Simply Connected Space - Properties

Properties

A surface (two-dimensional topological manifold) is simply connected if and only if it is connected and its genus is 0. Intuitively, the genus is the number of "handles" of the surface.

If a space X is not simply connected, one can often rectify this defect by using its universal cover, a simply connected space which maps to X in a particularly nice way.

If X and Y are homotopy equivalent and X is simply connected, then so is Y.

Note that the image of a simply connected set under a continuous function need not to be simply connected. Take for example the complex plane under the exponential map, the image is C - {0}, which clearly is not simply connected.

The notion of simple connectedness is important in complex analysis because of the following facts:

• If U is a simply connected open subset of the complex plane C, and f : U → C is a holomorphic function, then f has an antiderivative F on U, and the value of every line integral in U with integrand f depends only on the end points u and v of the path, and can be computed as F(v) - F(u). The integral thus does not depend on the particular path connecting u and v.
• The Riemann mapping theorem states that any non-empty open simply connected subset of C (except for C itself) is conformally equivalent to the unit disk.

The notion of simply connectedness is also a crucial condition in the Poincaré lemma.

In Lie theory, simple connectedness is prerequisite for working of important Baker–Campbell–Hausdorff formula.