Unitary Group - Topology

Topology

The unitary group U(n) is endowed with the relative topology as a subset of M(n, C), the set of all n × n complex matrices, which is itself homeomorphic to a 2n2-dimensional Euclidean space.

As a topological space, U(n) is both compact and connected. The compactness of U(n) follows from the Heine–Borel theorem and the fact that it is a closed and bounded subset of M(n, C). To show that U(n) is connected, recall that any unitary matrix A can be diagonalized by another unitary matrix S. Any diagonal unitary matrix must have complex numbers of absolute value 1 on the main diagonal. We can therefore write

A path in U(n) from the identity to A is then given by

The unitary group is not simply connected; the fundamental group of U(n) is infinite cyclic for all n:

The first unitary group U(1) is topologically a circle, which is well known to have a fundamental group isomorphic to Z, and the inclusion map U(n) → U(n+1) is an isomorphism on π₁. (It has quotient the Stiefel manifold.)

The determinant map det: U(n) → U(1) induces an isomorphism of fundamental groups, with the splitting U(1) → U(n) inducing the inverse.