Homotopy

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.

• 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 ${\displaystyle \operatorname {id} _{\mathbb {R} ^{n}}}$ and ${\displaystyle p_{0}}$, the projection of ${\displaystyle \mathbb {R} ^{n}}$ onto the origin. This can be described as ${\displaystyle H(t,\cdot )=t\cdot p_{0}+(1-t)\cdot \operatorname {id} _{\mathbb {R} ^{n}}}$.
• There is a homotopy equivalence between ${\displaystyle S^{1}}$ 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.
• For any ${\displaystyle 0\leq k<n}$, ${\displaystyle \mathbb {R} ^{n}-\mathbb {R} ^{k}\simeq S^{n-k-1}}$ by writing ${\displaystyle \mathbb {R} ^{n}}$ as ${\displaystyle \mathbb {R} ^{k}\times \mathbb {R} ^{n-k}}$ and 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 ${\displaystyle X}$. [4]

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.

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.

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 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.[5]

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.

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.

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.[6]

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.

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.

Since the relation of two functions ${\displaystyle f,g\colon X\to Y}$ 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 ${\displaystyle \partial ([0,1]^{n})}$ 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 ${\displaystyle \partial ([0,1]^{n})}$.

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.

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).