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I studied Bott periodicity theorem for unitary group $U(n)$ and ortogonl group O$(n)$ using Milnor's book "Morse Theory". Is there a method, using this theorem, to calculate $\pi_{k}(S^{n})$? (For example $U(1) \simeq S^1$, so $\pi_1(S^1)\simeq \mathbb{Z}$).

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It depends on the values of $k$ and $n$. It is trivial to compute the groups for $0 \leq k \leq n$, but rather non-trivial for $k > n$. –  user02138 Jun 29 '13 at 15:19
    
ok... for $k < n$ are all trivial. I look for to find non trivial groups –  ArthurStuart Jun 29 '13 at 15:36
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up vote 4 down vote accepted

In general, no. However there is a strong connection between Bott Periodicity and the stable homotopy groups of spheres. It turns out that $\pi_{n+k}(S^{n})$ is independent of $n$ for all sufficiently large $n$ (specifically $n \geq k+2$). We call the groups

$\pi_{k}^{S} = \lim \pi_{n+k}(S^{n})$

the stable homotopy groups of spheres. There is a homomorphism, called the stable $J$-homomorphism

$J: \pi_{k}(SO) \rightarrow \pi_{k}^{S}$.

The Adams conjecture says that $\pi_{k}^{S}$ is a direct summand of the image of $J$ with the kernel of another computable homomorphism. By Bott periodicity we know the homotopy groups $\pi_{k}(SO)$ and the definition of $J$, so the Bott Periodicity theorem is an important step in computations of stable homotopy groups of spheres (a task which is by no means complete).

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