Homotopy theory

In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology, but nowadays is learned as an independent discipline. Besides algebraic topology, the theory has also been used in other areas of mathematics such as algebraic geometry (e.g., A1 homotopy theory) and category theory (specifically the study of higher categories).

Spaces and maps
In homotopy theory and algebraic topology, the word "space" denotes a topological space. In order to avoid pathologies, one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being compactly generated, or Hausdorff, or a CW complex.

In the same vein as above, a "map" is a continuous function, possibly with some extra constraints.

Often, one works with a pointed space—that is, a space with a "distinguished point", called a basepoint. A pointed map is then a map which preserves basepoints; that is, it sends the basepoint of the domain to that of the codomain. In contrast, a free map is one which needn't preserve basepoints.

Homotopy
Let I denote the unit interval. A family of maps indexed by I, $$h_t : X \to Y$$ is called a homotopy from $$h_0$$ to $$h_1$$ if $$h : I \times X \to Y, (t, x) \mapsto h_t(x)$$ is a map (e.g., it must be a continuous function). When X, Y are pointed spaces, the $$h_t$$ are required to preserve the basepoints. A homotopy can be shown to be an equivalence relation. Given a pointed space X and an integer $$n \ge 1$$, let $$\pi_n(X) = [S^n, X]_*$$ be the homotopy classes of based maps $$S^n \to X$$ from a (pointed) n-sphere $$S^n$$ to X. As it turns out, $$\pi_n(X)$$ are groups; in particular, $$\pi_1(X)$$ is called the fundamental group of X.

If one prefers to work with a space instead of a pointed space, there is the notion of a fundamental groupoid (and higher variants): by definition, the fundamental groupoid of a space X is the category where the objects are the points of X and the morphisms are paths.

Cofibration and fibration
A map $$f: A \to X$$ is called a cofibration if given (1) a map $$h_0 : X \to Z$$ and (2) a homotopy $$g_t : A \to Z$$, there exists a homotopy $$h_t : X \to Z$$ that extends $$h_0$$ and such that $$h_t \circ f = g_t$$. In some loose sense, it is an analog of the defining diagram of an injective module in abstract algebra. The most basic example is a CW pair $$(X, A)$$; since many work only with CW complexes, the notion of a cofibration is often implicit.

A fibration in the sense of Serre is the dual notion of a cofibration: that is, a map $$p : X \to B$$ is a fibration if given (1) a map $$Z \to X$$ and (2) a homotopy $$g_t : Z \to B$$, there exists a homotopy $$h_t: Z \to X$$ such that $$h_0$$ is the given one and $$p \circ h_t = g_t$$. A basic example is a covering map (in fact, a fibration is a generalization of a covering map). If $$E$$ is a principal G-bundle, that is, a space with a free and transitive (topological) group action of a (topological) group, then the projection map $$p: E \to X$$ is an example of a fibration.

Classifying spaces and homotopy operations
Given a topological group G, the classifying space for principal G-bundles ("the" up to equivalence) is a space $$BG$$ such that, for each space X,
 * $$[X, BG] = $$ {principal G-bundle on X} / ~ $$, \,\, [f] \mapsto f^* EG$$

where Brown's representability theorem guarantees the existence of classifying spaces.
 * the left-hand side is the set of homotopy classes of maps $$X \to BG$$,
 * ~ refers isomorphism of bundles, and
 * = is given by pulling-back the distinguished bundle $$EG$$ on $$BG$$ (called universal bundle) along a map $$X \to BG$$.

Spectrum and generalized cohomology
The idea that a classifying space classifies principal bundles can be pushed further. For example, one might try to classify cohomology classes: given an abelian group A (such as $$\mathbb{Z}$$),
 * $$[X, K(A, n)] = \operatorname{H}^n(X; A)$$

where $$K(A, n)$$ is the Eilenberg–MacLane space. The above equation leads to the notion of a generalized cohomology theory; i.e., a contravariant functor from the category of spaces to the category of abelian groups that satisfies the axioms generalizing ordinary cohomology theory. As it turns out, such a functor may not be representable by a space but it can always be represented by a sequence of (pointed) spaces with structure maps called a spectrum. In other words, to give a generalized cohomology theory is to give a spectrum.

A basic example of a spectrum is a sphere spectrum: $$S^0 \to S^1 \to S^2 \to \cdots$$

Key theorems

 * Seifert–van Kampen theorem
 * Homotopy excision theorem
 * Freudenthal suspension theorem (a corollary of the excision theorem)
 * Landweber exact functor theorem
 * Dold–Kan correspondence
 * Eckmann–Hilton argument - this shows for instance higher homotopy groups are abelian.
 * Universal coefficient theorem

Obstruction theory and characteristic class
See also: Characteristic class, Postnikov tower, Whitehead torsion

Specific theories
There are several specific theories
 * simple homotopy theory
 * stable homotopy theory
 * chromatic homotopy theory
 * rational homotopy theory
 * p-adic homotopy theory
 * equivariant homotopy theory

Homotopy hypothesis
One of the basic questions in the foundations of homotopy theory is the nature of a space. The homotopy hypothesis asks whether a space is something fundamentally algebraic.

Concepts

 * fiber sequence
 * cofiber sequence

Simplicial homotopy theory

 * Simplicial homotopy