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 map
is called a homotopy from the map to the map , where . Intuitively, we may think of as a path from the map to the map . Indeed, a homotopy can be shown to be an equivalence relation. When X, Y are pointed spaces, the are required to preserve the basepoints.
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, for , are groups called homotopy groups; in particular, is called the fundamental group of X. Every group is the fundamental group of some space.[1]
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 is called a homotopy equivalence if there is another map such that and are both homotopic to the identities. Two spaces are said to be homotopy equivalent if there is a homotopy equivalence between them. A homotopy equivalence class of spaces is then called a homotopy type. There is a weaker notion: a map is said to be a weak homotopy equivalence if is an isomorphism for each and each choice of a base point. (Note the existence of such a map is more strict than saying that have isomorphic homotopy set/groups.) In general, a homotopy equivalence is a weak homotopy equivalence but the converse need not be true.
CW complex
[edit]A CW complex is a space that has a filtration whose union is and such that
- is a discrete space, called the set of 0-cells (vertices) in .
- Each is obtained by attaching several n-disks, n-cells, to via maps ; i.e., the boundary of an n-disk is identified with the image of in .
- A subset is open if and only if is open for each .
For example, a sphere has two cells: one 0-cell and one -cell, since can be obtained by collapsing the boundary of the n-disk to a point. Also, every compact manifold has a homotopy type of a CW complex.[citation needed]
Remarkably, Whitehead's theorem says that for CW complexes, a weak homotopy equivalence and a homotopy equivalence are the same thing.
Cofibration and fibration
[edit]A map is called a cofibration if given:
- A map , and
- A homotopy
such that , there exists a homotopy that extends and such that . An example is a neighborhood deformation retract; that is, contains a mapping cylinder neighborhood of a closed subspace and the inclusion (e.g., a tubular neighborhood of a closed submanifold).[2] In fact, a cofibration can be characterized as a neighborhood deformation retract pair.[3] Another basic example is a CW pair ; i.e., is a subcomplex of . Many often work only with CW complexes and the notion of a cofibration there is then 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 such that , there exists a homotopy that extends and such that . 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 K-theory is an example of a generalized cohomology theory.
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]Abstract homotopy theory is an axiomatic approach to homotopy theory. Such axiomatization is useful for non-traditional applications of homotopy theory. One approach to axiomatization is by Quillen's model categories. A model category is a category with a choice of three classes of maps called weak equivalences, cofibrations and fibrations, subject to the axioms that are reminiscent of facts in algebraic topology. For example, the category of (reasonable) topological spaces has a structure of a model category where a weak equivalence is a weak homotopy equivalence, a cofibration a certain retract and a fibration a Serre fibration.[4] Another example is the category of non-negatively graded chain complexes over a fixed base ring.[5]
Concepts
[edit]Simplicial homotopy theory
[edit]See also: Algebraic homotopy
See also
[edit]References
[edit]- ^ May, Ch 4. § 5.
- ^ Hatcher, Example 0.15.
- ^ May, Ch 6. § 4.
- ^ Dwyer & Spalinski, Example 3.5.
- ^ Dwyer & Spalinski, Example 3.7.
- 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.
- https://ncatlab.org/nlab/show/homotopical+algebra
- Homotopy Theories and Model Categories by W.G. Dwyer and J. Spalinski in Handbook of Algebraic Topology edited by I.M. James
- Hatcher, Allen. "Algebraic topology".
- Edwin Spanier, Algebraic topology
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
- D. Quillen, Homotopical algebra, Lectures Notes in Math. vol. 43, Springer Verlag, 1967.
- https://ncatlab.org/nlab/show/homotopy+theory
External links
[edit]"homotopy theory". ncatlab.org.