# Difference between revisions of "Fundamental group"

m (split off from algebraic topology article) |
m (add category) |
||

Line 13: | Line 13: | ||

If <math>\circ,\cdot</math> share the same unit <math>e</math> (such that <math>a \cdot e = e \cdot a = a \circ e = e \circ a = a</math>) then <math>\cdot = \circ</math> and both are abelian. | If <math>\circ,\cdot</math> share the same unit <math>e</math> (such that <math>a \cdot e = e \cdot a = a \circ e = e \circ a = a</math>) then <math>\cdot = \circ</math> and both are abelian. | ||

+ | |||

+ | [[Category:Topology]] | ||

+ | [[Category:Algebraic Topology]] |

## Revision as of 00:09, 13 December 2009

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 with . Call this collection . Now define an equivalence relation on by saying that if there is a continuous function 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 given by for all . 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.

More generally, if is an h-space, then is abelian, for there is a second multiplication on given by , which is "compatible" with the concatenation in the following respect:

We say that two binary operations on a set are compatible if, for every ,

If share the same unit (such that ) then and both are abelian.