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I am learning about the AHSS (for complex K-theory) by trying to compute the K-theory of some spaces. I have heard that the AHSS is functorial (maps of spaces induce maps of spectral sequences). Is there a nice example of spaces $X$, $Y$ and a map $X \rightarrow Y$ such that we know the K-theory of one of these spaces and the induced map on their AHSS's helps us to compute the K-theory of the other space? In most of the situations I can think of, either the AHSS of one of the spaces collapses at $E_2$, making the map between the AHSS's useless, or I don't know very much about the AHSS of either space beyond $E_2$, making it hard to begin.

I would also be very interested in where I can find some interesting calculations with the AHSS.

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up vote 6 down vote accepted

Well, functoriality implies that differentials in AHSS are (higher) operations — and these operations can (sometimes) be identified by considering some special cases (and using functoriality, of course). Example: $d_3$ coincides with $Sq^3$.

One example where K(X) is computed using K(Y) and functoriality of AHSS (well, and some geometrical realization of the exact sequence $0\to\mathbb Z\to\mathbb Z\to\mathbb Z/2\to0$) is computation of $K(\mathbb RP^{2n+1})$ from $K(\mathbb CP^n)$.

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