# How are characteristic classes morphisms of infinite loop spaces? (if they are)

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 $$bspin \rightarrow bso \rightarrow bo$$.)
• 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. – domenico fiorenza Sep 28 '19 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