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?


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 $bspin \rightarrow bso \rightarrow bo$.)

  • 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

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


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