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== The General Situation ==

== The General Situation ==

In many topological spaces, there are loops which can not be contracted to a point: a few examples were given above.  The topological group is a structure which serves to describe the set of loops in a space, where two loops that can be continuously deformed into each other are treated as equivalent.   

In many topological spaces, there are loops which can not be contracted to a point: a few examples were given above.  The fundamental group is a structure which serves to describe the set of loops in a space, where two loops that can be continuously deformed into each other are treated as equivalent.   

As an example, consider loops in the circle.  Let us pass to another analogy for now: a particle moves around in a topological space, and we wish to describe its path as simply as possible, in such a way that two paths that may be deformed into each other (i.e., are homotopic), are given the same description.  One path might be described as "go around the circle once in a clockwise direction".  It does not matter at what speed the particle moves, or whether it turns around before reversing itself and completing a cycle: such paths may all be deformed to one another, and to describe such a path up to homotopy it suffices to give the preceding description. Similarly, "go around the circle twice in a clockwise direction" describes another class of paths, not equivalent to going around once.  Thus, to describe a class of loops in a circle, it suffices to say how many times a path goes around the circle.  By writing a path going <math>n</math> times around clockwise as <math>n</math>, and <math>n</math> times counterclockwise as <math>-n</math>, to describe a homotopy class of paths it is sufficient merely to give an integer counting the number of times the path goes around the circle!

As an example, consider loops in the circle.  Let us pass to another analogy for now: a particle moves around in a topological space, and we wish to describe its path as simply as possible, in such a way that two paths that may be deformed into each other (i.e., are homotopic), are given the same description.  One path might be described as "go around the circle once in a clockwise direction".  It does not matter at what speed the particle moves, or whether it turns around before reversing itself and completing a cycle: such paths may all be deformed to one another, and to describe such a path up to homotopy it suffices to give the preceding description. Similarly, "go around the circle twice in a clockwise direction" describes another class of paths, not equivalent to going around once.  Thus, to describe a class of loops in a circle, it suffices to say how many times a path goes around the circle.  By writing a path going <math>n</math> times around clockwise as <math>n</math>, and <math>n</math> times counterclockwise as <math>-n</math>, to describe a homotopy class of paths it is sufficient merely to give an integer counting the number of times the path goes around the circle!