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Concept# Homotopy theory

Summary

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 studied as an independent discipline. Besides algebraic topology, the theory has also been used in other areas of mathematics such as algebraic geometry (e.g., A1 homotopy theory) and (specifically the study of ).
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
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 where the are the points of X and the morphisms are paths.
A map is called a cofibration if given (1) a map and (2) a homotopy , there exists a homotopy that extends and such that . To some loose sense, it is an analog of the defining diagram of an injective module in abstract algebra.

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Suspension (topology)

In topology, a branch of mathematics, the suspension of a topological space X is intuitively obtained by stretching X into a cylinder and then collapsing both end faces to points. One views X as "suspended" between these end points. The suspension of X is denoted by SX or susp(X). There is a variation of the suspension for pointed space, which is called the reduced suspension and denoted by ΣX. The "usual" suspension SX is sometimes called the unreduced suspension, unbased suspension, or free suspension of X, to distinguish it from ΣX.

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In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the Freudenthal suspension theorem, which states that given any pointed space , the homotopy groups stabilize for sufficiently large. In particular, the homotopy groups of spheres stabilize for . For example, In the two examples above all the maps between homotopy groups are applications of the suspension functor.

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