nLab
nilpotent homotopy type

Contents

Context

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Definition

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

  1. its fundamental group is a nilpotent group

  2. the canonical action of π 1(X)\pi_1(X) on all the higher homotopy groups π n2(X)\pi_{n \geq 2}(X) is nilpotent.

The first condition means that π 1(X)\pi_1(X) is isomorphic to an iterated central extension of abelian groups.

The second condition means that each π n2(X)\pi_{n \geq 2}(X) admits a sequence of subgroups

*=G kG 1=π n(X) \ast = G_k \hookrightarrow \cdots \hookrightarrow G_1 = \pi_n(X)

such that for all ii

  1. G i+1G iG_{i+1} \hookrightarrow G_i is a normal subgroup;

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

  3. each G iG_i is closed under the action of π 1(X)\pi_1(X);

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

This implies that given any element aπ n2(X)a \in \pi_{n \geq 2}(X), then after acting on it at most kk times with elements from π 1(X)\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

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

Last revised on April 19, 2018 at 03:11:26. See the history of this page for a list of all contributions to it.