In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from Ancient 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 (/həˈmɒtəpiː/,^[1] hə-MO-tə-pee; /ˈhoʊmoʊˌtoʊpiː/,^[2] HOH-moh-toh-pee) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.^[3]

In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra.

Formal definition

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 ${\displaystyle H:X\times \to Y}$ from the product of the space X with the unit interval to Y such that ${\displaystyle H(x,0)=f(x)}$ and ${\displaystyle H(x,1)=g(x)}$ for all ${\displaystyle x\in X}$.

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 ${\displaystyle f,g:X\to Y}$ is a family of continuous functions ${\displaystyle h_{t}:X\to Y}$ for ${\displaystyle t\in }$ such that ${\displaystyle h_{0}=f}$ and ${\displaystyle h_{1}=g}$, and the map ${\displaystyle (x,t)\mapsto h_{t}(x)}$ is continuous from ${\displaystyle X\times }$ to ${\displaystyle Y}$. The two versions coincide by setting ${\displaystyle h_{t}(x)=H(x,t)}$. It is not sufficient to require each map ${\displaystyle h_{t}(x)}$ to be continuous.^[4]

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 h_t(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.

Properties

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.

Examples

• If ${\displaystyle f,g:\mathbb {R} \to \mathbb {R} ^{2}}$ are given by ${\displaystyle f(x):=\left(x,x^{3}\right)}$ and ${\displaystyle g(x)=\left(x,e^{x}\right)}$, then the map ${\displaystyle H:\mathbb {R} \times \to \mathbb {R} ^{2}}$ given by ${\displaystyle H(x,t)=\left(x,(1-t)x^{3}+te^{x}\right)}$ is a homotopy between them.
• More generally, if ${\displaystyle C\subseteq \mathbb {R} ^{n}}$ is a convex subset of Euclidean space and ${\displaystyle f,g:\to C}$ are paths with the same endpoints, then there is a linear homotopy^[5] (or straight-line homotopy) given by

  ${\displaystyle {\begin{aligned}H:0,1\times 0,1&\longrightarrow C\\(s,t)&\longmapsto (1-t)f(s)+tg(s).\end{aligned}}}$

• Let ${\displaystyle \operatorname {id} _{B^{n}}:B^{n}\to B^{n}}$ be the identity function on the unit n-disk; i.e. the set ${\displaystyle B^{n}:=\left\{x\in \mathbb {R} ^{n}:\|x\|\leq 1\right\}}$. Let ${\displaystyle c_{\vec {0}}:B^{n}\to B^{n}}$ be the constant function ${\displaystyle c_{\vec {0}}(x):={\vec {0}}}$ which sends every point to the origin. Then the following is a homotopy between them:

  Zdroj:https://en.wikipedia.org?pojem=Homotopy
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