Homotopy theory

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.

Let I denote the unit interval. A family of maps indexed by I, ${\displaystyle h_{t}:X\to Y}$ is called a homotopy from ${\displaystyle h_{0}}$ to ${\displaystyle h_{1}}$ if ${\displaystyle h:I\times X\to Y,(t,x)\mapsto h_{t}(x)}$ is a map (e.g., it must be a continuous function). When X, Y are pointed spaces, the ${\displaystyle h_{t}}$ are required to preserve the basepoints. A homotopy can be shown to be an equivalence relation. Given a pointed space X and an integer ${\displaystyle n\geq 1}$, let ${\displaystyle \pi _{n}(X)=[S^{n},X]_{*}}$ be the homotopy classes of based maps ${\displaystyle S^{n}\to X}$ from a (pointed) n-sphere ${\displaystyle S^{n}}$ to X. As it turns out, ${\displaystyle \pi _{n}(X)}$ are groups; in particular, ${\displaystyle \pi _{1}(X)}$ 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.

A map ${\displaystyle f:A\to X}$ is called a cofibration if given (1) a map ${\displaystyle h_{0}:X\to Z}$ and (2) a homotopy ${\displaystyle g_{t}:A\to Z}$, there exists a homotopy ${\displaystyle h_{t}:X\to Z}$ that extends ${\displaystyle h_{0}}$ and such that ${\displaystyle h_{t}\circ f=g_{t}}$. To 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 ${\displaystyle (X,A)}$; 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 ${\displaystyle p:X\to B}$ is a fibration if given (1) a map ${\displaystyle Z\to X}$ and (2) a homotopy ${\displaystyle g_{t}:Z\to B}$, there exists a homotopy ${\displaystyle h_{t}:Z\to X}$ such that ${\displaystyle h_{0}}$ is the given one and ${\displaystyle p\circ h_{t}=g_{t}}$. A basic example is a covering map (in fact, a fibration is a generalization of a covering map). If ${\displaystyle E}$ 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 ${\displaystyle p:E\to X}$ is an example of a fibration.

Given a topological group G, the classifying space for principal G-bundles ("the" up to equivalence) is a space ${\displaystyle BG}$ such that, for each space X,

${\displaystyle [X,BG]=}$ {principal G-bundle on X} / ~ ${\displaystyle ,\,\,[f]\mapsto f^{*}EG}$

where

• the left-hand side is the set of homotopy classes of maps ${\displaystyle X\to BG}$,
• ~ refers isomorphism of bundles, and
• = is given by pulling-back the distinguished bundle ${\displaystyle EG}$ on ${\displaystyle BG}$ (called universal bundle) along a map ${\displaystyle X\to BG}$.

Brown's representability theorem guarantees the existence of classifying spaces.

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 ${\displaystyle \mathbb {Z} }$),

${\displaystyle [X,K(A,n)]=\operatorname {H} ^{n}(X;A)}$

where ${\displaystyle K(A,n)}$ 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: ${\displaystyle S^{0}\to S^{1}\to S^{2}\to \cdots }$

• fiber sequence
• cofiber sequence

• Simplicial homotopy