Calculating the cohomology of $\mathbb{R}P^{n}$ using the cup product

Question

I'm currently working on the calculation of the cohomology of $\mathbb{R}P^{n}$ using the cup product as done in Hatcher's Algebraic Topology here (page 220). I am moderately comfortable with most of the technical machinery used, but I am confused at the very start where he says that we can induct on $n$ to get the result. I'm confused what exactly the inductive hypothesis is here. Are we supposing only that the cup product of a generator of $H^{i}(\mathbb{R}P^{n})$ with a generator of $H^{j}(\mathbb{R}P^{n})$ gives a generator of $H^{n}(\mathbb{R}P^{n})$ whenever $i+j=n$? If so, how do we know $\textit{a priori}$ that the cohomology has one generator in each degree? On the other hand, if we are assuming that fact as part of our inductive hypothesis, how do we get the base case? I can get the base case from cellular homology and using the universal coefficient theorem, but Hatcher doesn't seem to be appealing to that.

I think answers to the above questions should answer the following question, but how exactly is that sufficient to prove the result anyway?

I'm sorry if this is a stupid question, but I'm just having trouble following the logic of the argument, rather than the technical tools used.

Thanks

Answer 1

After reading some of the material before the proof, I'll guess that Hatcher is assuming that we have already computed the cohomology groups, and we are now only interested in the ring structure.

I find that Hatcher's writing alternates between being too expansive and too dense. I suspect that he finds the writing to be "just right". :)

Answer 2

As others said, I think Hatcher assumed you know the additive group structure of $H^{*}(\mathbb{RP}^{n})$ already. You can of course compute it in a number of ways including simplical cohomology and start from $H^{1}(\mathbb{RP}^{1})$, then induct "up". You may be right you should not allow yourself to use universal coefficient theorem or Poincare duality, since then the result is almost obvious. In fact, the proof in page 221 heavily used tools ranging from celluar cohomology, excision, Kunneth formula. So you should not be too worried!