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I'm not aware of another natural geometric object. I'm particularly interested in the case when $n=2m$ is even, and i'm really only. 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).

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Welcome to the language barrier between physicists and mathematicians I'm looking for a reference/proof where i can understand the irreps of $so(n)$ Physicists prefer to use hermitian operators, while mathematicians are not biased towards hermitian operators

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. The generators of $so(n)$ are pure imaginary antisymmetric $n \\times n$ matrices I have known the data of $\\pi_m(so(n))$ from this table

A son had recently visited his mom and found out that the two digits that form his age (eg :24) when reversed form his mother's age (eg Later he goes back to his place and finds out that this whole 'age' reversed process occurs 6 times And if they (mom + son) were lucky it would happen again in future for two more times. 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

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 If we restrict $\operatorname {pin}_n (\mathbb r)$ group to $\operatorname {spin}_n (\mathbb r.