nLab simple topological space

Idea

A topological space $X$, or rather its homotopy type, is called simple if

1. the fundamental group $\pi_1(X)$ is abelian;

2. its canonical action on all higher homotopy groups $\pi_{\bullet \geq 2}(X)$ is trivial.

Examples

• Every connected H-space is nilpotent

and thus, in particular,

• every loop space is nilpotent

  (since all its connected components are homotopy equivalent to the unit component, which is a connected H-space).

(See May-Ponto 12, p. 49 (77 of 542))

Properties

Simple spaces are also nilpotent spaces (May-Ponto 12, p. 49 (77 of 542)).

Related concepts

• simply connected topological space

• nilpotent space

• H-space

References

• Peter May, Kate Ponto, More Concise Algebraic Topology, University of Chicago Press (2012) (pdf)