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

open subset , closed subset , neighbourhood

topological space , locale

base for the topology , neighbourhood base

finer/coarser topology

closure , interior , boundary

separation , sobriety

continuous function , homeomorphism

uniformly continuous function

embedding

open map , closed map

sequence , net , sub-net , filter

convergence

category Top

Universal constructions

Extra stuff, structure, properties

nice topological space

metric space , metric topology , metrisable space

Kolmogorov space , Hausdorff space , regular space , normal space

sober space

compact space , proper map

sequentially compact , countably compact , locally compact , sigma-compact , paracompact , countably paracompact , strongly compact

compactly generated space

second-countable space , first-countable space

contractible space , locally contractible space

connected space , locally connected space

simply-connected space , locally simply-connected space

cell complex , CW-complex

pointed space

topological vector space , Banach space , Hilbert space

topological group

topological vector bundle , topological K-theory

topological manifold

Examples

empty space , point space

discrete space , codiscrete space

Sierpinski space

order topology , specialization topology , Scott topology

Euclidean space

cylinder , cone

sphere , ball

circle , torus , annulus , Moebius strip

polytope , polyhedron

projective space (real , complex )

classifying space

configuration space

path , loop

mapping spaces : compact-open topology , topology of uniform convergence

Zariski topology

Cantor space , Mandelbrot space

Peano curve

line with two origins , long line , Sorgenfrey line

K-topology , Dowker space

Warsaw circle , Hawaiian earring space

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents
Idea
A connected topological space $x$ is called nilpotent if

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

the action of $\pi_1(X)$ on the higher homotopy groups is nilpotent 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.

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.
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