Simply connected space

Summary

In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected[1]) if it is path-connected and every path between two points can be continuously transformed into any other such path while preserving the two endpoints in question. Intuitively, this corresponds to a space that has no disjoint parts and no holes that go completely through it, because two paths going around different sides of such a hole cannot be continuously transformed into each other. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial.

Definition and equivalent formulations edit

 
This shape represents a set that is not simply connected, because any loop that encloses one or more of the holes cannot be contracted to a point without exiting the region.

A topological space   is called simply connected if it is path-connected and any loop in   defined by   can be contracted to a point: there exists a continuous map   such that   restricted to   is   Here,   and   denotes the unit circle and closed unit disk in the Euclidean plane respectively.

An equivalent formulation is this:   is simply connected if and only if it is path-connected, and whenever   and   are two paths (that is, continuous maps) with the same start and endpoint (  and  ), then   can be continuously deformed into   while keeping both endpoints fixed. Explicitly, there exists a homotopy   such that   and  

A topological space   is simply connected if and only if   is path-connected and the fundamental group of   at each point is trivial, i.e. consists only of the identity element. Similarly,   is simply connected if and only if for all points   the set of morphisms   in the fundamental groupoid of   has only one element.[2]

In complex analysis: an open subset   is simply connected if and only if both   and its complement in the Riemann sphere are connected. The set of complex numbers with imaginary part strictly greater than zero and less than one furnishes an example of an unbounded, connected, open subset of the plane whose complement is not connected. It is nevertheless simply connected. A relaxation of the requirement that   be connected leads to an exploration of open subsets of the plane with connected extended complement. For example, a (not necessarily connected) open set has a connected extended complement exactly when each of its connected components is simply connected.

Informal discussion edit

Informally, an object in our space is simply connected if it consists of one piece and does not have any "holes" that pass all the way through it. For example, neither a doughnut nor a coffee cup (with a handle) is simply connected, but a hollow rubber ball is simply connected. In two dimensions, a circle is not simply connected, but a disk and a line are. Spaces that are connected but not simply connected are called non-simply connected or multiply connected.

 
A sphere is simply connected because every loop can be contracted (on the surface) to a point.


The definition rules out only handle-shaped holes. A sphere (or, equivalently, a rubber ball with a hollow center) is simply connected, because any loop on the surface of a sphere can contract to a point even though it has a "hole" in the hollow center. The stronger condition, that the object has no holes of any dimension, is called contractibility.

Examples edit

 
A torus is not a simply connected surface. Neither of the two colored loops shown here can be contracted to a point without leaving the surface. A solid torus is also not simply connected because the purple loop cannot contract to a point without leaving the solid.
  • The Euclidean plane   is simply connected, but   minus the origin   is not. If   then both   and   minus the origin are simply connected.
  • Analogously: the n-dimensional sphere   is simply connected if and only if  
  • Every convex subset of   is simply connected.
  • A torus, the (elliptic) cylinder, the Möbius strip, the projective plane and the Klein bottle are not simply connected.
  • Every topological vector space is simply connected; this includes Banach spaces and Hilbert spaces.
  • For   the special orthogonal group   is not simply connected and the special unitary group   is simply connected.
  • The one-point compactification of   is not simply connected (even though   is simply connected).
  • The long line   is simply connected, but its compactification, the extended long line   is not (since it is not even path connected).

Properties edit

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

A universal cover of any (suitable) space   is a simply connected space which maps to   via a covering map.

If   and   are homotopy equivalent and   is simply connected, then so is  

The image of a simply connected set under a continuous function need not be simply connected. Take for example the complex plane under the exponential map: the image is   which is not simply connected.

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

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

The notion of simple connectedness is also a crucial condition in the Poincaré conjecture.

See also edit

References edit

  1. ^ "n-connected space in nLab". ncatlab.org. Retrieved 2017-09-17.
  2. ^ Ronald, Brown (June 2006). Topology and Groupoids. Academic Search Complete. North Charleston: CreateSpace. ISBN 1419627228. OCLC 712629429.
  • Spanier, Edwin (December 1994). Algebraic Topology. Springer. ISBN 0-387-94426-5.
  • Conway, John (1986). Functions of One Complex Variable I. Springer. ISBN 0-387-90328-3.
  • Bourbaki, Nicolas (2005). Lie Groups and Lie Algebras. Springer. ISBN 3-540-43405-4.
  • Gamelin, Theodore (January 2001). Complex Analysis. Springer. ISBN 0-387-95069-9.
  • Joshi, Kapli (August 1983). Introduction to General Topology. New Age Publishers. ISBN 0-85226-444-5.