My question is: What is the singular homology and cohomology of $SU(N)/SO(N)$ i.e. $H_n(SU(N)/SO(N),\mathbb{Z})$ and $H^n(SU(N)/SO(N),\mathbb{Z})$? Thank you!
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I'm not sure what is know about it in general, since the integral of cohomology of $SO(n)$ is quite intricate. I would suggest learning about the Serre and Eilenberg-Moore spectral sequences and making use of the fibration sequence $SU(n)/SO(n)\rightarrow BSO(n)\rightarrow BSU(n)$. I'd also suggest seeking the result for mod 2 coefficients first. You might like to try Bruner, May and Catanzaro's notes "Characteristic classes" for information on what the complexification map does to the Chern classes. – Tyrone Apr 01 '18 at 09:18
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1The calculation for $N = 3$ can be found here. – Michael Albanese May 16 '18 at 23:21