Tinyboss said:Because R is the universal cover of U(1), and e^ix is a covering map.
U(1) is a group that represents the unit circle in the complex plane. It is a fundamental group of a simply connected space, meaning that all loops in the space are contractible to a point. This property allows for a unique mapping from the simply connected space to U(1) through R.
A map from a simply connected space to U(1) can be thought of as a rotation in the complex plane. This rotation can be represented by a real number, which is where the factor of R comes in. The real number represents the angle of rotation, and thus the map factors through R.
A simply connected space is important because it allows for a unique mapping to U(1) through R. As mentioned before, all loops in a simply connected space are contractible to a point, which means that there is only one way to map the space to U(1) through R.
Yes, it is possible for a map from a simply connected space to factor through a different group. However, U(1) is a particularly useful group in this context because it is the fundamental group of a simply connected space, making the mapping easier and more intuitive to understand.
This type of mapping is commonly used in physics and mathematics to represent rotations and symmetries. It is also used in other fields such as computer graphics and engineering to describe transformations in space. Understanding the factors involved in this mapping can help in solving various problems and equations in these fields.