Can you give me a reference book where homotopy groups of O(n) are calculated?


1 Answer 1


In practice, this isn't really possible. There are fibre sequences

$$O(n-1) \to O(n) \to S^{n-1}$$

which allow you to inductively compute the homotopy groups of $O(n)$ in terms of the homotopy of $S^{k}$, for $k < n$. But the latter is one of the main open questions in homotopy theory.

Of course, real Bott periodicity tells you the homotopy groups of $O = \lim_{n\to \infty} O(n)$. By the previous fibre sequence, this is the same as $\pi_k(O(n))$ for $n>k+1$ -- the homotopy groups stabilise at that point -- since $\pi_k(S^{n-1}) = 0$ in that range. But the higher homotopy of $O(n)$ for a fixed $n$ is less tractable.

A good reference for what you can do with the fibre sequence above (and others like it) is Mimura-Toda's Topology of Lie Groups, I and II.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.