homotopy groups

The homotopy groupsMathworldPlanetmath are an infinite series of (covariant) functorsMathworldPlanetmath πn indexed by non-negative integers from based topological spacesPlanetmathPlanetmath to groups for n>0 and sets for n=0. πn(X,x0) as a set is the set of all homotopy classes of maps of pairs (Dn,Dn)(X,x0), that is, maps of the disk into X, taking the boundary to the point x0. Alternatively, these can be thought of as maps from the sphere Sn into X, taking a basepoint on the sphere to x0. These sets are given a group structureMathworldPlanetmath by declaring the productPlanetmathPlanetmathPlanetmathPlanetmath of 2 maps f,g to simply attaching two disks D1,D2 with the right orientation along part of their boundaries to get a new disk D1D2, and mapping D1 by f and D2 by g, to get a map of D1D2. This is continuousPlanetmathPlanetmath because we required that the boundary go to a , and well defined up to homotopyMathworldPlanetmath.

If f:XY satisfies f(x0)=y0, then we get a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath of homotopy groups f*:πn(X,x0)πn(Y,y0) by simply composing with f. If g is a map DnX, then f*([g])=[fg].

More algebraically, we can define homotopy groups inductively by πn(X,x0)πn-1(ΩX,y0), where ΩX is the loop spaceMathworldPlanetmath of X, and y0 is the constant path sitting at x0.

If n>1, the groups we get are abelian.

Homotopy groups are invariant under homotopy equivalenceMathworldPlanetmathPlanetmath, and higher homotopy groups (n>1) are not changed by the taking of covering spaces.

Some examples are:


πm(Sn)=0 if m<n.

πn(S1)=0 if n>1.

πn(M)=0 for n>1 where M is any surface of nonpositive Euler characteristicMathworldPlanetmath (not a sphere or projective planeMathworldPlanetmath).

Title homotopy groups
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