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The case of $V_1(\mathbb{R}^n) = \mathbb{R}P^{n-1}$ is in Michikazu Fujii, MR 219060 $K_{O}$-groups of projective spaces, Osaka J. Math. 4 (1967), 141--149. Perhaps some of the methods there will g …

Surprisingly, the group is uncountable. See Adams, J. F.; Clarke, F. W. Stable operations on complex $K$-theory. Illinois J. Math. 21 (1977), no. 4, 826–829.

If $Y$ is a connected CW-complex of finite type which is both an H-space and a co-H-space, then $Y$ has the homotopy type of $S^1$, $S^3$, $S^7$ or a point. This is a result of Robert West: Robert W. …

This follows from the fact that $(C_\ast(X),\partial)$ and $(H_\ast(X),0)$ are weakly equivalent chain complexes when coefficients are taken in a field (which is surpressed from the notation). The rea …

The answer is yes, if you replace the wedge axiom with the stronger direct limit axiom $h_{i}(X) = \mathrm{lim}\ h_{i}(X_{\alpha})$, where $X$ is the direct limit of subcomplexes $X_{\alpha}$. As …

Let $X$ be a connected space, and let $\lbrace x_\lambda\rbrace$ be a homogeneous basis for $H_\ast(X;\mathbb{F}_2)$. Then $$H_\ast(QX;\mathbb{F}_2) = \mathbb{F}_2 [Q^I x_{\lambda} \mid I\mbox{ admis …

You will find a careful proof of this in Eldon Dyer's monograph "Cohomology theories". It is true for any multiplicative cohomology theory $h^*$ and smooth maps $i:X\to Y$ and $j:Y\to Z$ which are $h^ …

Since for any two spectra $E,F$ we have $$ E_n(F)=[\mathbb{S}^n,E\wedge F]\cong [\mathbb{S}^n,F\wedge E] = F_n(E), $$ you may as well ask about $H_n(KU;\mathbb{Z})$, the integral homology of the compl …

The fact that singular cohomology agrees with Cech cohomology of a good cover $\mathcal{U}$ is a sort of generalised Mayer-Vietoris principle, as explained in Section II.8 and III.15 of Bott and Tu's …

The characteristic classes of the virtual bundle $E-F$ seem to have all the properties you asked for. If $c$ is a characteristic class of vector bundles satisfying a sum formula $$ c(E\oplus F) = c(E …

The natural extension homomorphism is defined in footnote 4 on page 7 of the expository paper by Gregory Landweber you are looking at. In that reference, K-theory with compact supports is defined for …

Following up on Johannes' comment, let's say you have a multiplicative fibration, meaning that $E$ and $B$ are $H$-spaces with multiplications $\mu\colon E\times E\to E$ and $\mu'\colon B\times B\to B …