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Different definitions of the complex Grassmannian

I have come across two different definitions of what I suspect is the same object. Both are called the complex Grassmannian: 1: $U(n)/U(k)\times U(n-k)$ 2: $SU(n)/(S(U(k)\times U(n-k))$ What is the ...
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222 views

Writing down cohomology groups of the complex Grassmannian

I am studying a homogeneous space and would like to know its cohomology groups. Using some sequences and fibrations I have figured out some of these groups, but largely in terms of the cohomology ...