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 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) Noncommutative geometry (e.g., KK-theory and Noncommutative topology)
Concepts
Spaces and maps 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 weak 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. The Cartesian product of two pointed spaces X , Y {\displaystyle X,Y} are not naturally pointed. A substitute is the smash product X ∧ Y {\displaystyle X\wedge Y} which is characterized by the adjoint relation
Map ( X ∧ Y , Z ) = Map ( X , Map ( Y , Z ) ) {\displaystyle \operatorname {Map} (X\wedge Y,Z)=\operatorname {Map} (X,\operatorname {Map} (Y,Z))} , that is, a smash product is an analog of a tensor product in abstract algebra (see tensor-hom adjunction). Explicitly, X ∧ Y {\displaystyle X\wedge Y} is the quotient of X × Y {\displaystyle X\times Y} by the wedge sum X ∨ Y {\displaystyle X\vee Y} .
Homotopy
Let I denote the unit interval [ 0 , 1 ] {\displaystyle [0,1]} . A map
h : X × I → Y {\displaystyle h:X\times I\to Y}
is called a homotopy from the map h 0 {\displaystyle h_{0}} to the map h 1 {\displaystyle h_{1}} , where h t ( x ) = h ( x , t ) {\displaystyle h_{t}(x)=h(x,t)} . Intuitively, we may think of h {\displaystyle h} as a path from the map h 0 {\displaystyle h_{0}} to the map h 1 {\displaystyle h_{1}} . Indeed, a homotopy can be shown to be an equivalence relation. When X, Y are pointed spaces, the maps h t {\displaystyle h_{t}} are required to preserve the basepoint and the homotopy h {\displaystyle h} is called a based homotopy. A based homotopy is the same as a (based) map X ∧ I + → Y {\displaystyle X\wedge I_{+}\to Y} where I + {\displaystyle I_{+}} is I {\displaystyle I} together with a disjoint basepoint. Given a pointed space X and an integer n ≥ 0 {\displaystyle n\geq 0} , let π n X = [ S n , X ] {\displaystyle \pi _{n}X=[S^{n},X]} be the homotopy classes of based maps S n → X {\displaystyle S^{n}\to X} from a (pointed) n-sphere S n {\displaystyle S^{n}} to X. As it turns out,
for n ≥ 1 {\displaystyle n\geq 1} , π n X {\displaystyle \pi _{n}X} are groups called homotopy groups; in particular, π 1 X {\displaystyle \pi _{1}X} is called the fundamental group of X, for n ≥ 2 {\displaystyle n\geq 2} , π n X {\displaystyle \pi _{n}X} are abelian groups by the Eckmann–Hilton argument,
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