N connected

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

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:

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

Examples

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 : X → Y is n-connected if and only if:

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:

If the group on the right π n − 1 ( F f ) vanishes, then the map on the left is a surjection.

Low-dimensional examples:

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 ↪ 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 ↪ 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 ↪ X to be 1-connected, it must be:

One-to-one on π 0 ( A ) → π 0 ( X ) means that if there is a path connecting two points a , b ∈ A by passing through X, there is a path in A connecting them, while onto π 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 π n − 1 ( A ) → π 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 → N , into a more general topological space, such as the space of all continuous maps between two associated spaces X ( M ) → 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.