Non-contractible closed paths in U(2)

Question

How can I know if a closed path in the group manifold of U(2) is non-contractible? For SO(3) there is a simple geometric way to know when a loop is non-contractible (see for instance section 1 of http://www.damtp.cam.ac.uk/user/examples/D18S.pdf ). However, I could not find a reference which analyses non-contractible loops for U(2). In particular, I am interested in an U(2) which is a subgroup of SU(3) and considering closed paths in the U(2) group manifold of the form $ h(t) = \exp( 2it A) $ where t is a parameter such that $0 \leq t \leq 2 \pi $ and A is an element of the Lie algebra associated to U(2). An important observation is that $ U(2) = [SU(2) \times U(1)] / Z_2 $ and not just $ SU(2) \times U(1) $.

Answer 1

In fact $(z,A)\mapsto{\rm diag}(z,1,\cdots,1)A$ is a homeomorphism $U(1)\times SU(n)\to U(n)$, although it's not a group homomorphism. Note $z$ may be recovered from $g\in U(n)$ as $z=\det g$.

The projection $U(1)\times SU(n)\to U(1)$ corresponds to $\det:U(n)\to U(1)$, which induces an isomorphism $\pi_1(U(n))\to\pi_1(U(1))$ because $SU(n)$ is simply connected. One can thus detect the homotopy class of a path in $U(n)$ by applying the determinant to it and looking at the image in $U(1)$. Since $ \det(\exp(X))= \exp(\mathrm{tr}(X))$, the $1$-parameter subgroup $t\mapsto\exp(tX)$ will have kernel $\mathbb{Z}$ (thus, a loop) iff $\mathrm{tr}(X)=2\pi n$ for some integer $n\in\mathbb{Z}$, in which case the restriction to $[0,1]$ is homotopic to the $n$th power of the loop $z\mapsto\mathrm{diag}(z,1,\cdots,1)$, a generator for $\pi_1(U(n))\cong\mathbb{Z}$.