Homotopy

In topology, two continuous functions from one topological space to another are called homotopic (from Greek ὁμός homós 'same, similar' and τόπος tópos 'place') if one can be 'continuously deformed' into the other, such a deformation being called a homotopy between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In topology, two continuous functions from one topological space to another are called homotopic (from Greek ὁμός homós 'same, similar' and τόπος tópos 'place') if one can be 'continuously deformed' into the other, such a deformation being called a homotopy between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. Formally, a homotopy between two continuous functions f and g from a topological space X to a topological space Y is defined to be a continuous function H : X × [ 0 , 1 ] → Y {displaystyle Hcolon X imes o Y} from the product of the space X with the unit interval to Y such that H ( x , 0 ) = f ( x ) {displaystyle H(x,0)=f(x)} and H ( x , 1 ) = g ( x ) {displaystyle H(x,1)=g(x)} for all x ∈ X {displaystyle xin X} . If we think of the second parameter of H as time then H describes a continuous deformation of f into g: at time 0 we have the function f and at time 1 we have the function g. We can also think of the second parameter as a 'slider control' that allows us to smoothly transition from f to g as the slider moves from 0 to 1, and vice versa. An alternative notation is to say that a homotopy between two continuous functions f , g : X → Y {displaystyle f,gcolon X o Y} is a family of continuous functions h t : X → Y {displaystyle h_{t}colon X o Y} for t ∈ [ 0 , 1 ] {displaystyle tin } such that h 0 = f {displaystyle h_{0}=f} and h 1 = g {displaystyle h_{1}=g} , and the map ( x , t ) ↦ h t ( x ) {displaystyle (x,t)mapsto h_{t}(x)} is continuous from X × [ 0 , 1 ] {displaystyle X imes } to Y {displaystyle Y} . The two versions coincide by setting h t ( x ) = H ( x , t ) {displaystyle h_{t}(x)=H(x,t)} . It is not sufficient to require each map h t ( x ) {displaystyle h_{t}(x)} to be continuous. The animation that is looped above right provides an example of a homotopy between two embeddings, f and g, of the torus into R3. X is the torus, Y is R3, f is some continuous function from the torus to R3 that takes the torus to the embedded surface-of-a-doughnut shape with which the animation starts; g is some continuous function that takes the torus to the embedded surface-of-a-coffee-mug shape. The animation shows the image of ht(x) as a function of the parameter t, where t varies with time from 0 to 1 over each cycle of the animation loop. It pauses, then shows the image as t varies back from 1 to 0, pauses, and repeats this cycle. Continuous functions f and g are said to be homotopic if and only if there is a homotopy H taking f to g as described above.Being homotopic is an equivalence relation on the set of all continuous functions from X to Y. This homotopy relation is compatible with function composition in the following sense: if f1, g1 : X → Y are homotopic, and f2, g2 : Y → Z are homotopic, then their compositions f2 ∘ f1 and g2 ∘ g1 : X → Z are also homotopic. Given two spaces X and Y, we say they are homotopy equivalent, or of the same homotopy type, if there exist continuous maps f : X → Y and g : Y → X such that g ∘ f is homotopic to the identity map idX and f ∘ g is homotopic to idY. The maps f and g are called homotopy equivalences in this case. Every homeomorphism is a homotopy equivalence, but the converse is not true: for example, a solid disk is not homeomorphic to a single point (since there is no bijection between them), although the disk and the point are homotopy equivalent (since you can deform the disk along radial lines continuously to a single point). As another example, the Möbius strip and an untwisted strip are homotopy equivalent (since you can deform both strips continuously to a circle) but not homeomorphic. Spaces that are homotopy equivalent to a point are called contractible.