nLab nilpotent topological space



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



Paths and cylinders

Homotopy groups

Basic facts




A connected topological space XX (or rather its homotopy type) 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 a nilpotent module 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.


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 May-Ponto 12, p. 49 (77 of 542))

(See Hilton 82, Section 3).


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


See also

See also

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 September 15, 2020 at 15:03:58. See the history of this page for a list of all contributions to it.