Homotopy Theory

In General > s.a. Bott's Periodicity Theorem; Hurewicz Theorem [relation to homology].
* Idea: A partial way to classify topological spaces X is to take some space Y and divide the set of continuous maps from Y to X into homotopy classes (two maps are homotopic if they can be continuiusly deformed into each other); One usually standardizes the procedure by taking Y = Sn.
$Homotopy of paths: An equivalence relation; Two paths σ: I → X and τ: I → X are homotopic rel(0,1) if there is a map F: I × I → X, such that...; > s.a. lorentzian geometry.$ Homotopy of maps:
@ General references: Hu 59; Adams 74 [stable]; Gray 75; Whitehead 78; Goerss & Jardine 99 [simplicial]; Munson & Volić 15 [cubical]; Schwede 18 [III]; Richter 20 [and category theory].
@ Controlled simple homotopy: Chapman 83.
@ Generalizations: Heller 88; Hayashi PTP(89) [abhomotopy groups]; Maumary & Ojima mp/00 [and supersymmetry]; Bergner Top(07) [homotopy theory of homotopy theories]; Riehl 14 [categorical homotopy theory]; Park a1510 [homotopy probability theory]; > s.a. algebraic topology [more refs].
> Online resources: see Wikipedia page and category page; s.a. Stable Homotopy Theory page.

Homotopy Groups > s.a. fundamental group.
\$ Def: Higher homotopy groups can be recursively defined by πn(X, x0):= πn−10, C) for n ≥ 2, but there is another, simpler definition.
* Zeroth: The zeroth homotopy group π0(X) is related to the connectedness of X.
* First: The first homotopy group π1(X) is also called fundamental group of X.
* Properties: All πn(X) for n ≥ 2 are Abelian, while π1 in general is not; One which is not, e.g., is

π1(D2−{p}−{q}) = π1(figure 8) .

* Results: If X is contractible, πq(X) = 0 for all q.

Examples of (Higher) Homotopy Groups
* In general: The computation of homotopy groups is often not easy, and one can use the Hurewicz theorem to reduce it to the easier calculation of homology, or other tools like bundle theory.
*
For spheres:

πq(S1) = 0   for   q ≥ 2 ,   πq(Sn) = 0   for   q < n ,   πn(Sn) = $$\mathbb Z$$ ,

while πq(Sn) for q > n is non-trivial and not known [for a non-trivial map S3 → S2, @ Hu 59, ch3].
* For projective space:

πq($$\mathbb R$$Pn) = πq(Sn)   for   q ≥ 2,   and all n.

* For various Lie groups:

π2(SU(n)) = 0   (this holds in general for all semisimple groups) ,
π4(SU(2)) =$$\mathbb Z$$2 ,
π4(SU(n)) = 0   for   n > 2 ,
πq(U(n)) = $$\mathbb Z$$, 0, ... (period 2)   (q = 1: electromagnetism; q = 5, chiral Lagrangian) ,
πq(O(n)) = $$\mathbb Z$$2, 0, $$\mathbb Z$$, 0, 0, 0, $$\mathbb Z$$, $$\mathbb Z$$2, ... (period 8)   (q = 1, spin) ,
πq(Sp(n)) = 0, 0, $$\mathbb Z$$, $$\mathbb Z$$2, $$\mathbb Z$$2, 0, $$\mathbb Z$$, 0, ... (period 8)   (q = 3, instantons; q = 4, Witten) .

* For other spaces: For coset spaces of orthogonal groups, πq(O(n)/O(nk)) = 0 if 0 ≤ qnk–1.
@ References: in Iyanaga & Kawada 80 [for Lie groups]; Ghane et al T&A(11) [for separable metric spaces].

And Physics > s.a. causal structures [chronological homotopy theory]; gauge theory; topological field theory [homotopy quantum field theory].
* Applications: The theory is used to study and classify defects and textures in an ordered medium (e.g., point, line or ring defects in spin lattices), or in cosmology (as in topological defects).
@ References: Mermin RMP(79) [and defects in ordered media]; Nash & Sen 83.
> Related topics: see supersymmetry in field theory.