Homotopy Groups and the Classification of Defects

Homotopy groups $π_{n} (X)$ measure the inequivalent ways an $n$ -sphere can be mapped into a space $X$ — the " $n$ -dimensional holes". They are the tool that classifies topological field configurations: a topological charge lives in $π_{n} (vacuum manifold)$ . This page develops the homotopy groups physics needs and tabulates the key values; the field-theory application is defects-and-solitons.md. It complements the cohomology of de-rham.md.

Homotopy of maps

Two continuous maps are homotopic if one can be continuously deformed into the other. The $n$ -th homotopy group $π_{n} (X, x_{0})$ is the set of homotopy classes of based maps $(S^{n}, *) \to (X, x_{0})$ , with group operation "concatenation":

$π_{0} (X)$ — path components (a set, a group only if $X$ is a group);
$π_{1} (X)$ — the fundamental group, loops up to deformation (generally non-abelian);
$π_{n} (X)$ for $n \geq 2$ — higher homotopy, always abelian.

A space is $n$ -connected if $π_{k} = 0$ for $k \leq n$ ; simply connected means $π_{1} = 0$ (already used for covering groups).

The values physics needs

$X$	$π_{1}$	$π_{2}$	$π_{3}$	note
$S^{1}$	$Z$	$0$	$0$	winding number
$S^{2}$	$0$	$Z$	$Z$	( $π_{3} (S^{2}) = Z$ = Hopf)
$S^{n}$	$0$	$0 (n > 2)$	—	$π_{n} (S^{n}) = Z$
$U (1)$	$Z$	$0$	$0$	phase / EM
$S U (N) (N \geq 2)$	$0$	$0$	$Z$	$π_{3} (G) = Z$ for simple $G$
$SO (3)$	$Z_{2}$	$0$	$Z$	spinor sign

Two structural facts do most of the work:

$π_{n} (S^{n}) = Z$ — the degree/winding number of a self-map of the sphere.
$π_{3} (G) = Z$ for every compact simple Lie group — the reason 4D instantons ( $π_{3}$ of the gauge group) exist for any non-abelian gauge theory.

Fibrations and the long exact sequence

The main computational tool: a fiber bundle $F \to E \to B$ (fiber-bundles.md) induces a long exact sequence of homotopy groups $\dots \to π_{n} (F) \to π_{n} (E) \to π_{n} (B) \to π_{n - 1} (F) \to \dots .$ From it one computes homotopy groups of coset/vacuum manifolds $G / H$ — exactly the spaces that arise in symmetry breaking. For example $π_{2} (G / H) ≅ π_{1} (H)$ when $G$ is simply connected, which is how monopoles are counted below.

Topological charge

When field configurations are maps $ϕ : S^{n} \to X$ (into a vacuum manifold $X$ ), their homotopy class is a conserved topological charge $Q_{top} = [ϕ] \in π_{n} (X),$ conserved because a finite-energy continuous evolution cannot jump between homotopy classes. This is precisely the $Q_{top} \in π_{n} (vacuum manifold)$ asserted in advanced/topological.md; the homotopy group here supplies its rigorous meaning, and the characteristic-class integrals compute it. Which $n$ (hence which kind of defect) applies is the subject of defects-and-solitons.md.

References

Hatcher, Algebraic Topology, Ch. 4 (free online).
Nakahara, Geometry, Topology and Physics, Ch. 4.
Mermin, "The topological theory of defects in ordered media", Rev. Mod. Phys. 51, 591 (1979).