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For a given topological group $G$ there are natural transformations $$K^* \leftarrow K^*_G \overset a\to H^{**}(EG \times_G -;\mathbb Q)$$ from equivariant K-theory, the first forgetting the $G$-struc …

In October 2016, Atiyah famously posted a preprint to the arXiv, "The Non-Existent Complex 6-Sphere" containing a very brief proof $S^6$ admits no complex structure, which I immediately read and reali …

Let $G$ be a compact, connected Lie group. There is an Atiyah–Hirzebruch spectral sequence $$H^*(BG;K^*) \implies K^*(BG)$$ connecting $H^*BG$, which generally contains torsion, with $K^*BG \cong \ …

This is very late and you've no doubt learned this in the last five years, but for completeness, the relation is indeed that they are linked by completion and the Chern character, as suggested in one …

For any interested latecomers who somehow discover this question in the future, I've found a very low-tech answer, bootstrapping from the low-tech answer to Is a Lie group equivariantly formal under c …