nLab n-connected space

Idea

An $n$-connected space is a generalisation of the pattern:

• space,
• inhabited space,
• path-connected space,
• simply connected space,
• etc.

For the general concept see at n-connected object of an (infinity,1)-topos.

Definition

A topological space $X$ is $n$-connected or $n$-simply connected if its homotopy groups are trivial up to degree $n$.

More explicitly, $X$ is precisely $k$-connected if every continuous map to $X$ from the $k$-sphere extends to a continuous map to $X$ from the $k+1$-disk. Then $X$ is $n$-(simply) connected if $X$ is precisely $k$-connected for $-1 \leq k \leq n$.

Special cases

• Any space is $(-2)$-simply connected.

• A space is $(-1)$-simply connected precisely if it has an element; that is if it is inhabited.

• A space is $0$-simply connected precisely if it is path-connected.

• A space is $1$-simply connected precisely if it is simply connected.

• A space is $\infty$-simply connected precisely if it is weakly contractible.

Terminology

The traditional terminology is ‘$n$-connected’, but this violates the rule that ‘$1$-foo’ should mean the same as ‘foo’. This can be fixed by saying ‘$n$-simply connected’ instead, which also has the advantage of stressing that we are extending the change from connected to simply connected spaces.

Properties

An $n$-connected topological space is precisely an n-connected object in the (∞,1)-topos ∞Gpd, presented by the model category Top of topological spaces.

Related concepts

• n-connected continuous function