Homotopy
In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.
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, for all points x in X, H(x,0)=f(x)
and H(x,1)=g(x).
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 o f1 and g2 o g1 : X → Z are homotopic as well.
Given two spaces X and Y, we say they are homotopy equivalent if there exist continuous maps f : X → Y and g : Y → X such that g o f is homotopic to the identity map idX and f o g is homotopic to idY.
The maps f and g are called homotopy equivalences in this case.
Intuitively, two spaces X and Y are homotopy equivalent if they can be transformed into one another by bending, shrinking and expanding operations. For example, a solid disk or solid ball is homotopy equivalent to a point, and R2 - {(0,0)} is homotopy equivalent to the unit circle S1. Those spaces that are homotopy equivalent to a point are called contractible.
Homotopy equivalence is important because in algebraic topology most concepts cannot distinguish homotopy equivalent spaces: if X and Y are homotopy equivalent, then
More abstractly, one can appeal to category theory concepts. One can define the homotopy category, whose objects are topological spaces, and whose morphisms are homotopy classes of continuous maps. Two topological spaces X and Y are isomorphic in this category if and only if they are homotopy-equivalent.
A homotopy invariant is any function on spaces, (or on mappings), that respects the relation of homotopy equivalence (resp. homotopy); such invariants are constitutive of homotopy theory. Of course one could have foundational objection to a function whose domain is the collection of all topological spaces.
In practice homotopy theory is carried out by working with CW complexes, for technical convenience; or in some other reasonable category.
Especially 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 K if there exists a homotopy H : X × [0,1] → Y between f and g such that H(k,t) = f(k) for all k∈K and t∈[0,1].
In case the two given continuous functions f and g from the
topological space X to the topological space Y are homeomorphisms, one can ask whether they can be connected 'through homeomorphisms'. 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 a homeomorphism.
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. The intuitive idea of deforming one to the other should correspond to a path of homeomorphisms: an isotopy starting with the identity homeomorphism of three-dimensional space, and ending at a homeomorphism h such that h moves K1 to K2.
In the standard sphere Sn we chose a base point a. For a space
X with base point b, we define πn(X) to be the homotopy classes of maps f : Sn → X that map the base point
a to the base point b. In particular, the equivalence clases are given by
homotopies that are constant on the basepoint of the sphere. Equivalently, we can define πn(X) to be the group of homotopy classes of maps g : [0,1]n → X from the n-cube to X that take the boundary of the n-cube to b.
For , the homotopy classes actually form a homotopy group. To define the group operation, recall that for the fundamental group, the product f * g of two loops f and g is defined by setting (f * g)(t) = f(2t) if t is in [0,1/2] and (f * g)(t) = g(2t-1) if t is in [1/2,1]. The idea of composition in the fundamental group is that of following the first path and the second in succession, or, equivalently, setting their two domains together. The concept of composition that we want for the n-th homotopy group is the same, except that now the domains that we stick together are cubes, and we must glue them along a face. We therefore define the sum of maps f, g :[0,1]n → X by the formula (f + g)(t1, t2, ...tn) = f(2t1, t2, ...tn) for t1 in [0,1/2] and (f + g)(t1, t2, ...tn) = g(2t1-1, t2, ...tn) for t1 in [1/2,1]. For the definition if terms of spheres, this corresponds with defining the sum f + g of maps Sn → X by taking the wedge sum of two n-spheres, defining a map h from the wedge sum to X to be f on the first sphere and g on the second, and defining f + g to be the composition of the map from Sn to the wedge sum of two n-spheres that collapses the equator, with h.
If , then πn is Abelian. (For a proof of this, note that in two dimensions or greater, two homotopies can be "rotated" around each other.)
There are also relative homotopy groups πn(X,A) for a pair (X,A).
Let p : E → B be a basepoint preserving Serre fibration, that is, a map possessing the homotopy lifting property with respect to CW complexes. Then there is a long exact sequence of homotopy groups
. . .→ πn(F) → πn(E) → πn(B) → πn-1(F) → . . . → π0(E) → π0(B) → 0
Here the maps involving π0 are not group homomorphisms because the π0 are not groups, but they are exact in the sense that the image equals the kernel.
Example: the Hopf fibration. Let B equal S2 and E equal S3. Let p be the Hopf fibration, which has fiber S1. From the long exact sequence
. . .→ πn(S1) → πn(S3) → πn(S2) → πn-1(S1) → . . .
and the fact that πn(S1) = 0 for , we find that πn(S3) = πn(S2) for . In particular, π3(S2) = π3(S3) = Z.
Formal definitions
Homotopy equivalence of spaces
Homotopy-invariant properties
Homotopy category and homotopy invariants
Relative homotopy
Isotopy
Homotopy groups
Long exact sequence
The long exact sequence of a fibration
The long exact sequence of relative homotopy classes
Incomplete