nLab
n-connected space

nn-(simply) connected spaces

Idea

An nn-connected space is a generalisation of the pattern:

Definition

A topological space XX is nn-connected or nn-simply connected if its homotopy groups are trivial up to degree nn.

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

Special cases

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

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

  • A space is 00-simply connected precisely if it is path-connected.

  • A space is 11-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 ‘nn-connected’, but this violates the rule that ‘11-foo’ should mean the same as ‘foo’. This can be fixed by saying ‘nn-simply connected’ instead, which also has the advantage of stressing that we are extending the change from connected to simply connected spaces.

Properties

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

Revised on July 6, 2013 09:10:22 by Toby Bartels (141.0.9.56)