Difference between revisions of "Algebraic topology"
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== Higher Homotopy Groups == | == Higher Homotopy Groups == | ||
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== Homology and Cohomology == | == Homology and Cohomology == | ||
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Revision as of 15:49, 6 January 2008
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
Homology and Cohomology
This article is a stub. Help us out by expanding it.
