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 when it is non-empty, path-connected, and its first n homotopy groups vanish identically, that is

\pi _{i}(X)\simeq 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)\simeq 0,\quad 0\leq i\leq n.

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 $f\colon X\to Y$ 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 n^th 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

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

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.