Fundamental group of the special orthogonal group SO(n)
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).
lie groups - Lie Algebra of SO (n) - Mathematics Stack Exchange
Apr 24, 2017 · Welcome to the language barrier between physicists and mathematicians. Physicists prefer to use hermitian operators, while mathematicians are not biased towards hermitian operators. …
Prove that the manifold $SO (n)$ is connected
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 ...
Homotopy groups O(N) and SO(N): $\\pi_m(O(N))$ v.s. $\\pi_m(SO(N))$
Oct 3, 2017 · I have known the data of $\\pi_m(SO(N))$ from this Table: $$\\overset{\\displaystyle\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\quad\\textbf{Homotopy …
Dimension of SO (n) and its generators - Mathematics Stack Exchange
Nov 18, 2015 · The generators of $SO(n)$ are pure imaginary antisymmetric $n \\times n$ matrices. How can this fact be used to show that the dimension of $SO(n)$ is $\\frac{n(n-1 ...
orthogonal matrices - Irreducible representations of $SO (N ...
Sep 21, 2020 · 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 ...
Are $SO(n)\\times Z_2$ and $O(n)$ isomorphic as topological groups?
Oct 19, 2019 · I am doing Exercise 4-16 in Armstrong's Basic Topology. The question is : are $SO (n)\times Z_2$ and $O (n)$ isomorphic as topological groups? (I have proved the ...
Why $\\operatorname{Spin}(n)$ is the double cover of $SO(n)$?
Dec 16, 2024 · You can let $\text {Spin} (n)$ act on $\mathbb {S}^ {n-1}$ through $\text {SO} (n)$. Since $\text {Spin} (n-1)\subset\text {Spin} (n)$ maps to $\text {SO} (n-1)\subset\text {SO} (n)$, you could …
Q. 18 from A first course in probability by Sheldon Ross
Apr 12, 2024 · 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 …
Calculating tangent spaces of Lie groups such as $SO(n)$ or $O(n)$
Sep 13, 2017 · I was asking myself what was, given either $O(n)$ or $SO(n)$ as manifolds immersed in $\\mathbb{R}^k$ for some natural $k$, the tangent space at a generic point $p ...