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Jul 19, 2013 at 16:42 comment added Eric Peterson Maybe you've already come across this, but the structure of $H_*(K(\mathbb{Z}/2, *); \mathbb{Z}/2)$ is known very explicitly as a Hopf ring. You can find this in the tail of Steve Wilson's book Brown-Peterson homology: An introduction and sampler. (He does it for an odd prime, but the even case is actually dramatically simpler.) Just reading that will give you a compact description of the behavior in all degrees, and I suspect that anything you'd mean by "nice" could be teased out of that.
Jul 19, 2013 at 10:15 history asked Ulrich Pennig CC BY-SA 3.0