Examples
- If are given by and , then the map given by is a homotopy between them.
- More generally, if is a convex subset of Euclidean space and are paths with the same endpoints, then there is a linear homotopy5 (or straight-line homotopy) given by
- Let be the identity function on the unit n-disk; i.e. the set . Let be the constant function which sends every point to the origin. Then the following is a homotopy between them:
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 idX and f ∘ g is homotopic to idY. 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 .6 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 idX (not only homotopic to it), and f ∘ g is equal to idY.7 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 and an untwisted (closed) strip are homotopy equivalent, since you can deform both strips continuously to a circle. But they are not homeomorphic.
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:
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
Properties
Lifting and extension properties
Main article: Homotopy lifting property
If we have a homotopy and a cover and we are given a map such that ( is called a lift of ), then we can lift all to a map such that . 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
Main article: Homotopy group
Since the relation of two functions 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 , the unit interval [0, 1] crossed with itself n times, and we take its boundary as a subspace, then the equivalence classes form a group, denoted , where is in the image of the subspace .
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 , it is also called the fundamental group.
Homotopy category
Main article:
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.9
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.
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.
