Homotopy

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 ${\displaystyle H:\mathbb{R}\times [0,1]\to {\mathbb{R}}^2}$ given by ${\displaystyle H(x,t)=\left(x,(1-t)x^3+t{e}^x\right)}$ is a homotopy between them.
• More generally, if ${\displaystyle C\subseteq {\mathbb{R}}^n}$ is a convex subset of Euclidean space and ${\displaystyle f,g:[0,1]\to C}$ are paths with the same endpoints, then there is a linear homotopy (or straight-line homotopy) given by

  ${\displaystyle \begin{array}{cc}H:[0,1]\times [0,1]& ⟶C\\ (s,t)& ⟼(1-t)f(s)+tg(s).\end{array}}$

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

  ${\displaystyle \begin{array}{cc}H:B^n\times [0,1]& ⟶B^n\\ (x,t)& ⟼(1-t)x.\end{array}}$

Homotopy equivalence

Given two topological spaces X and Y, a homotopy equivalence between X and Y is a pair of continuous maps f : X → Y and g : Y → X, such that g ∘ f is homotopic to the identity map id_X and f ∘ g is homotopic to id_Y. If such a pair exists, then X and Y are said to be homotopy equivalent, or of the same homotopy type. This relation of homotopy equivalence is often denoted ${\displaystyle \simeq }$. Intuitively, two spaces X and Y are homotopy equivalent if they can be transformed into one another by bending, shrinking and expanding operations. Spaces that are homotopy equivalent to a point are called contractible.

Homotopy equivalence vs. homeomorphism

A homeomorphism is a special case of a homotopy equivalence, in which g ∘ f is equal to the identity map id_X (not only homotopic to it), and f ∘ g is equal to id_Y. Therefore, if X and Y are homeomorphic, then they are homotopy equivalent, but the converse is not necessarily true. Some examples:

• A solid disk is homotopy equivalent to a single point, since you can deform the disk along radial lines continuously to a single point; however, they are not homeomorphic, since there is no bijection between them (since one is an infinite set, while the other is finite).
• The Möbius strip and an untwisted (closed) strip are homotopy equivalent, since you can deform both strips continuously to a circle. But they are not homeomorphic.

Examples

• The first example of a homotopy equivalence is ${\displaystyle {\mathbb{R}}^n}$ with a point, denoted ${\displaystyle {\mathbb{R}}^n\simeq \{0\}}$. The part that needs to be checked is the existence of a homotopy ${\displaystyle H:I\times {\mathbb{R}}^n\to {\mathbb{R}}^n}$ between ${\displaystyle {\mathrm{id}}_{{\mathbb{R}}^n}}$ and ${\displaystyle p_0}$, the projection of ${\displaystyle {\mathbb{R}}^n}$ onto the origin. This can be described as ${\displaystyle H(t,\cdot )=t\cdot p_0+(1-t)\cdot {\mathrm{id}}_{{\mathbb{R}}^n}}$.
• There is a homotopy equivalence between ${\displaystyle S^1}$ (the 1-sphere) and ${\displaystyle {\mathbb{R}}^2-\{0\}}$.
  • More generally, ${\displaystyle {\mathbb{R}}^n-\{0\}\simeq S^{n-1}}$.
• Any fiber bundle ${\displaystyle \pi :E\to B}$ with fibers ${\displaystyle F_b}$ homotopy equivalent to a point has homotopy-equivalent total and base spaces. This generalizes the previous two examples since ${\displaystyle \pi :{\mathbb{R}}^n-\{0\}\to S^{n-1}}$ is a fiber bundle with fiber ${\displaystyle {\mathbb{R}}_{>0}}$.
• Every vector bundle is a fiber bundle with a fiber homotopy equivalent to a point.
• ${\displaystyle {\mathbb{R}}^n-{\mathbb{R}}^k\simeq S^{n-k-1}}$ for any ${\displaystyle 0\le k<n}$, by writing ${\displaystyle {\mathbb{R}}^n-{\mathbb{R}}^k}$ as the total space of the fiber bundle ${\displaystyle {\mathbb{R}}^k\times ({\mathbb{R}}^{n-k}-\{0\})\to ({\mathbb{R}}^{n-k}-\{0\})}$, then applying the homotopy equivalences above.
• If a subcomplex ${\displaystyle A}$ of a CW complex ${\displaystyle X}$ is contractible, then the quotient space ${\displaystyle X/A}$ is homotopy equivalent to ${\displaystyle X}$.
• A deformation retraction is a homotopy equivalence.

