homotopy groups

The homotopy groups are an infinite series of (covariant) functors $\pi_{n}$ indexed by non-negative integers from based topological spaces to groups for $n>0$ and sets for $n=0$ . $\pi_{n}(X,x_{0})$ as a set is the set of all homotopy classes of maps of pairs $(D^{n},\partial D^{n})\to(X,x_{0})$ , that is, maps of the disk into $X$ , taking the boundary to the point $x_{0}$ . Alternatively, these can be thought of as maps from the sphere $S^{n}$ into $X$ , taking a basepoint on the sphere to $x_{0}$ . These sets are given a group structure by declaring the product of 2 maps $f, g$ to simply attaching two disks $D_{1},D_{2}$ with the right orientation along part of their boundaries to get a new disk $D_{1}\cup D_{2}$ , and mapping $D_{1}$ by $f$ and $D_{2}$ by $g$ , to get a map of $D_{1}\cup D_{2}$ . This is continuous because we required that the boundary go to a , and well defined up to homotopy.

If $f:X\to Y$ satisfies $f(x_{0})=y_{0}$ , then we get a homomorphism of homotopy groups $f^{*}:\pi_{n}(X,x_{0})\to\pi_{n}(Y,y_{0})$ by simply composing with $f$ . If $g$ is a map $D^{n}\to X$ , then $f^{*}([g])=[f\circ g]$ .

More algebraically, we can define homotopy groups inductively by $\pi_{n}(X,x_{0})\cong\pi_{n-1}(\Omega X,y_{0})$ , where $\Omega X$ is the loop space of $X$ , and $y_{0}$ is the constant path sitting at $x_{0}$ .

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

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

Some examples are:

$\pi_{n}(S^{n})=\mathbb{Z}$ .

$\pi_{m}(S^{n})=0$ if $m<n$ .

$\pi_{n}(S^{1})=0$ if $n>1$ .

$\pi_{n}(M)=0$ for $n>1$ where $M$ is any surface of nonpositive Euler characteristic (not a sphere or projective plane).

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