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Can someone concisely list all characteristic classes (i.e., the cohomology classes $H^*(BX,A)$ of the corresponding classifying spaces) for the most relevant structure groups $X$ such as $O(n)$, $SO(n)$, $U(n)$, $SU(n)$ and finite abelian groups $A$ such as $\mathbb{Z}_2$ and $\mathbb{Z}$, and others if you feel like? Additionally, the relations between the different classes would be interesting, such as restriction of the $\mathbb{Z}$-classes to $\mathbb{Z}_2$ classes.

I know that the $\mathbb{Z}_2$-valued classes for $O(n)$ are polynomials of Stiefel-Whitney classes, but I'm having trouble finding the results for $\mathbb{Z}$-valued classes. Anyways, the results seem quite scattered, and I think it would be good if there was an easy-access place where all (or at least the most important ones of) the classes are summarized. If such a place exists already, I'd be happy with a link of course.

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    $\begingroup$ For $U(n)$ and $SU(n)$ it's known that all the classes are polynomials in the Chern classes. For abelian groups this is group cohomology. For integral cohomology of finite cyclic groups everything is polynomial in the obvious class in $H^2( \mathbb Z/n, \mathbb Z)$ and integral cohomology of other abelian groups can be computed from this and the Kunneth formula. $\endgroup$
    – Will Sawin
    Dec 13, 2021 at 19:26
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    $\begingroup$ SO and O: jstor.org/stable/2044298 $\endgroup$
    – mme
    Dec 13, 2021 at 19:41
  • $\begingroup$ Stong's "Notes on Cobordism Theory" might be of use. $\endgroup$
    – pancini
    Dec 13, 2021 at 23:19

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