# 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.

$\pi_{n}(S^{1})=0$ 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 HomotopyGroups 2013-03-22 12:15:28 2013-03-22 12:15:28 bwebste (988) bwebste (988) 13 bwebste (988) Definition msc 54-00 higher homotopy groups EilenbergMacLaneSpace HomotopyDoubleGroupoidOfAHausdorffSpace QuantumFundamentalGroupoids CohomologyGroupTheorem