homotopy groups HomotopyGroups 2002-02-02 03:37:46 2003-08-13 20:23:40 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here 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 \PMlinkescapetext{fixed point}, 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).