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added a couple "problems", still needs work
::<math>H : [0,1] \times [0,1] \to X</math>
::<math>H : [0,1] \times [0,1] \to X</math>
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_0(t)</math>, <math>H(t,0)=\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>s</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>. 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 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.
'''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
:: <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>.
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
:: <math>H(t,s) = \begin{cases} H_1(t,2s) & \textrm{if $0 \leq s \leq 1/2$,} \\ H_2(t,2s-1) & \textrm{if $1/2 \leq s \leq 1$} \end{cases} </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.
== The General Situation ==
== The General Situation ==
:: <math>(\gamma^{-1})(t) = \gamma(1-t)</math>.
:: <math>(\gamma^{-1})(t) = \gamma(1-t)</math>.
To see how this works in the preceding case of the circle, let <math>\gamma^n</math> be a path which goes around the circle <math>n</math> times. The composition <math>\gamma^m * \gamma^n</math> is the path which goes around <math>m</math> times, and then <math>n</math> more: this is exactly the loop <math>\gamma^{m+n}</math>. This indicates that the group operation on <math>\pi_1(S^1) \cong \mathbb Z</math> is just the usual operation of addition on <math>\mathbb Z</math>.
To see how this works in the preceding case of the circle, let <math>\gamma^n</math> be a path which goes around the circle <math>n</math> times. The composition <math>\gamma^m * \gamma^n</math> is the path which goes around <math>m</math> times, and then <math>n</math> more: this is exactly the loop <math>\gamma^{m+n}</math>. This indicates that the group operation on <math>\pi_1(S^1) \cong \mathbb Z</math> is just the usual operation of addition on <math>\mathbb Z</math>. To prove carefully that paths going around the circle different numbers of times are not homotopic to each other requires a bit more machinery than is developed in this article.
== Examples ==
== Examples ==
where <math>i_1,i_2,i_3,\ldots</math> and <math>j_1,j_2,j_3,\ldots</math> are sequences of integers.
where <math>i_1,i_2,i_3,\ldots</math> and <math>j_1,j_2,j_3,\ldots</math> are sequences of integers.
The multiplication of two loops written in this form is obtained by concatenating the two words describing them. The inverse of a word is obtained by reversing it and switching the signs on all of the exponents. The group of such words on two letters <math>a,b</math> with these operations is termed the [[free group]] on two generators, and usually denoted by <math>F_2</math> or <math>\mathbb Z * \mathbb Z</math>.
The multiplication of two loops written in this form is obtained by concatenating the two words describing them. The inverse of a word is obtained by reversing it and switching the signs on all of the exponents. The group of such words on two letters <math>a,b</math> with these operations is termed the [[free group]] on two generators, and usually denoted by <math>F_2</math> or <math>\mathbb Z * \mathbb Z</math>.
'''Problem:''' Describe the fundamental group of the topological space obtained by gluing three circles together at a single point.
'''Solution:''' The fundamental group <math>\pi_1(S_1 \vee S_1 \vee S_1)</math> is described in a similar manner to that of the example. Let <math>a</math>, <math>b</math>, and <math>c</math> denote loops around each of the three circles. An element of the fundamental group is just a "word" on these three letters, describing the order in which a path goes around each of the three circles.
The fundamental groups of a few other familiar topological spaces are given by:
The fundamental groups of a few other familiar topological spaces are given by: