think you can use a linear combination of the Adams operations on K/p = K ^ M(p) to make a map that's idempotent on homotopy, and then use that to split a summand off?
