Exact sequence for the Spin group I read that the $Spin(n)$ is the double cover of $SO(n)$ such that the following sequence is exact
$$ 1 \to \mathbb{Z}_2 \to Spin(n) \to SO(n) \to 1 $$
My first question is what information does this sequence exactly convey? I am familiar with the exact sequences one encounters in algebraic topology, for example Meyer-Vietoris type of sequences where I get information on how to compute homology groups of a more complicated space from easier ones, but I am not sure what I should understand from the sequence above. 
My second question is, that since $Spin(n)$ is the double cover of $SO(n)$ is it true that $Spin(n) \cong SO(n)/\mathbb{Z}_2$ just las $SU(2) \cong SO(3)\mathbb{Z}_2$?
 A: You can get a lot of information from this sequence by combining it with other background knowledge. For example, once you know that $Spin(n)$ is (for $n \ge 3$) even the universal cover of $SO(n)$, you know that the representation theory of $Spin(n)$ is the same as the representation theory of the Lie algebra $\mathfrak{so}(n)$, and that among these representations the representations of $SO(n)$ are the ones where this central copy of $\mathbb{Z}_2$ acts trivially. The ones that don't have this property include the spin representations, which describe spinors. Note that when $n = 3$ we have $Spin(3) \cong SU(2)$. 
On the algebraic topology side, because $\mathbb{Z}_2$ is central, this sequence actually deloops to a fiber sequence
$$BSpin(n) \to BSO(n) \to B^2 \mathbb{Z}_2$$
which turns out to tell you that an oriented smooth manifold has a spin structure if and only if a certain characteristic class in $H^2(-, \mathbb{Z}_2)$ vanishes. This class turns out to be the Stiefel-Whitney class $w_2$. 
