nLab
nilpotent homotopy type

Definition

A topological space/simplicial set/homotopy type/infinity-groupoid $X$ is nilpotent if

1. its fundamental group is a nilpotent group

2. the canonical action of $\pi_1(X)$ on all the higher homotopy groups $\pi_{n \geq 2}(X)$ is nilpotent.

The first condition means that $\pi_1(X)$ is isomorphic to an iterated central extension of abelian groups.

The second condition means that each $\pi_{n \geq 2}(X)$ admits a sequence of subgroups

$$\ast = G_k \hookrightarrow \cdots \hookrightarrow G_1 = \pi_n(X)$$

such that for all $i$

1. $G_{i+1} \hookrightarrow G_i$ is a normal subgroup;

2. the quotient $G_i/G_{i+1}$ is an abelian group;

3. each $G_i$ is closed under the action of $\pi_1(X)$;

4. the induced action on $G_i/G_{i+1}$ is trivial.

This implies that given any element $a \in \pi_{n \geq 2}(X)$, then after acting on it at most $k$ times with elements from $\pi_1(X)$ the result is zero.

Properties

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

• localization of a space;

• fracture theorem.

Related concepts

• finite homotopy type

• p-local homotopy type

• p-complete homotopy type

References

Review includes

• Emily Riehl, def. 14.4.9 in Categorical homotopy theory, new mathematical monographs 24, Cambridge University Press 2014 (published version)

See also

• Wikipedia, Nilpotent space