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One of the most important problems of topology (and [[algebraic topology]] in particular) is the precise and rigorous characterization of the "holes" in a space.  This characterization is in terms of some [[groups]] known as the homotopy groups and homology groups.  The most important of these is the 1st homotopy group, known as the '''fundamental group'''.

One of the most important problems of topology (and [[algebraic topology]] in particular) is the precise and rigorous characterization of the "holes" in a space.  This characterization is in terms of some [[group|groups]] known as the homotopy groups and homology groups.  The most important of these is the 1st homotopy group, known as the '''fundamental group'''.

The concept of [[homotopy]], on which the fundamental group is based, is a type of equivalence between paths.  Two paths are homotopic if one can be continuously deformed/moved/stretched into the other.  This stretching can't pass over a hole, so a characterization of homotopic paths yields a characterization of holes.

The concept of [[homotopy]], on which the fundamental group is based, is a type of equivalence between paths.  Two paths are said to be homotopic if one can be continuously deformed/moved/stretched into the other.  This stretching can't pass over a hole, so a characterization of homotopic paths yields a characterization of holes.

Given a [[topological space]] <math>X</math>, the fundamental group is an associated algebraic object which describes the paths that are not homotopic, and hence describes the set of "holes" in the space.

Given a [[topological space]] <math>X</math>, the fundamental group is an associated algebraic object which describes the paths that are not homotopic, and hence describes the set of "holes" in the space.

The fundamental group is useful to distinguish between different topological spaces: for example, to prove rigorously that a surface with two holes cannot be smoothly deformed into a surface with three holes, one can check that these two spaces have different fundamental groups and so must be distinct.  The fundamental group captures information about the set of holes in a space by looking at the set of loops drawn on the space.

The fundamental group is useful to distinguish between different topological spaces: for example, to prove rigorously that a surface with two holes cannot be smoothly deformed into a surface with three holes, one can check that these two spaces have different fundamental groups and so must be distinct.  The fundamental group captures information about the set of holes in a space by looking at the set of loops drawn in the space.

== 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.  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:

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.  A 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, and so the torus is not simply connected.  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.

===As Homotopy===

===As Homotopy===

The notion of [[homotopy]] makes the above definition of simple connectedness precise.  A "loop in <math>X</math>", earlier represented by a rubber band, is now a [[continuous function]] <math>\gamma : [0,1] \to X</math> satisfying <math>\gamma(0)=\gamma(1)=*</math> of the unit interval into <math>X</math>.  It is generally convenient to fix a "basepoint" <math>* \in X</math>, which may be any point in the topological space <math>X</math>, and assume that <math>\gamma(0)=\gamma(1)=*</math>.  The choice of basepoint does not change the fundamental group (up to isomorphism).  In terms of the analogy of a rubber band, <math>\gamma(t)</math> is the point in <math>X</math> where the point a proportion <math>t</math> along the rubber band is initiall placed.

The notion of [[homotopy]] makes the above definition of simple connectedness precise.  A "loop in <math>X</math>", earlier represented by a rubber band, is now a [[continuous function]] <math>\gamma : [0,1] \to X</math> satisfying <math>\gamma(0)=\gamma(1)=*</math> of the unit interval into <math>X</math>.  It is generally convenient to fix a "basepoint" <math>* \in X</math>, which may be any point in the topological space <math>X</math>, and assume that <math>\gamma(0)=\gamma(1)=*</math>.  The choice of basepoint does not change the fundamental group (up to isomorphism).  In terms of the analogy of a rubber band, the function <math>\gamma</math> is defined so that <math>\gamma(t)</math> is the point in <math>X</math> where the point a proportion <math>t</math> along the rubber band is initiall placed.

Two paths <math>\gamma_0,\gamma_1</math> are said to be homotopic if there is a continuous function

Two paths <math>\gamma_0,\gamma_1</math> are said to be homotopic if there is a continuous function

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>.  This relation is often written as <math>\gamma_0 \sim \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>.  This relation is often written as <math>\gamma_0 \sim \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 loop based at <math>*</math> is homotopic to the constant path.  This definition serves to make the intuition in terms of shrinking rubber bands precise.

'''Problem:''' Prove that the relation of homotopy of paths is an [[equivalence relation]] on the set of paths.

'''Problem:''' Prove that the relation of homotopy of paths is an [[equivalence relation]] on the set of paths.

