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.
Applications to other fields of mathematics
[edit]Besides algebraic topology, the theory has also been used in other areas of mathematics such as:
- Algebraic geometry (e.g., A1 homotopy theory)
- Category theory (specifically the study of higher categories)
Concepts
[edit]Spaces and maps
[edit]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
[edit]Let I denote the unit interval. A family of maps indexed by I, is called a homotopy from to if is a map (e.g., it must be a continuous function). When X, Y are pointed spaces, the are required to preserve the basepoints. A homotopy can be shown to be an equivalence relation. Given a pointed space X and an integer , let be the homotopy classes of based maps from a (pointed) n-sphere to X. As it turns out, are groups; in particular, 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
[edit]A map is called a cofibration if given:
- A map , and
- A homotopy .
There exists a homotopy that extends and such that . 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 ; 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 is a fibration if given (1) a map and (2) a homotopy , there exists a homotopy such that is the given one and . A basic example is a covering map (in fact, a fibration is a generalization of a covering map). If 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 is an example of a fibration.
Classifying spaces and homotopy operations
[edit]Given a topological group G, the classifying space for principal G-bundles ("the" up to equivalence) is a space such that, for each space X,
- {principal G-bundle on X} / ~
where
- the left-hand side is the set of homotopy classes of maps ,
- ~ refers isomorphism of bundles, and
- = is given by pulling-back the distinguished bundle on (called universal bundle) along a map .
Brown's representability theorem guarantees the existence of classifying spaces.
Spectrum and generalized cohomology
[edit]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 ),
where 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:
Key theorems
[edit]- 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
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See also: Characteristic class, Postnikov tower, Whitehead torsion
Localization and completion of a space
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Specific theories
[edit]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
[edit]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.
Abstract homotopy theory
[edit]Concepts
[edit]Model categories
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Simplicial homotopy theory
[edit]See also
[edit]References
[edit]- 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
[edit]- Cisinski's notes
- http://ncatlab.org/nlab/files/Abstract-Homotopy.pdf
- Math 527 - Homotopy Theory Spring 2013, Section F1, lectures by Martin Frankland
External links
[edit]"homotopy theory". ncatlab.org.