nLab
nilpotent topological space

Contents

Context

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

Idea

A connected topological space xx is called nilpotent if

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

  2. the action of π 1(X)\pi_1(X) on the higher homotopy groups is nilpotent in that the sequence N 0,nπ n(X)N_{0,n} \coloneqq \pi_n(X), N k+1,n{gnn|nN k,n,gπ 1(X)}N_{k+1,n} \coloneqq \{g n - n | n \in N_{k,n}, g \in \pi_1(X)\} terminates.

Properties

Rational nilpotent spaces

The central statement of rational homotopy theory says that the rational homotopy type of nilpotent topological spaces of finite type is equivalently reflected in nilpotent L-infinity algebras of finite type.

References

The rational homotopy theory of nilpotent topological spaces is discussed in

  • Aldridge Bousfield, V. K. A. M. Gugenheim, section 9.1 of On PL deRham theory and rational homotopy type , Memoirs of the AMS, vol. 179 (1976)

  • Joseph Neisendorfer, Lie algebras, coalgebras and rational homotopy theory for nilpotent spaces, Pacific J. Math. Volume 74, Number 2 (1978), 429-460. (euclid)

Discussion in homotopy type theory:

  • Luis Scoccola, Nilpotent Types and Fracture Squares in Homotopy Type Theory (arXiv:1903.03245)

Last revised on June 17, 2019 at 05:21:29. See the history of this page for a list of all contributions to it.