n-connected

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This article is about the concept in algebraic topology. For the concept in graph theory, see Connectivity (graph theory).

In the mathematical branch of algebraic topology, specifically homotopy theory, n-connectedness is a way to say that a space vanishes or that a map is an isomorphism "up to dimension n, in homotopy".

n-connected space[edit]

A topological space X is said to be n-connected when it is non-empty, path-connected, and its first n homotopy groups vanish identically, that is

\pi_i(X) \equiv 0~, \quad 1\leq i\leq n ,

where the left-hand side denotes the i-th homotopy group.

The requirements of being non-empty and path-connected can be interpreted as (−1)-connected and 0-connected, respectively, which is useful in defining 0-connected and 1-connected maps, as below. The 0-th homotopy set can be defined as:

\pi_0(X,*) := [(S^0,*), (X,*)].

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 (have a chosen base point), which cannot be done if X is empty.

A topological space X is path-connected if and only if its 0-th 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 (a discrete set of two points) 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

\pi_i(X) \equiv 0, \quad 0\leq i\leq n.

Examples[edit]

  • A space X is (−1)-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.

Thus, the term "n-connected" is a natural generalization of being non-empty, path-connected, or simply connected.

It is obvious from the definition that an n-connected space X is also i-connected for all i < n.

n-connected map[edit]

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 (n − 1)-connected space. In terms of homotopy groups, it means that a map f\colon X \to Y is n-connected if and only if:

  • \pi_i(f)\colon \pi_i(X) \overset{\sim}{\to} \pi_i(Y) is an isomorphism for i < n, and
  • \pi_n(f)\colon \pi_n(X) \twoheadrightarrow \pi_n(Y) is a surjection.

The last condition is frequently confusing; it is because the vanishing of the (n − 1)-st homotopy group of the homotopy fiber Ff corresponds to a surjection on the nth homotopy groups, in the exact sequence:

\pi_n(X) \overset{\pi_n(f)}{\to} \pi_n(Y) \to \pi_{n-1}(Ff).

If the group on the right \pi_{n-1}(Ff) vanishes, then the map on the left is a surjection.

Low-dimensional examples:

  • A connected map (0-connected map) is one that is onto path components (0th homotopy group); this corresponds to the homotopy fiber being non-empty.
  • A simply connected map (1-connected map) is one that is an isomorphism on path components (0th homotopy group) and onto the fundamental group (1st homotopy 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 x_0 \hookrightarrow X 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[edit]

This is instructive for a subset: an n-connected inclusion A \hookrightarrow X 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 A \hookrightarrow X to be 1-connected, it must be:

  • onto \pi_0(X),
  • one-to-one on \pi_0(A) \to \pi_0(X), and
  • onto \pi_1(X).

One-to-one on \pi_0(A) \to \pi_0(X) means that if there is a path connecting two points a, b \in A by passing through X, there is a path in A connecting them, while onto \pi_1(X) 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 \pi_{n-1}(A) \to \pi_{n-1}(X) only implies that any element of \pi_{n-1}(A) 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 (so also onto \pi_n(X)) means that (up to dimension n−1) 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 such that the inclusion of the k-skeleton in n-connected (for n>k) – such as the inclusion of a point in the n-sphere – means that any cells in dimension between k and n are not affecting the homotopy type from the point of view of low dimensions.

Applications[edit]

The concept of n-connectedness is used in the Hurewicz theorem which describes the relation between singular homology and the higher homotopy groups.

In geometric topology, cases when the inclusion of a geometrically-defined space, such as the space of immersions M \to N, into a more general topological space, such as the space of all continuous maps between two associated spaces X(M) \to X(N), are n-connected are said to satisfy a homotopy principle or "h-principle". There are a number of powerful general techniques for proving h-principles.

See also[edit]