# Homotopy Groups of de Sitter and Anti-de Sitter?

Given $n$-dimensional de Sitter or Anti-de 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?

• What's your interest on these homotopy groups? – 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.