Null-homotopy

A function ${\displaystyle f}$ is said to be null-homotopic if it is homotopic to a constant function. (The homotopy from ${\displaystyle f}$ to a constant function is then sometimes called a null-homotopy.) For example, a map ${\displaystyle f}$ from the unit circle ${\displaystyle S^1}$ to any space ${\displaystyle X}$ is null-homotopic precisely when it can be continuously extended to a map from the unit disk ${\displaystyle D^2}$ to ${\displaystyle X}$ that agrees with ${\displaystyle f}$ on the boundary.

It follows from these definitions that a space ${\displaystyle X}$ is contractible if and only if the identity map from ${\displaystyle X}$ to itself—which is always a homotopy equivalence—is null-homotopic.

Invariance

Homotopy equivalence is important because in algebraic topology many concepts are homotopy invariant, that is, they respect the relation of homotopy equivalence. For example, if X and Y are homotopy equivalent spaces, then:

• X is path-connected if and only if Y is.
• X is simply connected if and only if Y is.
• The (singular) homology and cohomology groups of X and Y are isomorphic.
• If X and Y are path-connected, then the fundamental groups of X and Y are isomorphic, and so are the higher homotopy groups. (Without the path-connectedness assumption, one has π₁(X, x₀) isomorphic to π₁(Y, f(x₀)) where f : X → Y is a homotopy equivalence and x₀ ∈ X.)

An example of an algebraic invariant of topological spaces which is not homotopy invariant is compactly supported homology (which is, roughly speaking, the homology of the compactification, and compactification is not homotopy invariant).

Variants

Relative homotopy

In order to define the fundamental group, one needs the notion of homotopy relative to a subspace. These are homotopies which keep the elements of the subspace fixed. Formally: if f and g are continuous maps from X to Y and K is a subset of X, then we say that f and g are homotopic relative to K if there exists a homotopy H : X × [0, 1] → Y between f and g such that H(k, t) = f(k) = g(k) for all k ∈ K and t ∈ [0, 1]. Also, if g is a retraction from X to K and f is the identity map, this is known as a strong deformation retract of X to K. When K is a point, the term pointed homotopy is used.

Isotopy

When two given continuous functions f and g from the topological space X to the topological space Y are embeddings, one can ask whether they can be connected 'through embeddings'. This gives rise to the concept of isotopy, which is a homotopy, H, in the notation used before, such that for each fixed t, H(x, t) gives an embedding.

A related, but separate, concept is that of ambient isotopy.

Requiring that two embeddings be isotopic is a stronger requirement than that they be homotopic. For example, the map from the interval [−1, 1] into the real numbers defined by f(x) = −x is not isotopic to the identity g(x) = x. Any homotopy from f to the identity would have to exchange the endpoints, which would mean that they would have to 'pass through' each other. Moreover, f has changed the orientation of the interval and g has not, which is impossible under an isotopy. However, the maps are homotopic; one homotopy from f to the identity is H: [−1, 1] × [0, 1] → [−1, 1] given by H(x, y) = 2yx − x.

Two homeomorphisms (which are special cases of embeddings) of the unit ball which agree on the boundary can be shown to be isotopic using Alexander's trick. For this reason, the map of the unit disc in ${\displaystyle {\mathbb{R}}^2}$ defined by f(x, y) = (−x, −y) is isotopic to a 180-degree rotation around the origin, and so the identity map and f are isotopic because they can be connected by rotations.

In geometric topology—for example, in knot theory—the idea of isotopy is used to construct equivalence relations. For example, when should two knots be considered the same? We take two knots, K₁ and K₂, in three-dimensional space. A knot is an embedding of a one-dimensional space, the "loop of string" (or the circle), into this space, and this embedding gives a homeomorphism between the circle and its image in the embedding space. One may try to define knot equivalence based on isotopy instead of the more restricted property of ambient isotopy. That is, two knots are isotopic when there exists a continuous function starting at t = 0 giving the K₁ embedding, ending at t = 1 giving the K₂ embedding, with all intermediate values corresponding to embeddings. However, this definition would make every knot equivalent to the unknot, as the knotted portions can be "contracted" down to a straight line. The problem is that, while continuous, this is not an injective function of the Euclidean space that the knot is embedded in. An ambient isotopy, studied in this context, is an isotopy of the larger space, considered in light of its action on the embedded submanifold. Knots K₁ and K₂ are considered equivalent when there is a continuous [0, 1]-indexed family of maps which moves K₁ to K₂ via homeomorphisms of the Euclidean space.

