9
$\begingroup$

The direct sum of real vector bundles endows $BO=\mathrm{colim} BO(n)$ with a natural structure of abelian group up to homotopy. The same applies to the classifying spaces of all groups in the Whitehead tower of $O$, i.e., one has a natural structure of abelian group up to homotopy on $BSO$, $BSpin$, $BString$, etc.

As $w(E\oplus F)=w(E)\cup w(F)$ one sees that $$ w_1\colon BO \to K(\mathbb{Z}/2\mathbb{Z},1) $$ is a morphism of abelian groups up to homotopy, and that similarly $$ w_2\colon BSO \to K(\mathbb{Z}/2\mathbb{Z},2) $$ is a morphism of abelian groups up to homotopy. One can even make a step further and see $$ BSO \to BO\xrightarrow{w_1} K(\mathbb{Z}/2\mathbb{Z},1) $$ and $$ BSpin \to BSO\xrightarrow{w_2} K(\mathbb{Z}/2\mathbb{Z},2) $$ as ``short exact sequences of abelian groups up to homotopy''.

One can be more ambitious here. Not only $BO$ is an abelian group up to homotopy, but it is an $\infty$-loop space, i.e. $BO=\Omega^\infty bo$ for a certain connective spectrum $bo$. The same applies to $BSO$, $BSpin$,etc., and it also applies to $K(\mathbb{Z}/2\mathbb{Z},n)$ as $K(\mathbb{Z}/2\mathbb{Z},n)=\Omega^\infty\Sigma^n H\mathbb{Z}/2$. So one may hope that the above sequences are actually infinitely deloopable and come from fibrations $$ bso \to bo\xrightarrow{\Omega^{-\infty}w_1} \Sigma H\mathbb{Z}/2 $$ and $$ bspin \to bso\xrightarrow{\Omega^{-\infty}w_2} \Sigma^2 H\mathbb{Z}/2 $$ of connective spectra. Versions of this latter statement seem to appear in the literature, at least in the form "$BSO \to BO\xrightarrow{w_1} K(\mathbb{Z}/2\mathbb{Z},1)$ is a fibration of infinite loop spaces" which however I am only able to give a precise meaning by interpreting it as above. For instance one finds: "Recall that $BSpin^c$ participates in a fibration of infinite loop spaces $K(\mathbb{Z},2)\to BSpin^c\to BSO\xrightarrow{bw_2}K(\mathbb{Z},3)$'' in section 7 of Ando-Blumberg-Gepner's Twists of K-theory and TMF.

My question is:

  • Is it true that the above are indeed fibrations of connective spectra inducing the usual fibrations of topological spaces via $\Omega^\infty$?

  • Where can I find a rigorous proof of this statement?

$\endgroup$
10
$\begingroup$

Your sequences are all arise in the following standard way. Suppose $x$ is an $(n-1)$--connected spectrum and let $X = \Omega^{\infty} x$. One always has a fibration sequence $$y \rightarrow x \rightarrow \Sigma^n H\pi_n(X)$$ and applying $\Omega^\infty$ to this yields a fibration sequence of spaces $$Y \rightarrow X \rightarrow K(\pi_n(X),n).$$

$Y$ is the $n$--connected cover of $X$.

(In your situation, one has successive covers $bspin \rightarrow bso \rightarrow bo$.)

$\endgroup$
  • 1
    $\begingroup$ Ah, sure! For any spectrum $x$ one has the fiber sequence $x_{>n} \to x \to x_{\leq n}$ given by the standard ($n$-connected,$n$-truncated) $t$-structure on spectra. If $x$ is $(n-1)$-connected then $x_{\leq n}=\Sigma^n H\pi_n(x) = \Sigma^n H\pi_n(\Omega^\infty x)$. Then one applies $\Omega^\infty$. I was missing the obvious here. $\endgroup$ – domenico fiorenza Sep 28 at 5:52
11
$\begingroup$

Yes it is true. You have correctly interpreted the intended meaning of the phrase ``fibration of infinite loop spaces''. One early reference is chapter I of $E_{\infty}$ ring spaces and $E_{\infty}$ ring spectra, available at http://www.math.uchicago.edu/~may/BOOKS/e_infty.pdf

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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