# Homology of compact symmetric spaces

Could anyone point me to a table giving the homology of all compact symmetric spaces, i.e. $G/U$ where $G$ is a compact Lie group and $U$ the fixed points of an involution of $G$? I'd be happy even with rational coefficients, although torsion information would be great. Thanks.

• what makes you think that this list can even be finitely generated? e.g. are you aware there being only finitely many involutions (modulo conjugation) for the sequence of groups $SO(n), n=1,2,3, \ldots$ ? – J. Martel Feb 9 '14 at 1:41
• Yes, the symmetric spaces were classified by E. Cartan. See en.wikipedia.org/wiki/Symmetric_space#Classification_result for the list. However, I haven't seen anywhere a list of their homologies. Probably it was done in the 1950s, but I couldn't find it in e.g. the Borel-Hirzebruch papers. – Edgardo Feb 9 '14 at 2:37
• Have you checked in a book like Joe Wolf's "Spaces of Constant Curvature" – Marty Feb 9 '14 at 3:32
• Your question asks for $G/U$ where you've declared $U$ to be the fixed point set of an involution on $G$. This is much more general than Cartan's symmetric spaces. E.g. implies nothing about the quotient being locally homogeneous nor the involution being an isometry for the Killing form. – J. Martel Feb 9 '14 at 3:54
• Probably <a href="mathoverflow.net/questions/78717/… question </a> is relevant. Anyhow there is a book by Toda and Mimura (in Japanese, unfortunately) with a table of cohomology of classical homogeneous spaces. – user43326 Feb 9 '14 at 7:00