Simply connected

A geometrical object is called simply connected if it consists of one piece and doesn't have any "holes" or "handles". For instance, a doughnut is not simply connected, but a ball (even a hollow one) is. A circle is not simply connected but a disk and a line is.

Formal definition and equivalent formulations

A topological space X is called simply connected if it is path-connected and any continuous map f : S¹ -> X (where S¹ denotes the unit circle in Euclidean 2-space) can be contracted to a point in the following sense: there exists a continuous map F : D² -> X (where D² denotes the unit disk[?] in Euclidean 2-space) such that F restricted to S¹ is f.

An equivalent formulation is this: X is simply connected if and only if it is path connected, and whenever p : [0,1] → X and q : [0,1] → X are two paths (i.e.: continuous maps) with the same start and endpoint (p(0) = q(0) and p(1) = q(1)), then p and q are homotopic relative {0,1}. Intuitively, this means that p can be "continuously deformed" to get q while keeping the endpoints fixed. Hence the term simply connected: for any two given points in X, there is one and "essentially" only one path connecting them.

A third way to express the same: X is simply connected if and only if X is path-connected and the fundamental group of X is trivial, i.e. consists only of the identity element.

Examples

• The Euclidean plane R² is simply connected, but R² minus the origin (0,0) is not. If n>2, then both Rⁿ and Rⁿ minus the origin is simply connected.
• Analogously: the n-dimensional sphere Sⁿ is simply connected if and only if n≥2.
• A torus, the Möbius band and the Klein bottle are not simply connected.
• Every topological vector space is simply connected; this includes Banach spaces and Hilbert spaces.
• The special orthogonal group SO(n,R) is not simply connected for n≥2; the special unitary group SU(n) is simply connected.
• The long line L is simply connected, but its compactification, the extended long line L* is not (since it is not even path connected).

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 "holes" or "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.

The notion of simply 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 path 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(v). The integral thus does not depend on the particular path connecting u and v.
• The Riemann mapping theorem states that any two such non-empty open simply connected subsets of C can be conformally and bijectively mapped to the unit disk.