'''Solution:''' First we need to check reflexivity, that <math>\gamma \sim \gamma</math> for any path <math>\gamma</math>.  A homotopy from <math>\gamma</math> to itself is provided by

'''Solution:''' First we need to check reflexivity, that <math>\gamma \sim \gamma</math> for any path <math>\gamma</math>.  A homotopy from <math>\gamma</math> to itself is provided by

:: <math>H(t,s) = \gamma(t)</math> for all <math>s</math>.  This clearly satisfies <math>H(t,0)=H(t,1)=\gamma(t)</math>, as required.

<math>H(t,s) = \gamma(t)</math> for all <math>s</math>.  This clearly satisfies <math>H(t,0)=H(t,1)=\gamma(t)</math>, as required.

An equivalence relation must also satisfy reflexivity, so that <math>\gamma_0 \sim \gamma_1</math> implies that <math>\gamma_1 \sim \gamma_0</math>.  To prove that homotopy of loops satisfy this, suppose that <math>\gamma_0 \sim \gamma_1</math>.  Then there is a homotopy <math>H_0(t,s) : [0,1] \times [0,1] \to X</math> satisfying the requisite conditions.  A homotopy from <math>\gamma_1</math> to <math>\gamma_0</math> is provided by running <math>H_0</math> in reverse: set <math>H(t,s) = H_0(t,1-s)</math>.

An equivalence relation must also satisfy reflexivity, so that <math>\gamma_0 \sim \gamma_1</math> implies that <math>\gamma_1 \sim \gamma_0</math>.  To prove that homotopy of loops satisfy this, suppose that <math>\gamma_0 \sim \gamma_1</math>.  Then there is a homotopy <math>H_0(t,s) : [0,1] \times [0,1] \to X</math> satisfying the requisite conditions.  A homotopy from <math>\gamma_1</math> to <math>\gamma_0</math> is provided by running <math>H_0</math> in reverse: set <math>H(t,s) = H_0(t,1-s)</math>.

The final requirement is that of transitivity, that <math>\gamma_0 \sim \gamma_1</math> and <math>\gamma_1 \sim \gamma_2</math> implies that <math>\gamma_0 \sim \gamma_2</math>.  This is the intuitive fact that if one path may be deformed to another, and that one may be deformed some third path, then the original path may be deformed directly to the third path.  To prove this, let <math>H_0</math> be a homotopy from <math>\gamma_0</math> to <math>\gamma_1</math>, and <math>H_1</math> a homotopy from <math>\gamma_1</math> to <math>\gamma_2</math>.  Define a new homotopy by  

The final requirement is that of transitivity, that <math>\gamma_0 \sim \gamma_1</math> and <math>\gamma_1 \sim \gamma_2</math> implies that <math>\gamma_0 \sim \gamma_2</math>.  This is the intuitive fact that if one path may be deformed to another, and that one may be deformed some third path, then the original path may be deformed directly to the third path.  To prove this, let <math>H_0</math> be a homotopy from <math>\gamma_0</math> to <math>\gamma_1</math>, and <math>H_1</math> a homotopy from <math>\gamma_1</math> to <math>\gamma_2</math>.  Define a new homotopy by  

This is a homotopy from <math>\gamma_0</math> to <math>\gamma_2</math>.

This is a homotopy from <math>\gamma_0</math> to <math>\gamma_2</math>.

This exercise proves that homotopy is an equivalence relation on loops in <math>X</math>. The elements of the fundamental group are defined to be loops in <math>X</math>, modulo this equivalence relation of homotopy.

This exercise proves that homotopy is an equivalence relation on loops in <math>X</math>, whether or not <math>X</math> is simply connected. The elements of the fundamental group are defined to be homotopy equivalence class of loops in <math>X</math> based at <math>*</math>: thus two loops represent different elements of the fundamental group if and only if they are not homotopic to each other.  In a simply connected space, all loops are homotopic and thus represent a single homotopy class, and so the fundamental group is the trivial group, with only one element.

== 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 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.   

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, and such 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 class of paths it is sufficient merely to give an integer!  

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!

The fundamental group <math>\pi_1(X)</math> is defined to be the space of loops in a space, modulo the relation of homotopy.  The preceding description indicates that <math>\pi_1(S^1) \cong \Z</math>, by an isomorphism which sends a class of paths to the integer describing the number of times these paths go around the circle.

The fundamental group <math>\pi_1(X)</math> is defined to be the space of loops in a space, modulo the relation of homotopy.  The preceding description indicates that <math>\pi_1(S^1) \cong \Z</math>, by an isomorphism which sends a class of paths to the integer describing the number of times these paths go around the circle.

