N-connected space

In the mathematical branch of algebraic topology, specifically homotopy theory, n-connectedness (sometimes, n-simple connectedness) generalizes the concepts of path-connectedness and simple connectedness. To say that a space is n-connected is to say that its first n homotopy groups are trivial, and to say that a map is n-connected means that it is an isomorphism "up to dimension n, in homotopy".

n-connected space

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

[math]\displaystyle{ \pi_i(X) \cong 0, \quad 1 \leq i \leq n, }[/math]

where [math]\displaystyle{ \pi_i(X) }[/math] denotes the i-th homotopy group and 0 denotes the trivial 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 0th homotopy set can be defined as:

[math]\displaystyle{ \pi_0(X, *) := \left[\left(S^0, *\right), \left(X, *\right)\right]. }[/math]

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 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 (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

[math]\displaystyle{ \pi_i(X) \simeq 0, \quad 0 \leq i \leq n. }[/math]

Examples

• 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.
• An n-sphere is (n − 1)-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 (n − 1)-connected space. In terms of homotopy groups, it means that a map [math]\displaystyle{ f\colon X \to Y }[/math] is n-connected if and only if:

• [math]\displaystyle{ \pi_i(f)\colon \pi_i(X) \mathrel{\overset{\sim}{\to}} \pi_i(Y) }[/math] is an isomorphism for [math]\displaystyle{ i \lt n }[/math], and
• [math]\displaystyle{ \pi_n(f)\colon \pi_n(X) \twoheadrightarrow \pi_n(Y) }[/math] 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 n^th homotopy groups, in the exact sequence:

[math]\displaystyle{ \pi_n(X) \mathrel{\overset{\pi_n(f)}{\to}} \pi_n(Y) \to \pi_{n-1}(Ff). }[/math]

If the group on the right [math]\displaystyle{ \pi_{n-1}(Ff) }[/math] 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 [math]\displaystyle{ x_0 \hookrightarrow X }[/math] 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 [math]\displaystyle{ A \hookrightarrow X }[/math] 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 inclus−ion map [math]\displaystyle{ A \hookrightarrow X }[/math] to be 1-connected, it must be:

• onto [math]\displaystyle{ \pi_0(X), }[/math]
• one-to-one on [math]\displaystyle{ \pi_0(A) \to \pi_0(X), }[/math] and
• onto [math]\displaystyle{ \pi_1(X). }[/math]

One-to-one on [math]\displaystyle{ \pi_0(A) \to \pi_0(X) }[/math] means that if there is a path connecting two points [math]\displaystyle{ a, b \in A }[/math] by passing through X, there is a path in A connecting them, while onto [math]\displaystyle{ \pi_1(X) }[/math] 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 [math]\displaystyle{ \pi_{n-1}(A) \to \pi_{n-1}(X) }[/math] only implies that any elements of [math]\displaystyle{ \pi_{n-1}(A) }[/math] 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 [math]\displaystyle{ \pi_n(X) }[/math]) 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 where the inclusion of the k-skeleton is n-connected (for n > k) – 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.

Applications

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 [math]\displaystyle{ M \to N, }[/math] into a more general topological space, such as the space of all continuous maps between two associated spaces [math]\displaystyle{ X(M) \to X(N), }[/math] 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

• Connected space
• Connective spectrum
• Path-connected
• Simply connected