Homotopy theory

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

Concepts

Spaces

In homotopy theory (as well as algebraic topology), one typically does not work with an arbitrary topological space to avoid pathologies in point-set topology. Instead, one assumes a space is a reasonable space; the meaning depends on authors but it can mean that a space is compactly generated Hausdorff space or is a CW complex. (In a sense, “what is a space” is not a settled matter in homotopy theory; cf. #Homotopy hypothesis below.)

Frequently, one works with a space X with some chosen point * in the space; such a space is called a based space. A map between based spaces are then required to preserve the base points. For example, if $I=[0,1]$ is the unit interval and 0 is the base point, then a map $f:I\to X$ is a path from the base point $*=f(0)$ to the point $f(1)$. The adjective “free” is used to indicate freeness of choice of base points; for example, a free path would be an arbitrary map $I\to X$ that does not necessarily preserve the base point (if any). A map between based spaces is also often called a based map, to emphasize that it is not a free map.

Homotopy

Main article: 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 based spaces, then $h_{t}$ are required to preserve the base points. A homotopy can be shown to be an equivalence relation. Given a based space X and an integer $n\geq 1$, let $\pi _{n}(X)=[S^{n},X]_{*}$ be the homotopy classes of based maps $S^{n}\to X$ from a (based) 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 based 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 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 $h_{t}\circ f=g_{t}$. 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 $(X,A)$; 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

• 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 $X\to BG$.

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

Spectrum and generalized cohomology

Main articles: Spectrum (algebraic topology) 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 groups that satisfies the axioms generalizing ordinary cohomology theory. As it turns out, such a space may not be representable by a space but by a sequence of (based) spaces is called 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

Obstruction theory

Postnikov tower stuff

Operations

Localization

Main article: Localization of a topological space

Specific theories

There are several specific theories

• stable homotopy theory
• chromatic homotopy theory
• rational homotopy theory
• p-adic homotopy theory
• equivariant homotopy theory

Homotopy hypothesis

Main article: 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; in other words,

Abstract homotopy theory

Concepts

• fiber sequence
• cofiber sequence

Simplicial homotopy theory

• simplicial homotopy

References

• May, J. A Concise Course in Algebraic Topology
• George William Whitehead (1978). Elements of homotopy theory. Graduate Texts in Mathematics. Vol. 61 (3rd ed.). New York-Berlin: Springer-Verlag. pp. xxi+744. ISBN 978-0-387-90336-1. MR 0516508. Retrieved September 6, 2011.
• Ronald Brown, Topology and groupoids (2006) Booksurge LLC ISBN 1-4196-2722-8.

Further reading

• Cisinski's notes
• http://ncatlab.org/nlab/files/Abstract-Homotopy.pdf

External links

• https://ncatlab.org/nlab/show/homotopy+theory