I think you meant to say that $H^2({\mathbb CP}^2,{\mathbb Z}_2)$ is ${\mathbb Z}_2$, not $0$.

It looks like there is a misprint in the formula and the exponent $p-1$ should be $q-1$. For example, we have $S^3 - S^0 = S^2 \times {\mathbb R}$ and $S^3 - S^1 = S^1 \times {\mathbb R}^2$.

Alternatively one could say that the line bundle over ${\mathbb {RP}}^{n-1}$ is nontrivial because its restriction over ${\mathbb {RP}}^1$ is nontrivial.

The paper of Hatcher referred to was published (in slightly modified form) as Section 2 of a paper by Pierre Lochak, Leila Schneps, and myself: "On the Teichmüller tower of mapping class groups", J. reine angew. Math. 521 (2000), 1-24. The proof of the result mentioned in Sean Lawton's answer was based heavily on parts of an earlier paper of Bill Thurston and myself.

@Ben Wieland: The vector space needs to be of countable dimension in order to have $CP^\infty$ as its projectivization. However, if the rational functions had a countable basis, this would imply that only countably many complex numbers occurred as poles of rational functions, a contradiction. This is almost counterintuitive: rational functions depend upon only countably many complex parameters (their coefficients) yet as a vector space their dimension is uncountable.

I seem to recall that Bill Thurston told me that these ideal triangulations of punctured-torus bundles are due to Troels Jorgensen. The earliest appearance of them in the literature that I am aware of is in a paper by Bill Floyd and myself, "Incompressible surfaces in punctured-torus bundles", Top. and its Appl. 13 (1982), 263-282. Gueritaud's paper seems to have a nice exposition, with nicer pictures than the hand-drawn ones in my paper with Floyd.

@ThiKu. Of the 28 exotic 7-spheres, only 16 are fiber bundles over $S^4$ with fiber $S^3$. I'm not sure who originally noted this, but it can be deduced from a paper of Itiro Tamura, "Remarks on differentiable structures on spheres" in J. Math. Soc. Japan. 13 (1961), 383-386.