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== Simply Connected Spaces ==

== Simply Connected Spaces ==

=== Intuition ===

=== Intuition ===

The easiest topological spaces to deal with, from the standpoint of the fundamental group, are the '''simply connected''' spaces.  A space is termed simply connected if every loop in the space is [[homotopy|homotopic]] to a constant loop.  The intuition behind this definition is simple: imagine a rubber band is immersed in the topological space.  In a simply connected space, the rubber band may contract to a point, while remaining entirely within the space.  For example, no matter how a rubber band is arranged on a [[sphere]], it may always be shrunk down while remaining on the sphere.  In contrast, it is possible to place a rubber band on a torus in such a way that no such contraction is possible.  A few other spaces which have the property of being simply connected are:

The easiest topological spaces to deal with, from the standpoint of the fundamental group, are the '''simply connected''' spaces.  A space is termed simply connected if every loop in the space is [[homotopy|homotopic]] to a constant loop.  The intuition behind this definition is simple: imagine a rubber band is immersed in the topological space.  IA simply connected space is one in which the rubber band may always be contracted to a point, while remaining entirely within the space.  For example, no matter how a rubber band is arranged on a [[sphere]], it may always be shrunk down while remaining on the sphere.  In contrast, it is possible to place a rubber band on a torus in such a way that no such contraction is possible.  A few other spaces which have the property of being simply connected are:

* Euclidean space <math>\mathbb R^n</math>.  This includes the cases familiar cases of the line (<math>\mathbb R^1</math>), the plane (<math>\mathbb R^2</math>), and 3-dimensional space (<math>\mathbb R^3</math>).

* Euclidean space <math>\mathbb R^n</math>.  This includes the cases familiar cases of the line (<math>\mathbb R^1</math>), the plane (<math>\mathbb R^2</math>), and 3-dimensional space (<math>\mathbb R^3</math>).

* [[Convex]] subsets of Euclidean space.  This includes, for example, the unit disk in the plane.

* [[Convex]] subsets of Euclidean space.  This includes, for example, the unit disk in the plane.

which satisfies

which satisfies

::<math>H(t,0)=\gamma_1(t)</math>, <math>H(t,1)=\gamma_2(t)</math>, <math>H(0,s)=H(1,s)=*</math>

::<math>H(t,0)=\gamma_1(t)</math>, <math>H(t,1)=\gamma_2(t)</math>, <math>H(0,s)=H(1,s)=*</math>

The intuition is this: for every value of <math>s \in [0,1]</math>, we get a path <math>\gamma_s(t) : [0,1] \to X</math>, and at <math>s=0</math> and <math>s=1</math>, these are the two paths in question.  Thus <math>H</math> provides a smoothly-varying set of paths in <math>X</math>, parametrized by a variable <math>t</math>, starting with <math>\gamma_0</math>, and ending with <math>\gamma_1</math>.   

The intuition is this: for every value of <math>s \in [0,1]</math>, we get a path <math>\gamma_s(t) : [0,1] \to X</math>, and at <math>s=0</math> and <math>s=1</math>, these are the two paths in question.  Thus <math>H</math> provides a smoothly-varying set of paths in <math>X</math>, parametrized by a variable <math>s</math>, starting with <math>\gamma_0</math>, and ending with <math>\gamma_1</math>.   

The constant path is defined by <math>\gamma_c(t) = *</math>, for all <math>t</math>.  A space is said to be simply-connected if every path based at <math>*</math> is homotopic to the constant path.  This definition serves to make the intuition in terms of shrinking rubber bands precise.

The constant path is defined by <math>\gamma_c(t) = *</math>, for all <math>t</math>.  A space is said to be simply-connected if every path based at <math>*</math> is homotopic to the constant path.  This definition serves to make the intuition in terms of shrinking rubber bands precise.