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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). I'm particularly interested in the case when $n=2m$ is even, and i'm really only. The generators of $so(n)$ are pure imaginary antisymmetric $n \\times n$ matrices

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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

I have known the data of $\\pi_m(so(n))$ from this table

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 in linear algebra right now and we're mostly just working with vector spaces, but they're introducing us to the basic concepts of fields and groups in preparation taking for abstract algebra la. I'm working through problem 4.16 in armstrong's basic topology, which has the following questions Prove that $o (n)$ is homeomorphic to $so (n) \times z_2$ Suppose that i have a group $g$ that is either $su(n)$ (special unitary group) or $so(n)$ (special orthogonal group) for some $n$ that i don't know