Computation of $H^∗ (Σ\mathbb RP^n , \mathbb F_2)$ as graded $\mathbb F_2$-algebras

Question

I'm trying to compute $H^*(\Sigma\mathbb{R}P^n,\mathbb{F}_2)$ as a $\mathbb{F}_2$ graded algebra. I know that the cup product is 0 since since it is for all suspensions, but I'm not sure how to use that or how to start.I'm trying to compute $H^1(\Sigma\mathbb{R}P^n,\mathbb{F}_2)$ and I think I should use properties of the suspension but I don't know them.

I have an answer but no way to know if this is true

I'm stuck and I don't know which direction to take some hints would help a lot!

Thanks !

**Edit: **

I have used mayer vietoris for all degrees above 1 but I dont know how to compute degress 0 and 1 cans someone help ?

Answer 1

Here is my answer thanks to the comments:

Write $\mathbb{R}P^n=X$ and $\Sigma X= X_- \cup X_+$ where $X_-$, $ X_+$ are two cones and their intersection is $X$ now we use Mayer Vietoris to get :

$$... \to H^{n-1}(X_-)\oplus H^{n-1}(X_+) \to H^{n-1}(X)\to H^{n}(\Sigma X)\to H^{n}(X_-)\oplus H^{n}(X_+) \to ...$$

Since $X_-$, $ X_+$ are cones, they are contractible hence $H^{n-1}(X_-)\oplus H^{n-1}(X_+)=0$ for all $n$ ( we work with reduced cohomology) so we get an isomorphism $H^{n-1}(X) \simeq H^{n}(\Sigma X)$.