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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). If we restrict $\operatorname {pin}_n (\mathbb r)$ group to $\operatorname {spin}_n (\mathbb r. Welcome to the language barrier between physicists and mathematicians

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Physicists prefer to use hermitian operators, while mathematicians are not biased towards hermitian operators I hope this resolves the first question 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 have known the data of $\\pi_m(so(n))$ from this table The generators of $so(n)$ are pure imaginary antisymmetric $n \\times n$ matrices 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. I'm not aware of another natural geometric object. Each of 20 families selected to take part in a treasure hunt consist of a mother, father, son, and daughter Assuming that they look for the treasure in pairs that are randomly chosen from the 80

Are $so (n)\times z_2$ and $o (n)$ isomorphic as topological groups

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