Similar language is used for the equivalent concept in contexts where one has a stronger notion of equivalence. For example, a path between two smooth embeddings is a smooth isotopy.

Timelike homotopy

On a Lorentzian manifold, certain curves are distinguished as timelike (representing something that only goes forwards, not backwards, in time, in every local frame). A timelike homotopy between two timelike curves is a homotopy such that the curve remains timelike during the continuous transformation from one curve to another. No closed timelike curve (CTC) on a Lorentzian manifold is timelike homotopic to a point (that is, null timelike homotopic); such a manifold is therefore said to be multiply connected by timelike curves. A manifold such as the 3-sphere can be simply connected (by any type of curve), and yet be timelike multiply connected.

Properties

Lifting and extension properties

If we have a homotopy ${\displaystyle H:X\times [0,1]\to Y}$ and a cover ${\displaystyle p:\overline{Y}\to Y}$ and we are given a map ${\displaystyle {\overline{h}}_0:X\to \overline{Y}}$ such that ${\displaystyle H_0=P\circ {\overline{h}}_0}$ (${\displaystyle {\overline{h}}_0}$ is called a lift of ${\displaystyle h_0}$), then we can lift all ${\displaystyle H}$ to a map ${\displaystyle \overline{H}:X\times [0,1]\to \overline{Y}}$ such that ${\displaystyle p\circ \overline{H}=H}$. The homotopy lifting property is used to characterize fibrations.

Another useful property involving homotopy is the homotopy extension property, which characterizes the extension of a homotopy between two functions from a subset of some set to the set itself. It is useful when dealing with cofibrations.

Groups

Since the relation of two functions ${\displaystyle f,g:X\to Y}$ being homotopic relative to a subspace is an equivalence relation, we can look at the equivalence classes of maps between a fixed X and Y. If we fix ${\displaystyle X=[0,1{]}^n}$, the unit interval [0, 1] crossed with itself n times, and we take its boundary ${\displaystyle \mathrm{\partial }([0,1{]}^n)}$ as a subspace, then the equivalence classes form a group, denoted ${\displaystyle {\pi }_n(Y,y_0)}$, where ${\displaystyle y_0}$ is in the image of the subspace ${\displaystyle \mathrm{\partial }([0,1{]}^n)}$.

We can define the action of one equivalence class on another, and so we get a group. These groups are called the homotopy groups. In the case ${\displaystyle n=1}$, it is also called the fundamental group.

Homotopy category

The idea of homotopy can be turned into a formal category of category theory. The homotopy category is the category whose objects are topological spaces, and whose morphisms are homotopy equivalence classes of continuous maps. Two topological spaces X and Y are isomorphic in this category if and only if they are homotopy equivalent. Then a functor on the category of topological spaces is homotopy invariant if it can be expressed as a functor on the homotopy category.

For example, homology groups are a functorial homotopy invariant: this means that if f and g from X to Y are homotopic, then the group homomorphisms induced by f and g on the level of homology groups are the same: H_n(f) = H_n(g) : H_n(X) → H_n(Y) for all n. Likewise, if X and Y are in addition path-connected, and the homotopy between f and g is pointed, then the group homomorphisms induced by f and g on the level of homotopy groups are also the same: π_n(f) = π_n(g) : π_n(X) → π_n(Y).

Applications

Based on the concept of the homotopy, computation methods for algebraic and differential equations have been developed. The methods for algebraic equations include the homotopy continuation method and the continuation method (see numerical continuation). The methods for differential equations include the homotopy analysis method.

Homotopy theory can be used as a foundation for homology theory: one can represent a cohomology functor on a space X by mappings of X into an appropriate fixed space, up to homotopy equivalence. For example, for any abelian group G, and any based CW-complex X, the set ${\displaystyle [X,K(G,n)]}$ of based homotopy classes of based maps from X to the Eilenberg–MacLane space ${\displaystyle K(G,n)}$ is in natural bijection with the n-th singular cohomology group ${\displaystyle H^n(X,G)}$ of the space X. One says that the omega-spectrum of Eilenberg-MacLane spaces are representing spaces for singular cohomology with coefficients in G. Using this fact, homotopy classes between a CW complex and a multiply connected space can be calculated using cohomology as described by the Hopf–Whitney theorem.

Recently, homotopy theory has been used to develop deep learning-based generative models like diffusion models and flow-based generative models. Perturbing the complex non-Gaussian states is a challenging task. Using deep learning and homotopy, such complex states can be transformed to Gaussian state and mildly perturbed to get transformed back to perturbed complex states.