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fundamental group
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(Definition)
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Let  be a pointed topological space (that is, a topological space with a chosen basepoint  ). Denote by
![$ [(S^1,1),(X,x_{0})]$](http://images.planetmath.org:8080/cache/objects/849/l2h/img3.png "$ [(S^1,1),(X,x_{0})]$") the set of homotopy classes of maps
 such that
 . Here,  denotes the basepoint
 . Define a product
![$ [(S^1,1),(X,x_{0})] \times [(S^1,1),(X,x_{0})] \to [(S^1,1),(X,x_{0})]$](http://images.planetmath.org:8080/cache/objects/849/l2h/img8.png "$ [(S^1,1),(X,x_{0})] \times [(S^1,1),(X,x_{0})] \to [(S^1,1),(X,x_{0})]$") by
![$ [\sigma][\tau]=[\sigma\tau]$](http://images.planetmath.org:8080/cache/objects/849/l2h/img9.png "$ [\sigma][\tau]=[\sigma\tau]$") , where
 means “travel along  and then  ”. This gives
![$ [(S^1,1),(X,x_{0})]$](http://images.planetmath.org:8080/cache/objects/849/l2h/img13.png "$ [(S^1,1),(X,x_{0})]$") a group structure and we define the fundamental group of  to be
![$ \pi_1(X,x_{0})= [(S^1,1),(X,x_{0})]$](http://images.planetmath.org:8080/cache/objects/849/l2h/img15.png "$ \pi_1(X,x_{0})= [(S^1,1),(X,x_{0})]$") .
In general, the fundamental group of a topological space depends upon the choice of basepoint. However, basepoints in the same path-component of the space will give isomorphic groups. In particular, this means that the fundamental group of a (non-empty) path-connected space is well-defined, up to isomorphism, without the need to specify a basepoint.
Here are some examples of fundamental groups of familiar spaces:
-
 for each
 .
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 .
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 , where  is the torus.
It can be shown that  is a functor from the category of pointed topological spaces to the category of groups. In particular, the fundamental group is a topological invariant, in the sense that if  is homeomorphic to  via a basepoint-preserving map, then
 is isomorphic to
 .
It can also be shown that two homotopically equivalent path-connected spaces have isomorphic fundamental groups.
Homotopy groups generalize the concept of the fundamental group to higher dimensions. The fundamental group is the first homotopy group, which is why the notation  is used.
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"fundamental group" is owned by yark. [ full author list (2) | owner history (1) ]
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(view preamble)
Cross-references: homotopy groups, homotopically equivalent, isomorphic, homeomorphic, topological invariant, category, category of pointed topological spaces, functor, torus, isomorphism, well-defined, path-connected, isomorphic groups, structure, group, product, maps, classes, homotopy, basepoint, topological space, pointed topological space
There are 40 references to this entry.
This is version 12 of fundamental group, born on 2001-11-14, modified 2006-10-07.
Object id is 849, canonical name is FundamentalGroup.
Accessed 8929 times total.
Classification:
| AMS MSC: | 55Q05 (Algebraic topology :: Homotopy groups :: Homotopy groups, general; sets of homotopy classes) | | | 20F34 (Group theory and generalizations :: Special aspects of infinite or finite groups :: Fundamental groups and their automorphisms) | | | 57M05 (Manifolds and cell complexes :: Low-dimensional topology :: Fundamental group, presentations, free differential calculus) |
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Pending Errata and Addenda
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