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 sphereto $x_0$. 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${\pi }_n(X,x_0)\cong {\pi }_{n-1}(\mathrm{\Omega }X,y_0)$, where $\mathrm{\Omega }X$ 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 .

${\pi }_n(S^1)=0$ 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).