homotopy groups
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 , 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 .
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).
Title | homotopy groups |
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Canonical name | HomotopyGroups |
Date of creation | 2013-03-22 12:15:28 |
Last modified on | 2013-03-22 12:15:28 |
Owner | bwebste (988) |
Last modified by | bwebste (988) |
Numerical id | 13 |
Author | bwebste (988) |
Entry type | Definition |
Classification | msc 54-00 |
Synonym | higher homotopy groups |
Related topic | EilenbergMacLaneSpace |
Related topic | HomotopyDoubleGroupoidOfAHausdorffSpace |
Related topic | QuantumFundamentalGroupoids |
Related topic | CohomologyGroupTheorem |