# Group or manifold ? [closed]

I have a question in seeing this $$U(n)=\frac{U(n)}{U(n-1)} * \frac{U(n-1)}{U(n-2)}*\cdots *\frac{U(2)}{U(1)}*U(1)$$

So, group U(n) is written as product of quotient spaces.

Is quotient space, for example $\frac{U(n)}{U(n-1)}$ , as topological space the same as quotient gropup i.e. set of cosets?

How to prove $\frac{U(n)}{U(n-1)}$ is diffeomorphic to some sphere ? Thx

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## closed as off topic by Martin Brandenburg, Dan Petersen, S. Carnahan♦Feb 21 '13 at 13:50

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This question would be more appropriate at math.stackexchange.com or one of the other sites listed in the FAQ. –  S. Carnahan Feb 21 '13 at 13:50

## 1 Answer

I'm not sure what you mean by the star product but yes, you see a quotient G/H as the set of cosets. There are topological requirements on H though, for instance being closed etc... at least for G/H to be a differentiable manifold, I'm not sure about topological manifold.

Finally, the idea to prove the sphere statement is to notice that U(n) acts transitively on a sphere of appropriate dimension, choose a point and prove that its stabilizer is U(n-1) somehow seen inside U(n) (e.g. lower block of a matrix in U(n))

You can find more details on both my answers in the Lie Groups chapter in Warner's book (Foundations of diff geom I think)

David

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Thx,star product is ordinary product of quotient groups (I don't know symbol for multiplication in latex so I wrote from keybord). –  vejn Feb 22 '13 at 15:08