Also, if i'm not mistaken, steenrod gives a more direct argument in topology of fibre bundles, but he might be using the long exact sequence of a fibration (which you mentioned). Are $so (n)\times z_2$ and $o (n)$ isomorphic as topological groups Welcome to the language barrier between physicists and mathematicians
Physicists prefer to use hermitian operators, while mathematicians are not biased towards hermitian operators What is the lie algebra and lie bracket of the two groups? The generators of $so(n)$ are pure imaginary antisymmetric $n \\times n$ matrices
The question really is that simple Prove that the manifold $so (n) \subset gl (n, \mathbb {r})$ is connected It is very easy to see that the elements of $so (n. I'm not aware of another natural geometric object.
I'm looking for a reference/proof where i can understand the irreps of $so(n)$ I'm particularly interested in the case when $n=2m$ is even, and i'm really only. So, the quotient map from one lie group to another with a discrete kernel is a covering map hence $\operatorname {pin}_n (\mathbb r)\rightarrow\operatorname {pin}_n (\mathbb r)/\ {\pm1\}$ is a covering map as @moishekohan mentioned in the comment I hope this resolves the first question
U(n) and so(n) are quite important groups in physics I thought i would find this with an easy google search
