Homotopy

The two dashed paths shown above are homotopic relative to their endpoints. The animation represents one possible 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 practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra.

Formal definition

A homotopy between two embeddings of the torus into R³: as "the surface of a doughnut" and as "the surface of a coffee mug". This is also an example of an isotopy.

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 from the product of the space X with the unit interval [0,1] to Y such that, if x ∈ X then H(x,0) = f(x) and H(x,1) = g(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 is a family of continuous functions h_t : X → Y for t ∈ [0,1] such that h₀ = f and h₁ = g, and the map (x,t) ↦ h_t(x) is continuous from X × [0,1] to Y. The two versions coincide by setting h_t(x) = H(x,t). It is not sufficient to require each map h_t(x) to be continuous.^[1]

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.

Homotopy equivalence

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 id_X and f ∘ g is homotopic to id_Y.

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.

Intuitively, two spaces X and Y are homotopy equivalent if they can be transformed into one another (i.e., made homeomorphic) by bending, shrinking and expanding operations.

Examples

The first example of a homotopy equivalence is $\mathbb {R} ^{n}$ with a point, denoted $\mathbb {R} ^{n}\simeq \{0\}$ . The part that needs to be checked is the existence of a homotopy $H:I\times \mathbb {R} ^{n}\to \mathbb {R} ^{n}$ which can be described as $H(t,x)=t\cdot p_{0}+(1-t)\cdot \operatorname {id}$ , where $p_{0}$ is the projection onto the origin.
There is a homotopy equivalence between $S^{1}$ and ${\mathbb {R}}^{2}-\{0\}$ .
More generally, $\mathbb {R} ^{n}-\{0\}\simeq S^{n-1}$ .
Any fiber bundle $\pi :E\to B$ with fibers $F_{b}$ homotopy equivalent to a point has homotopy equivalent total and base spaces. This generalizes the previous two examples since $\pi :\mathbb {R} ^{n}-\{0\}\to S^{n-1}$ is a fiber bundle with fiber ${\mathbb {R}}_{{>0}}$ .
Every vector bundle is a fiber bundle with a fiber homotopy equivalent to a point.
For any $0\leq k<n$ , $\mathbb {R} ^{n}-\mathbb {R} ^{k}\simeq S^{n-k-1}$ by writing $\mathbb {R} ^{n}$ as $\mathbb {R} ^{k}\times \mathbb {R} ^{n-k}$ and applying the homotopy equivalences above.

Null-homotopy

A function f is said to be null-homotopic if it is homotopic to a constant function. (The homotopy from f to a constant function is then sometimes called a null-homotopy.) For example, a map f from the unit circle S¹ to any space X is null-homotopic precisely when it can be continuously extended to a map from the unit disk D² to X that agrees with f on the boundary.

It follows from these definitions that a space X is contractible if and only if the identity map from 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:

If X is path-connected then so is Y.
If X is simply connected then so is Y.
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).

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 retract 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.

Groups

Since the relation of two functions f, g : X → 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 X = [0,1]ⁿ, the unit interval [0,1] crossed with itself n times, and we take its boundary ∂([0,1]ⁿ) as a subspace then the equivalence classes form a group, denoted π_n(Y,y₀), where y₀ is in the image of the subspace ∂([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 n = 1, it is also called the fundamental group.

Timelike

On a Lorentzian manifold, certain curves are distinguished as timelike. A timelike homotopy between two timelike curves is a homotopy such that each intermediate curve is timelike. 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.

Lifting property

If we have a homotopy H : X × [0,1] → Y and a cover p : Y → Y and we are given a map h₀ : X → Y such that H₀ = p ○ h₀ (h₀ is called a lift of h₀), then we can lift all H to a map H : X × [0,1] → Y such that p ○ H = H. The homotopy lifting property is used to characterize fibrations.

Extension property

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.

Isotopy

In case the 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.^[2]

A related, but different, 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 R² 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.

The unknot is not equivalent to the trefoil knot since one cannot be deformed into the other through a continuous path of homeomorphisms of the ambient space. Thus they are not ambient-isotopic.

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. The intuitive idea behind the notion of knot equivalence is that one can deform one embedding to another through a path of embeddings: 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. This corresponds to the definition of isotopy. 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 an ambient isotopy which moves K₁ to K₂. This is the appropriate definition in the topological category.

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.

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 ^[3] and the continuation method (see numerical continuation). The methods for differential equations include the homotopy analysis method.

References

↑ Path homotopy and separately continuous functions
↑ Weisstein, Eric Wolfgang. "Isotopy". MathWorld.
↑ Allgower, Eugene. "Introduction to Numerical Continuation Methods" (PDF). CSU. Retrieved 3 January 2013.

Sources

Armstrong, M.A. (1979). Basic Topology. Springer. ISBN 0-387-90839-0.
Hazewinkel, Michiel, ed. (2001) [1994], "Homotopy", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Hazewinkel, Michiel, ed. (2001) [1994], "Isotopy (in topology)", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Spanier, Edwin (December 1994). Algebraic Topology. Springer. ISBN 0-387-94426-5.

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