n-connected

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

A topological space X is said to be n-connected if and only if it is 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 requirement of being path-connected can also be expressed as 0-connectedness, when defining the 0th homotopy set as:

π₀(X, * ): = [(S⁰, * ),(X, * )];

this is only a pointed set, not a group, unless X is itself a topological group.

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

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

Examples

• As described above, a space X is 0-connected if and only if it is 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 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

The corresponding relative notion to the absolute notion of an n-connected space is an n-connected map, which is almost defined as a map whose homotopy cofiber Cf is an n-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 nth homotopy of the homotopy cofiber Cf corresponds surjection on the nth homotopy groups, in the exact sequence:

$\pi_n(X) \overset{\pi_n(f)}{\to} \pi_n(Y) \to \pi_n(Cf).$

If the group on the right π_n(Cf) vanishes, then the map on the left is a surjection.

For instance, a simply connected map (1-connected map) is one that is an isomorphism on path-components, and onto the fundamental group.

Interpretation

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 π₀(X),
• one-to-one on $\pi_0(A) \to \pi_0(X),$ and
• onto π₁(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 π₁(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 π_{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 π_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

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

• connected space
• simply connected
• path-connected
• homotopy group