The fundamental group of a globally symmetric space $M$ of compact type is known (see Loos [1], Borel [2]). The result can be formulated as follows: it is isomorphic to the quotient $$(*) \quad \pi_1(M) = \Gamma_M/\Gamma_0$$ where $\Gamma_M$ is the lattice of $M$, given by $$\Gamma_M = \{ \gamma \in \mathfrak{t}: \, \exp_{x_0}(\gamma) = x_0 \}$$ where $\mathfrak{t}$ is a maximal flat on the tangent space $T_{x_0}M$, and $\Gamma_0 = \Gamma_{\widetilde{M}}$, the lattice of the simply connected covering $\widetilde{M}$ of $M$ (which shares the same $T_{x_0}M$ and $\mathfrak{t}$ as $M$). The isomorphism is induced by the natural inclusion $\Gamma_M \to \pi_1(M)$ as homotopy classes of geodesic loops at $x_0$, $\gamma \mapsto [\exp_{x_0}(t\gamma)]$, $t \in [0,1]$.

Does $(*)$ holds more generally for a globally symmetric space $M$ which is only compact, not necessarily of compact type? The difference being that, in this case, $\widetilde{M}$ is not necessarily compact, but a product of simply connected euclidean and compact type symmetric spaces. Furthemore, the (connected component) of the isometry group need not be semisimple, but only compact. Finally, $\pi_1(M)$ can be infinite.

Nevertheless, the above constructions can be made for $M$ compact and give the correct fundamental group $\mathbb{Z}$ for $M = U(2)$: loosely speaking, only the semisimple part gets quotiented out.

Does $(*)$ holds in general for compact globally symmetric spaces? I could not find a proof in the literature, which only considers this question for compact type.

-Remark: This is loosely related to this question, however the question of identifying the fundamental group as lattice quotients was not raised there.

[1]: Borel, Semisimple Groups and Riemannian Symmetric Spaces, Hindustan Book Agency, 1998.

[2]: Loos, Symmetric spaces, vol.II - Compact spaces and Classification, W. A. Benjamin ,1969.



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy