Homotopical connectivity


In algebraic topology, homotopical connectivity is a property describing a topological space based on the dimension of its holes. In general, low homotopical connectivity indicates that the space has at least one low-dimensional hole. The concept of n-connectedness generalizes the concepts of path-connectedness and simple connectedness.
An equivalent definition of homotopical connectivity is based on the homotopy groups of the space. A space is n-connected if its first n homotopy groups are trivial.
Homotopical connectivity is defined for maps, too. A map is n-connected if it is an isomorphism "up to dimension n, in homotopy".

Definition using holes

All definitions below consider a topological space X.
A hole in X is, informally, a thing that prevents some suitably placed sphere from continuously shrinking to a point. Equivalently, it is a sphere that cannot be continuously extended to a ball. Formally,
  • A d-dimensional sphere in X is a continuous function.
  • A d-dimensional ball in X is a continuous function.
  • A d-dimensional-boundary hole in X is a d-dimensional sphere that is not nullhomotopic. Equivalently, it is a d-dimensional sphere that cannot be continuously extended to a -dimensional ball. It is sometimes called a -dimensional hole.
  • X is called n-connected if it contains no holes of boundary-dimension d ≤ n.'
  • The homotopical connectivity' of X'', denoted, is the largest integer n for which X is n-connected.
  • A slightly different definition of connectivity, which makes some computations simpler, is: the smallest integer d such that X contains a d-dimensional hole. This connectivity parameter is denoted by, and it differs from the previous parameter by 2, that is,.

    Examples

  • A 2-dimensional hole is a circle in X, that cannot be shrunk continuously to a point in X. An example is shown on the figure at the right. The yellow region is the topological space X; it is a pentagon with a triangle removed. The blue circle is a 1-dimensional sphere in X. It cannot be shrunk continuously to a point in X; therefore; X has a 2-dimensional hole. Another example is the punctured plane - the Euclidean plane with a single point removed,. To make a 2-dimensional hole in a 3-dimensional ball, make a tunnel through it. In general, a space contains a 1-dimensional-boundary hole if and only if it is not simply-connected. Hence, simply-connected is equivalent to 1-connected. X is 0-connected but not 1-connected, so . The lowest dimension of a hole is 2, so .
  • A 3-dimensional hole is shown on the figure at the right. Here, X is a cube with a ball removed. The 2-dimensional sphere cannot be continuously shrunk to a single point. X is simply-connected but not 2-connected, so. The smallest dimension of a hole is 3, so.
  • For a 1-dimensional hole we need to consider - the zero-dimensional sphere. What is a zero dimensional sphere? - For every integer d, the sphere is the boundary of the -dimensional ball. So is the boundary of, which is the segment . Therefore, is the set of two disjoint points. A zero-dimensional sphere in X is just a set of two points in X. If there is such a set, that cannot be continuously shrunk to a single point in X, this means that there is no path between the two points, that is, X is not path-connected; see the figure at the right. Hence, path-connected is equivalent to 0-connected. X is not 0-connected, so . The lowest dimension of a hole is 1, so .
  • A 0-dimensional hole is a missing 0-dimensional ball. A 0-dimensional ball is a single point; its boundary is an empty set. Therefore, the existence of a 0-dimensional hole is equivalent to the space being empty. Hence, non-empty is equivalent to -connected. For an empty space X, and, which is its smallest possible value.
  • A ball has no holes of any dimension. Therefore, its connectivity is infinite:.

    Homotopical connectivity of spheres

In general, for every integer d, The proof requires two directions:
  • Proving that, that is, cannot be continuously shrunk to a single point. This can be proved using the Borsuk–Ulam theorem.
  • Proving that, that is, that is, every continuous map for can be continuously shrunk to a single point.

    Definition using groups

A space X is called n-connected, for n ≥ 0, if it is non-empty, and all its homotopy groups of order d'' ≤ n are the trivial group: where denotes the i-th homotopy group and 0 denotes the trivial group. The two definitions are equivalent. The requirement for an n-connected space consists of requirements for all d ≤ n:
  • The requirement for d=−1 means that X should be nonempty.
  • The requirement for d=0 means that X should be path-connected.
  • The requirement for any d ≥ 1 means that X contains no holes of boundary dimension d. That is, every d-dimensional sphere in X is homotopic to a constant map. Therefore, the d-th homotopy group of X is trivial. The opposite is also true: If X has a hole with a d-dimensional boundary, then there is a d-dimensional sphere that is not homotopic to a constant map, so the d-th homotopy group of X is not trivial. In short, X has a hole with a d-dimensional boundary, if-and-only-if.The homotopical connectivity of X is the largest integer n for which X is n-connected.
The requirements of being non-empty and path-connected can be interpreted as -connected and 0-connected, respectively, which is useful in defining 0-connected and 1-connected maps, as below. The 0th homotopy set can be defined as:
This is only a pointed set, not a group, unless X is itself a topological group; the distinguished point is the class of the trivial map, sending S0 to the base point of X. Using this set, a space is 0-connected if and only if the 0th homotopy set is the one-point set. The definition of homotopy groups and this homotopy set require that X be pointed, which cannot be done if X is empty.
A topological space X is path-connected if and only if its 0th homotopy group vanishes identically, as path-connectedness implies that any two points x1 and x2 in X can be connected with a continuous path which starts in x1 and ends in x2, which is equivalent to the assertion that every mapping from S0 to X can be deformed continuously to a constant map. With this definition, we can define X to be n''-connected if and only if

Examples

  • A space X is -connected if and only if it is non-empty.
  • A space X is 0-connected if and only if it is non-empty and path-connected.
  • A space is 1-connected if and only if it is simply connected.
  • An n-sphere is -connected.

    ''n''-connected map

The corresponding relative notion to the absolute notion of an n-connected space is an n-connected map, which is defined as a map whose homotopy fiber Ff is an -connected space. In terms of homotopy groups, it means that a map is n-connected if and only if:
  • is an isomorphism for, and
  • is a surjection.
The last condition is frequently confusing; it is because the vanishing of the -st homotopy group of the homotopy fiber Ff corresponds to a surjection on the nth homotopy groups in the exact sequence
If the group on the right vanishes, then the map on the left is a surjection.
Low-dimensional examples:
  • A connected map is one that is onto path components ; this corresponds to the homotopy fiber being non-empty.
  • A simply connected map is one that is an isomorphism on path components and onto the fundamental group.
n-connectivity for spaces can in turn be defined in terms of n-connectivity of maps: a space X with basepoint x0 is an n-connected space if and only if the inclusion of the basepoint is an n-connected map. The single point set is contractible, so all its homotopy groups vanish, and thus "isomorphism below n and onto at n" corresponds to the first n homotopy groups of X vanishing.

Interpretation

This is instructive for a subset:
an n-connected inclusion is one such that, up to dimension n − 1, homotopies in the larger space X can be homotoped into homotopies in the subset A.
For example, for an inclusion map to be 1-connected, it must be:
  • onto
  • one-to-one on and
  • onto
One-to-one on means that if there is a path connecting two points by passing through X, there is a path in A connecting them, while onto means that in fact a path in X is homotopic to a path in A.
In other words, a function which is an isomorphism on only implies that any elements of that are homotopic in X are abstractly homotopic in A – the homotopy in A may be unrelated to the homotopy in X – while being n-connected means that homotopies in X can be pushed into homotopies in A.
This gives a more concrete explanation for the utility of the definition of n-connectedness: for example, a space where the inclusion of the k-skeleton is n-connected – such as the inclusion of a point in the n-sphere – has the property that any cells in dimensions between k and n do not affect the lower-dimensional homotopy types.