Difference between revisions of "Algebraic topology"
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Perhaps the simplest object of study in algebraic topology is the [[fundamental group]]. Let <math>X</math> be a [[path-connected]] topological space, and let <math>x\in X</math> be any point. Now consider all possible "loops" on <math>X</math> that start and end at <math>x</math>, i.e. all [[continuous function]]s <math>f:[0,1]\to X</math> with <math>f(0)=f(1)=x</math>. Call this collection <math>L</math>. Now define an [[equivalence relation]] <math>\sim</math> on <math>L</math> by saying that <math>p\sim q</math> if there is a continuous function <math>g:[0,1]\times[0,1]\to X</math> with <math>g(a,0)=p(a)</math>, <math>g(a,1)=q(a)</math>, and <math>g(0,b)=g(1,b)=x</math>. We call <math>g</math> a [[homotopy]]. Now define <math>\pi_1(X)=L/\sim</math>. That is, we equate any two elements of <math>L</math> which are equivalent under <math>\sim</math>. | Perhaps the simplest object of study in algebraic topology is the [[fundamental group]]. Let <math>X</math> be a [[path-connected]] topological space, and let <math>x\in X</math> be any point. Now consider all possible "loops" on <math>X</math> that start and end at <math>x</math>, i.e. all [[continuous function]]s <math>f:[0,1]\to X</math> with <math>f(0)=f(1)=x</math>. Call this collection <math>L</math>. Now define an [[equivalence relation]] <math>\sim</math> on <math>L</math> by saying that <math>p\sim q</math> if there is a continuous function <math>g:[0,1]\times[0,1]\to X</math> with <math>g(a,0)=p(a)</math>, <math>g(a,1)=q(a)</math>, and <math>g(0,b)=g(1,b)=x</math>. We call <math>g</math> a [[homotopy]]. Now define <math>\pi_1(X)=L/\sim</math>. That is, we equate any two elements of <math>L</math> which are equivalent under <math>\sim</math>. | ||
| − | Unsurprisingly, the fundamental group is a group. The [[identity]] is the [[equivalence class]] containing the map <math>1:[0,1]\to X</math> given by <math>1(a)=x</math> for all <math>a\in[0,1]</math>. The [[inverse]] of a map <math>h</math> is the map <math>h^{-1}</math> given by <math>h^{-1}(a)=h(1-a)</math>. We can compose maps as follows: <math>g\cdot h(a)=\begin{cases} g(2a) & 0\le a\le 1/2, \\ h(2a-1) & 1/2\le a\le 1.\end{cases}</math> One can check that this is indeed [[well-defined]]. | + | Unsurprisingly, the fundamental group is a group. The [[identity]] is the [[equivalence class]] containing the map <math>1:[0,1]\to X</math> given by <math>1(a)=x</math> for all <math>a\in[0,1]</math>. The [[Function/Introduction#The_Inverse_of_a_Function | inverse]] of a map <math>h</math> is the map <math>h^{-1}</math> given by <math>h^{-1}(a)=h(1-a)</math>. We can compose maps as follows: <math>g\cdot h(a)=\begin{cases} g(2a) & 0\le a\le 1/2, \\ h(2a-1) & 1/2\le a\le 1.\end{cases}</math> One can check that this is indeed [[well-defined]]. |
Note that the fundamental group is not in general [[abelian group|abelian]]. For example, the fundamental group of a figure eight is the [[free group]] on two [[generator]]s, which is not abelian. However, the fundamental group of a circle is <math>{\mathbb{Z}}</math>, which is abelian. | Note that the fundamental group is not in general [[abelian group|abelian]]. For example, the fundamental group of a figure eight is the [[free group]] on two [[generator]]s, which is not abelian. However, the fundamental group of a circle is <math>{\mathbb{Z}}</math>, which is abelian. | ||
Revision as of 08:17, 18 July 2006
Algebraic topology is the study of topology using methods from abstract algebra. In general, given a topological space, we can associate various algebraic objects, such as groups and rings.
Fundamental Groups
Perhaps the simplest object of study in algebraic topology is the fundamental group. Let 
be a path-connected topological space, and let 
be any point. Now consider all possible "loops" on that start and end at 
, i.e. all continuous functions ![$f:[0,1]\to X$](http://latex.artofproblemsolving.com/e/2/3/e23ac345d4603fbc14345da777eeece650ae1812.png)
with 
. Call this collection 
. Now define an equivalence relation 
on by saying that 
if there is a continuous function ![$g:[0,1]\times[0,1]\to X$](http://latex.artofproblemsolving.com/7/8/5/7856594758f5af46cda3cad2ccbf2bf8da35fdaf.png)
with 
, 
, and 
. We call 
a homotopy. Now define 
. That is, we equate any two elements of which are equivalent under .
Unsurprisingly, the fundamental group is a group. The identity is the equivalence class containing the map ![$1:[0,1]\to X$](http://latex.artofproblemsolving.com/7/d/a/7da75d7aadb2dddcfd6393787f20a4085a3c1e49.png)
given by 
for all ![$a\in[0,1]$](http://latex.artofproblemsolving.com/e/3/7/e3777765ea228b1655f3649991e57ddcc3253cdd.png)
. The inverse of a map 
is the map 
given by 
. We can compose maps as follows: 
One can check that this is indeed well-defined.
Note that the fundamental group is not in general abelian. For example, the fundamental group of a figure eight is the free group on two generators, which is not abelian. However, the fundamental group of a circle is 
, which is abelian.
Higher Homotopy Groups
(I know next to nothing about these. Please fill in if you know about them.)
Homology and Cohomology
(This is for when I'm feeling braver. Or, better yet, when someone else is feeling braver.)
