Recovering algebraic information about $\mathbb R^n$ from the topology of $\mathbb R P^n$. One way to define $\mathbb R P^n$ is by identifying it with all $1$-dimensional (linear algebraic) subspaces of $\mathbb R^{n+1}$. 
My question is vague because I know very little about this subject, so I'm hoping that others can both clarify the question and provide potential answers or references:
We have (for example) that $\mathbb R P^2$ is nonorientable, it can't be embedded in $\mathbb R^3$, it is compact we can calculate its homology, it fits into the sequence $S^0 \to S^n \to \mathbb R P^n$ etc.
Basically, my point is that there are some interesting things to say about the shape or geometry of $\mathbb R P^n$, and I was wondering if there is a useful way to recover or build up some correspondence between these facts and algebraic ones regarding the $1$-subspaces of $\mathbb R$. 
More explicitly: what are examples of theorems regarding $\mathbb R^n$ as a vector space purely from studying the topological data $\mathbb R P^n$
 A: This is an elaboration of my comment about certain algebra structures on $\mathbb{R}^n$ and their relationship with the topology of $\mathbb{R}P^{n-1}$. 
The connection is this: suppose $A$ is a finite-dimensional division algebra of dimension $n$ over $\mathbb{R}$.  The multiplication on $A$ is a bilinear map $\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$.  By linearity, this descends to a map $\mathbb{R}P^{n-1} \times \mathbb{R}P^{n-1} \to \mathbb{R}P^{n-1}$.  Moreover, this map is axial, i.e., its restriction to $* \times \mathbb{R}P^{n-1}$ or to $\mathbb{R}P^{n-1} \times *$ is homotopic to the identity.  

Theorem (Hopf-Stiefel).
  An axial map $\mathbb{R}P^{n-1} \times \mathbb{R}P^{n-1} \to \mathbb{R}P^{n-1}$ exists only when $n$ is a power of two.  

Its proof is simple.  On mod 2 cohomology, the map induces a ring homomorphism $\mathbb{Z}/2[x]/(x^n) \to \mathbb{Z}/2[a]/(a^n) \otimes \mathbb{Z}/2[b]/(b^n)$.   The axial condition implies that $x \mapsto a + b$.  Therefore, $(a+b)^n \equiv 0 \pmod{2, a^n, b^n}$.  Expanding this using the binomial theorem, this means that $\binom{n}{i}$ is even for $0 < i < n$, which only happens when $n$ is a power of two.  
Of course, it's not so surprising that this is a special case of a condition about bilinear maps preserving a quadratic form on Euclidean spaces where the dimensions are allowed to vary.  
Here is another theorem of this type, proved using the topological $K$-theory of $\mathbb{R}P^{n-1}$ instead of ordinary cohomology. 

Theorem (Atiyah). 
  An axial map $\mathbb{R}P^{n-1} \times \mathbb{R}P^{n-1} \to \mathbb{R}P^{n-1}$ exists only when $2^{\lfloor (n-1)/2 \rfloor - i + 1}$ divides $\binom{n+i-1}{i}$ for $0 < i \leq \lfloor (n-1)/2 \rfloor$.

In this situation, this is stronger than the Hopf-Stiefel condition.  As a corollary, the Atiyah condition shows 

Corollary. 
  Every finite-dimensional division algebra over $\mathbb{R}$ must have dimension $1$, $2$, $4$, or $8$.  

A good reference for this is chapter 12 of Compositions of quadratic forms by D.B. Shapiro, which is available on his webpage. 
Finally, I can't resist pointing out that these conditions admit generalizations to arbitrary fields that are not of characteristic $2$, due to Dan Dugger and Dan Isaksen.  The strategies are the same, but they use things like motivic cohomology and algebraic $K$-theory of "deleted quadrics", which play the role of real projective spaces there.  
