6
$\begingroup$

When studying the stable homotopy of $BG^{\wedge}_p$, with $G$ a finite group, authors know that this abuse of notation is not dangerous because $\Sigma^{\infty}BG^{\wedge}_p$ and $(\Sigma^{\infty}BG)^{\wedge}_p$ are stable homotopy equivalent. However, that abuse is also common when $G$ is a compact Lie group, I do not know any reference that justifies this abuse unlike the finite group case (in this case $\Sigma^{\infty}BG^{\wedge}_p$ and $(\Sigma^{\infty}BG)^{\wedge}_p$ are not always stable homotopy equivalent).

What is more, some result such as the following from a Chun-Nip Lee's article:

Let $Y_1, Y_2$ be connected CW complexes such that both $H^{\ast}(Y_1, \mathbb{F}_p )$ and $H^{\ast}(Y_2, \mathbb{F}_p )$ are finitely generated as graded rings. Then the image of the ring homomorphism $\{(Y_1)^{\wedge}_p,(Y_2)^{\wedge}_p\}\rightarrow\mathrm{Hom}_{\mathbb{F}_p} (\widetilde{H}^{\ast}(Y_1, \mathbb{F}_p ), \widetilde{H}^{\ast}(Y_1, \mathbb{F}_p ))$ is finite.

What did the author mean by $(Y_i)^{\wedge}_p$?. In general, is there a criterion to know whether $\Sigma^{\infty}Y^{\wedge}_p\simeq (\Sigma^{\infty} Y)^{\wedge}_p$ or not?

Improve this question
$\endgroup$
4
  • 3
    $\begingroup$ See arxiv.org/abs/1712.07633 $\endgroup$
    – skd
    Commented Jul 15, 2020 at 18:24
  • $\begingroup$ @skd Thanks, that article proves that $\Sigma^{\infty} (BS^{1})^{\wedge}_p$ is not $p$-complete, what made me think that $BG^{\wedge p}$ in the stable homotopy context means $(\Sigma^{\infty} BG)^{\wedge p}$, although it seems counterintuitive to me. $\endgroup$
    – Victor TC
    Commented Jul 15, 2020 at 22:20
  • 1
    $\begingroup$ @VictorTC Unfortunately those kinds of notations are wildly inconsistent from paper to paper (especially when you go to somewhat old ones). There's not much to do but be extra careful. $\endgroup$
    – Denis Nardin
    Commented Jul 16, 2020 at 7:24
  • 1
    $\begingroup$ When in doubt interpret $\Sigma^\infty(X_p)$ as $(\Sigma^\infty X)_p$. Or, rather, interpret it as $(\Sigma^\infty X_p)_p$, which is equivalent to $(\Sigma^\infty X)_p$. So this is a functor that you can apply to $p$-complete spaces. (How generally are they equivalent? If you're using $\mathbb Z/p$-localization, it's easy to see that they're always equivalent. If you're using $p$-completion, I'm not sure if it's always equivalent, but usually you're using it because in that case it's equivalent to $\mathbb Z/p$-localization.) $\endgroup$
    – Ben Wieland
    Commented Jul 27, 2020 at 20:02

0

Reset to default

You must log in to answer this question.