Simply connected space
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A geometrical object is called simply connected if it consists of one piece and doesn't have any circle-shaped "holes" or "handles". Higher-dimensional holes are allowed. For instance, a doughnut (with hole) is not simply connected, but a ball (even a hollow one) is. A circle is not simply connected but a disk and a line are. The opposite is non-simply connected or, in a somewhat old-fashioned term, multiply connected.
Informally, suppose someone hands you an object made out of a strong, inflexible material, one that won't bend or break under any condition. Shake the object and turn it every direction you can think of. If anything falls off, rattles, spins, or otherwise moves separately of the object, it's not a "simple" object. Formally, such a simple object is called a connected space, but for our informal definition, we can just think of a simple object as being an object that's all one piece.
Once you have a simple object, take a piece of string and insert one end into the object at any point. Let that end of the string follow any path, leaving behind string everywhere it goes, and then emerge at the same spot it went in, so that you have a loop going through the object.
Now hold on to both ends of the string and maneuver the string inside the object until you are able to pull the loop out through the hole. You may need to feed in some extra string, but that's not a problem. If you can find any path inside the object that makes it impossible to get the loop of string out, the object is not simply connected. If no path from any point of entry gets the loop caught in the object, then it is simply connected.
Notice how this definition does not rule out higher-dimensional holes. For example, while a hollow ball has a 2-dimensional hole in its middle, any loop you tie around the ball you can shrink to a point. The stronger condition, that the object have no holes of any dimension, is called contractibility.
In algebraic topology this idea is made into a formal tool.
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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.
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 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 non-empty open simply connected subset of C (except for C itself) can be conformally and bijectively mapped to the unit disk.