Contents

# Contents

## Idea

A connected topological space $X$ (or rather its homotopy type) is called nilpotent if

1. its fundamental group $\pi_1(X)$ is a nilpotent group;

2. the action of $\pi_1(X)$ on the higher homotopy groups is a nilpotent module in that the sequence $N_{0,n} \coloneqq \pi_n(X)$, $N_{k+1,n} \coloneqq \{g n - n | n \in N_{k,n}, g \in \pi_1(X)\}$ terminates.

## Examples

Directly from the definition we have that:

and more generally

As a special case of this

and thus

• every loop space is nilpotent

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

(See Hilton 82, Section 3).

## Properties

Nilpotency is involved in sufficient conditions for many important constructions in (stable) homotopy theory, see for instance at