@DenisNardin Denis, I remembered, my mean is this: Is $S^2$ a Group?

Is $\pi _n (A \times B) = \pi _n (A) \times \pi _ n (B) $ and if not under what condition is true?

Leaving academia? @EricPeterson I hope you still drop by here once in awhile.

@DenisNardin @ArunDebray the notation was super confusing. It wasn't clear if HF_2^HF_2 was some exponential or a smash product.

@SaalHardali Free modules have the property that a map of R-modules is null homotopic if it is 0 in homotopy.

@AlirezaBadali This follows from category theory since maps into a product are pair of maps.

@SeanTilson So is $\pi _n (A \times B) = \pi _n (A) \times \pi _ n (B) $ true?

Is density a topological property?

@PiotrPstrągowski Perhaps also in arxiv.org/pdf/math/9810178.pdf, although it is a bit light on details!

@PiotrPstrągowski Drew's thesis is another good reference for this stuff :)

If $f:(X,T_1) \to (Y,T_2)$ be a continuous mapping and $H$ is dense in the $X$ so is $f(H)$ is dense in the $Y$?

@CharlesRezk Thanks for the answer! May I ask you, how do you think these charts were computed? The author says "by standard computations" but I'm not able to find any reference (the basic one one read in order to get used to ass) which shows any computation of the multiplicative structure.

@LuigiM That's an excellent question!

@DylanWilson Ha, thanks! I don't think I went too much further than Nave though

@LuigiM Mike Hopkins always recommended the paper by Mahwald-Milgram ("Operations which detect $Sq^4$ in connective $K$-theory ...") in order to learn about these kinds of computations, though experience suggests that that is actually a very hard paper to pick up and read if you don't already know what is going on.

The first chart on that page in the Freed-Hopkins paper is actually describing the calculation in the first part of section 4 of Mahowald-Milgram (e.g., Cor 4.6). If that helps ...

@CharlesRezk I will surely have a look at them! Thanks for the two papers