Is there a relation obeyed by the 2nd universal Chern class, analogous to the 2nd universal Stiefel-Whitney class?

Question

1. The Stiefel-Whitney classes.

• 1.1: There exist $w_j \in H^j(BO(n),\mathbb{Z}_2)$ for $j\in\{1,2,\ldots,n\}$ such that $H^*(BO(n),\mathbb{Z}_2) = \mathbb{Z}_2[w_1,w_2,\ldots,w_n]$.

• 1.2: The map $BO(n) \xrightarrow{Bf} B^1\mathbb{Z}_2$ obtained from the s.e.s. $SO(n)\rightarrow O(n)\xrightarrow{f}\mathbb{Z}_2$ corresponds to $w_1 \in H^1(BO(n),\mathbb{Z}_2)$.

• 1.3: The map $BSO(n) \rightarrow B^2\mathbb{Z}_2$ obtained as the connecting map for the long fiber sequence obtained from the s.e.s. $\mathbb{Z}_2\rightarrow\mathrm{Spin}(n)\rightarrow SO(n)$ corresponds to $(Bg)^*w_2 \in H^2(BSO(n),\mathbb{Z}_2)$, where $g : SO(n)\hookrightarrow O(n)$ is inclusion.

2. The Chern classes.

• 2.1: There exist $c_j \in H^{2j}(BU(n),\mathbb{Z})$ for $j\in\{1,2,\ldots,n\}$ such that $H^*(BU(n),\mathbb{Z}) = \mathbb{Z}[c_1,c_2,\ldots,c_n]$.

• 2.2: The map $BU(n) \xrightarrow{Bf} BS^1 = B^2\mathbb{Z}$ obtained from the s.e.s. $SU(n)\rightarrow U(n)\xrightarrow{f}S^1$ corresponds to $c_1 \in H^2(BU(n),\mathbb{Z})$.

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I was wondering, is there a construction "2.3" analogous to "1.3" which expresses a relation obeyed by $c_2 \in H^4(BU(n),\mathbb{Z})$ involving canonically defined morphisms? Perhaps this involves the next step in the Whitehead tower of $U(n)$, similarly to how 1.3 involves the Spin group? Apologies if my question is imprecise, but it seems like there should be a construction for $c_2$ analogous to that of $w_2$.

Answer 1

Yes, it involves the next step in the Whitehead tower. Here's one way to say it. For $n \ge 3$ we have that $BSU(n)$ is $3$-connected and satisfies $\pi_4(BSU(n)) = \mathbb{Z}$. So we can write down a map

$$c_2 : BSU(n) \to B^4 \mathbb{Z}$$

which is an isomorphism on $\pi_4$ whose homotopy fiber is the $4$-connected cover of $BSU(n)$. This map loops to a map $SU(n) \to B^3 \mathbb{Z}$ which is similarly an isomorphism on $\pi_3$ and whose homotopy fiber is the $3$-connected cover of $SU(n)$ (which can be called a string group). And regarded as a class in $H^4(BSU(n), \mathbb{Z})$ I believe but have not checked that it is the pullback of the universal second Chern class from $H^4(BU(n), \mathbb{Z})$ (a priori we might get some multiple, but even if we don't there's a sign ambiguity I'm not sure how to resolve).

We can play this game more generally passing through the Whitehead tower of any (reasonable) path-connected space $Y$ and we'll get a sequence of cohomology invariants taking values in $H^k(X, \pi_k(Y))$, each of which is generally only well-defined provided the previous invariants vanish, such that the first $k$ invariants of a map $f : X \to Y$ vanish iff $f$ lifts to the $k$-connected stage of the Whitehead tower of $Y$, so all of them vanish iff $f$ is null-homotopic, although if $X$ is a $d$-dimensional CW complex it's only necessary to check the first $d$.