== Group Structure ==

== Group Structure ==

The preceding definition is incomplete, because it does not specify the operation on <math>\pi_1(X)</math> which gives these classes of paths the structure of not just a set, but a group.  The definition of the group operation is simple: given two paths <math>\gamma_1</math> and <math>\gamma_2</math>, the sum <math>\gamma_1 * \gamma_2</math> is the path obtained by first following <math>\gamma_1</math>, and then following <math>\gamma_2</math>.  More precisely, set

The preceding definition is incomplete, because it does not specify the operation on <math>\pi_1(X)</math> which gives these classes of paths the structure of not just a set, but a group.  The definition of the group operation is simple: given two paths <math>\gamma_1</math> and <math>\gamma_2</math>, the product <math>\gamma_1 * \gamma_2</math> is the path obtained by first following <math>\gamma_1</math>, and then following <math>\gamma_2</math>.  More precisely, set

:: <math>(\gamma_1 * \gamma_2)(t) = \begin{cases} \gamma_1(2t) & \textrm{if $0 \leq t \leq 1/2$} \\ \gamma_2(2t-1) & \textrm{if $1/2 \leq t \leq 1$} \end{cases}</math>

:: <math>(\gamma_1 * \gamma_2)(t) = \begin{cases} \gamma_1(2t) & \textrm{if $0 \leq t \leq 1/2$} \\ \gamma_2(2t-1) & \textrm{if $1/2 \leq t \leq 1$} \end{cases}</math>

For this to be a group, it is necessary to see that there is an identity element with respect to the operation, and that every loop has an inverse.  It is clear that the constant loop at <math>*</math> serves as an identity, and the inverse of a loop <math>\gamma</math> is obtained simply by following <math>\gamma</math> in the opposite direction:

For this to be a group, it is necessary to see that there is an identity element with respect to the operation, and that every loop has an inverse.  It is clear that the constant loop at <math>*</math> serves as an identity, and the inverse of a loop <math>\gamma</math> is obtained simply by following <math>\gamma</math> in the opposite direction:

== Examples ==

== Examples ==

The fundamental groups of more complex spaces may be harder to describe.  Consider the figure eight <math>S^1 \vee S^1</math>, the space obtained by gluing two circles together at a point.  What is required to describe a class of paths in this space?  A path could be specified by saying "starting at the glue point, go around the left circle 5 times, then the right circle 2 times, then the left circle -3 times (i.e., three times counterclockwise)", etc.  This might be written as

The fundamental groups of more complex spaces may be harder to describe.  Consider the figure eight <math>S^1 \vee S^1</math>, the space obtained by gluing two circles together at a point.  What is required to describe a class of paths in this space?  A path could be specified by saying "starting at the gluing point, go around the left circle 5 times, then the right circle 2 times, then the left circle -3 times (i.e., three times counterclockwise)", etc.  This might be written as

:: <math>\gamma = a^5 b^2 a^{-3}</math>,

:: <math>\gamma = a^5 b^2 a^{-3}</math>,

where <math>a</math> denotes a path which goes around the left circle once clockwise, and <math>b</math> a path which goes around the right circle once clockwise.  More generally, any loop in the figure-eight can be specified by a "word"

where <math>a</math> denotes a path which goes around the left circle once clockwise, and <math>b</math> a path which goes around the right circle once clockwise.  More generally, any loop in the figure-eight can be specified by a "word"

== Applications ==

== Applications ==

The fundamental group is a basic object in the study of algebraic topology.  In its most basic form, it provides a way to prove that two topological spaces are not the same thing.  For example, the torus and the 2-sphere have different fundamental groups, and this proves that they not the same space.  Intuitively, the "hole" in a torus makes it different from a sphere with no holes.  This fact, while apparently obvious, is difficult to prove without the methods of algebraic topology, including the fundamental group.

The fundamental group is a basic object in the study of algebraic topology.  In its most basic form, it provides a way to prove that two topological spaces are not the same thing (more precisely, that they are not [[homeomorphic]] or indeed [[homotopy equivalent]]).  For example, the torus and the 2-sphere have different fundamental groups, and this proves that they not the same space.  Intuitively, the "hole" in a torus makes it different from a sphere with no holes.  This fact, while apparently obvious, is difficult to prove without the methods of algebraic topology, including the fundamental group.

The applications of the fundamental group within algebraic topology are many, and it is one of the most basic tools in the field.

The applications of the fundamental group within algebraic topology are many, and it is one of the most basic tools in the field.

== Higher Homotopy Groups ==

== Higher Homotopy Groups ==