CW structure on Unitary Group I computed  the CW structure on U(n) using morse theory.I want to verify my answer.So I was wondering if someone here can supply the answer as I can't find any source by googling.Also I want to know if there is any reference for this.
 A: A simply-connected compact Lie group $G$ has the same rational homotopy type (and rational cohomology ring) of as a product of odd-dimensional spheres $S^{2m_1+1}\times\cdots\times S^{2m_r+1}$ where the $m_i$ are 
invariants called exponents and $r$ is the rank of $G$. (The exponents are related to 
many algebraic invariants of $G$).
In your case $U(n)$ is diffeomorphic to $S^1 \times SU(n)$ and the exponents
of $SU(n)$ are $1,\ldots, n-1$. For instance, in case $n=3$, $SU(3)$ has
exponents $1$, $2$ and $U(3)\sim S^1 \times S^3 \times S^5$ has a cell decomposition with one cell in each dimension $0$, $1$, $3$, $4$, $5$, $6$, $8$, $9$.  
Classical references are A. Borel http://www.ams.org/journals/bull/1955-61-05/S0002-9904-1955-09936-1/home.html and H. Samelson
http://www.ams.org/journals/bull/1952-58-01/S0002-9904-1952-09544-6/home.html.
Edit: There is the maybe easier following argument. In general, for a 
(locally trivial) fiber bundle $\pi:E\to B$ with typical fiber $F$, where
$B$ and $F$ are CW-complexes, a cell decomposition for the total space $E$ is the same
as that of the direct product $B\times F$, since each cell $c$ of $B$ is contractible
and thus the bundle above it is trivial: $\pi^{-1}(c)\cong c\times F$. 
Now consider the (principal) fiber bundle
$SU(n-1)\to SU(n)\to S^{2n-1}$ for $n\geq2$ and apply induction.  
Edit 2: The minimal number of cells in a CW-decomposition equals the sum of the Betti numbers over $\mathbf Z_2$ (or any field), by the Morse inequalities. We need only to know that $G$ and $S^{2m_1+1}\times\cdots\times S^{2m_r+1}$ have the same $\mathbf Z_2$-Betti numbers, so the minimal number of cells in this case is $2^r$, where $r$ is the rank of $G$. 
