Does any one know why $d\_3: H^\* (X, K^0(point))\rightarrow H^{\*+3}(X,K^0(point))$ is actually extended $Sq^3$ to $\mathbb{Z} $ coefficient.



Third differential in Atiyah Hirzebruch spectral sequence
This follows from the following considerations:


1. This differential in the Atiyah-Hirzebruch spectral sequence must be a stable cohomology operation for general nonsense reasons (the first nonvanishing differential always is, no matter what the generalized cohomology theory is).
2. There are exactly two stable cohomology operations $H^\*(X) \to H^{\*+3}(X)$ with integer coefficients. One of them is zero, and the other is $\beta \circ Sq^2 \circ r$, where $r$ is reduction mod 2 and $\beta$ is the Bockstein from mod-2 cohomology to integer cohomology. This comes from a calculation of the cohomology of Eilenberg-Mac Lane spaces, which describe all possible cohomology operations; for n sufficiently large we have $H^{n+3}(K(\mathbb{Z},n)) = \mathbb{Z}/2$.
3. The $d\_3$ differential is not the zero cohomology operation. For this, it suffices to find one space for which this differential is nontrivial (and you can find this by actually calculating the complex K-groups). I believe that you can find this for $\mathbb{RP}^2 \times \mathbb{RP}^4$; perhaps someone more industrious can flesh this out?


