# 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
${\pi}_{n}(X,{x}_{0})\cong {\pi}_{n-1}(\mathrm{\Omega}X,{y}_{0})$, 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^{}).

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 |