Given $n$dimensional de Sitter or Antide Sitter space, $dS_n$ or $AdS_n$, what are the homotopy groups $\pi_m(dS_n)$ and $\pi_m(AdS_n)$ and how does one calculate such things?

$\begingroup$ What's your interest on these homotopy groups? $\endgroup$ – Fernando Muro Sep 15 '11 at 19:35
De Sitter space $dS_n$ is homeomorphic to $\mathbb{R} \times S^{n  1}$, so its homotopy groups are those of $S^{n  1}$, which are intractably complicated when $n > 2$.
Anti de Sitter space $AdS_n$ is homeomorphic to $\mathbb{R}^{n  1} \times S^1$, so its homotopy groups are those of $S^1$: $\pi_1(AdS_n) = \mathbb{Z}$, $\pi_m(AdS_n) = 0$ when $m > 1$.
The general rule for (finite dimensional) manifolds is that unless it has contractible universal cover (i.e., unless it is a $K(\pi, 1)$), there is no hope for computing its homotopy groups explicitly.

1$\begingroup$ It may be that what the OP calls anti de Sitter space is actually the universal cover of what you have called anti de Sitter space (cf. Hawking and Ellis, for instance). If so, then it is contractible. $\endgroup$ – José FigueroaO'Farrill Sep 15 '11 at 18:12