Homotopy theory

It originated as a topic in algebraic topology, but nowadays is learned as an independent discipline.In order to avoid pathologies, one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being compactly generated weak Hausdorff or a CW complex.In the same vein as above, a "map" is a continuous function, possibly with some extra constraints.In contrast, a free map is one which needn't preserve basepoints.In general, every manifold has the homotopy type of a CW complex;[3] in fact, Morse theory implies that a compact manifold has the homotopy type of a finite 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.Similar statements also hold for pairs and excisive triads.The above theorem justifies a common habit of working only with CW complexes.the inclusion (e.g., a tubular neighborhood of a closed submanifold).[7] In fact, a cofibration can be characterized as a neighborhood deformation retract pair.; many often work only with CW complexes and the notion of a cofibration there is then often implicit.A fibration in the sense of Hurewicz is the dual notion of a cofibration: that is, a mapA basic example of a fibration is a covering map as it comes with a unique path lifting.is a principal G-bundle over a paracompact space, that is, a space with a free and transitive (topological) group action of a (topological) group, then the projection map[12] While a cofibration is injective with closed image,[13] a fibration need not be surjective.is said to satisfy the lifting property[15] if for each commutative square diagram there is a mapA Serre fibration is a map satisfying the RLP for the inclusionsPrecisely, they are defined as[17] Because of the adjoint relation between a smash product and a mapping space, we have: These functors are used to construct fiber sequences and cofiber sequences.restrict to the category of CW complexes in the following weak sense: a theorem of Milnor says that ifhas the homotopy type of a CW complex, then so does its loop spaceThe 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.See also: Characteristic class, Postnikov tower, Whitehead torsion There are several specific theories 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.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 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.[20] Another example is the category of non-negatively graded chain complexes over a fixed base ring.of a simplicial set is a CW complex and the compositionIn fact, a simplicial set is the nerve of some category if and only if it satisfies the Segal conditions (a theorem of Grothendieck).Since simplicial sets are sort of abstract spaces (if not topological spaces), it is possible to develop the homotopy theory on them, which is called the simplicial homotopy theory.
mathematicshomotopiesalgebraic topologyAlgebraic geometryA1 homotopy theoryCategory theoryhigher categoriestopological spacepathologiescompactly generated weak HausdorffCW complexpointed spacesmash productadjoint relationtensor producttensor-hom adjunctionwedge sumHomotopyequivalence relationintegergroupshomotopy groupsfundamental groupabelian groupsEckmann–Hilton argumenthomotopy equivalencehomotopy typeweak homotopy equivalenceMorse theoryWhitehead's theoremhomotopy categoryCW approximationsingular chaincellular homologycofibrationneighborhood deformation retractmapping cylindertubular neighborhoodCW pairfibrationmapping path spacecovering mapprincipal G-bundlegroup actiontopologicallifting propertymodel categoriesright lifting propertyleft lifting propertySerre fibrationloop functorsuspension functorfiber sequencescofiber sequenceshomotopy fiberclassifying spaceprincipal G-bundlesBrown's representability theoremSpectrum (algebraic topology)Generalized cohomologyabelian groupEilenberg–MacLane spacecontravariant functorcategory of abelian groupsrepresentableK-theorysphere spectrumRing spectrumModule spectrumSeifert–van Kampen theoremHomotopy excision theoremFreudenthal suspension theoremLandweber exact functor theoremDold–Kan correspondenceabelianUniversal coefficient theoremDold–Thom theoremCharacteristic classPostnikov towerWhitehead torsionLocalization of a topological spacesimple homotopy theorystable homotopy theorychromatic homotopy theoryrational homotopy theoryHomotopy hypothesisfundamental groupoidcategoryobjectsmorphismsAlgebraic homotopySimplicial setsimplicial homotopy theorysimplicial complexsingular homologynerve of a categorySegal conditions ∞ {\displaystyle \infty } -category ∞ {\displaystyle \infty } -groupoidHighly structured ring spectrumHomotopy type theoryPursuing StacksShape theoryModuli stack of formal group lawsCrossed moduleUniversity of ChicagoUniversity of Chicago PressUniversity of EdinburghWhitehead, George WilliamTransactions of the American Mathematical SocietyAnnals of MathematicsCôte d'Azur UniversityUniversity of ToulouseUniversity of Illinois Urbana-ChampaignUniversity of Regina