*n*-connected

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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".

## Contents

*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 *S*^{0} 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 *x _{1}* and

*x*in

_{2}*X*can be connected with a continuous path which starts in

*x*and ends in

_{1}*x*, which is equivalent to the assertion that every mapping from

_{2}*S*(a discrete set of two points) to

^{0}*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

- 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

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 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 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 *x*_{0} is an *n*-connected space if and only if the inclusion of the basepoint 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 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 to be 1-connected, it must be:

One-to-one on means that if there is a path connecting two points by passing through *X,* there is a path in *A* connecting them, while onto 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 only implies that any element of 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 ) 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 into a more general topological space, such as the space of all continuous maps between two associated spaces 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.