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homotopy groups
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(Definition)
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The homotopy groups are an infinite series of (covariant) functors  indexed by non-negative integers from based topological spaces to groups for  and sets for  .
 as a set is the set of all homotopy classes of maps of pairs
 , that is, maps of the disk into  , taking the boundary to the point  . Alternatively, these can be thought of as maps from the sphere  into  ,
taking a basepoint on the sphere to  . These sets are given a group structure by declaring the product of 2 maps  to simply attaching two disks  with the right orientation along part of
their boundaries to get a new disk
 , and mapping  by  and  by  , to get a map of
 . This is continuous because we required that the boundary go to a fixed point, and well defined up to homotopy.
If  satisfies
 , then we get a homomorphism of homotopy groups
 by simply composing with  . If  is a map  , then
![$ f^*([g])=[f\circ g]$](http://images.planetmath.org:8080/cache/objects/1641/l2h/img25.png "$ f^*([g])=[f\circ g]$") .
More algebraically, we can define homotopy groups inductively by
 , where  is the loop space of  , and  is the constant path sitting at  .
If  , the groups we get are abelian.
Homotopy groups are invariant under homotopy equivalence, and higher homotopy groups ( ) are not changed by the taking of covering spaces.
Some examples are:
 .
 if  .
 if  .
 for  where  is any surface of nonpositive Euler characteristic (not a sphere or projective plane).
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"homotopy groups" is owned by bwebste. [ full author list (2) | owner history (1) ]
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(view preamble)
Cross-references: projective plane, Euler characteristic, surface, covering spaces, homotopy equivalence, invariant, abelian, path, loop space, homomorphism, well defined, continuous, mapping, orientation, right, product, structure, basepoint, sphere, point, boundary, maps, classes, homotopy, groups, based topological spaces, integers, indexed by, functors, series, infinite
There are 17 references to this entry.
This is version 9 of homotopy groups, born on 2002-02-02, modified 2003-08-13.
Object id is 1641, canonical name is HomotopyGroups.
Accessed 6681 times total.
Classification:
| AMS MSC: | 54-00 (General topology :: General reference works ) |
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Pending Errata and Addenda
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