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* =
S^{n}.

$ __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].

@ __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 π_{n}(*X*, *x*_{0}):= π_{n–1}(Ω_{0},* C*)
for *n* ≥ 2, but there is another, simpler definition.

* __Zeroth__: The zeroth homotopy group π_{0}(*X*)
is related to the connectedness of *X*.

* __First__: The first homotopy group π_{1}(*X*)
is also called fundamental group of *X*.

* __Properties__: All π_{n}(*X*)
for *n* ≥ 2 are Abelian, while π_{1} in
general is not; One which is not, e.g., is

π_{1}(D^{2}–{*p*}–{*q*})
= π_{1}(figure 8) .

* __Results__: If *X* is
contractible, π_{q}(*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__:

π_{q}(S^{1})
= 0 for *q* ≥ 2
, π_{q}(S^{n})
= 0 for *q* < *n* , π_{n}(S^{n})
= \(\mathbb Z\) ,

while π_{q}(S^{n})
for *q* > *n* is non-trivial and not known [for a non-trivial
map S^{3 }→ S^{2}, @ Hu 59, ch3].

* __For projective space__:

π_{q}(\(\mathbb R\)P^{n})
= π_{q}(S^{n}) for *q* ≥ 2, and
all *n*.

* __For various Lie groups__:

π_{2}(SU(*n*))
= 0 (this holds in general for all semisimple
groups) ,

π_{4}(SU(2))
=\(\mathbb Z\)_{2} ,

π_{4}(SU(*n*))
= 0 for *n* > 2 ,

π_{q}(U(*n*))
= \(\mathbb Z\), 0, ... (period 2) (*q* =
1: electromagnetism;
*q* = 5, chiral Lagrangian) ,

π_{q}(O(*n*))
= \(\mathbb Z\)_{2}, 0, \(\mathbb Z\),
0, 0, 0, \(\mathbb Z\), \(\mathbb Z\)_{2},
... (period 8) (*q* = 1, spin) ,

π_{q}(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, π_{q}(O(*n*)/O(*n*–*k*)) = 0 if 0
≤ *q* ≤ *n*–*k*–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]; 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.

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feb 2016