nLab nilpotent topological space

Idea

A connected topological space $X$ (or rather its homotopy type) is called nilpotent if

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

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

Examples

Directly from the definition we have that:

• every simply connected topological space is nilpotent;

and more generally

• every simple space is nilpotent.

As a special case of this

• every connected H-space is nilpotent

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).

(cf. May & Ponto 2012, p. 49 (77 of 542))

• For $G$ a nilpotent Lie group, the classifying space $B G$ is a nilpotent space.

(cf. Hilton 1982, Section 3).

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.

• fundamental theorem of dg-algebraic rational homotopy theory

Related concepts

• nilpotent element

• nilpotent Lie algebra, nilpotent L-infinity algebra

• nilpotent group, nilpotent module

• connected topological space

• simply-connected topological space

• simple topological space

• rational homotopy theory

  • Sullivan model

  • rational fiber lemma

• finite homotopy type

• p-local homotopy type

• p-complete homotopy type

References

The notion originates with:

• Emmanuel Dror; §4.3 in: A Generalization of the Whitehead Theorem, in: Symposium on Algebraic Topology, Lecture Notes in Mathematics 249, Springer (1971) 13-22 [doi:10.1007/BFb0060891, pdf]

• Aldridge K. Bousfield, Daniel M. Kan; §4.3 in: Homotopy Limits, Completions and Localizations, Lecture Notes in Mathematics 304, Springer(1972; 1987) [doi:10.1007/978-3-540-38117-4]

Further discussion:

• Peter Hilton: Nilpotency in group theory and topology, Publicacions de la Secció de Matemàtiques 26 3 (1982) 47-78 [jstor:43741908, numdam:SPHM_1976___3_A1_0]

• Peter May, Kate Ponto: More Concise Algebraic Topology, University of Chicago Press (2012) [pdf]

• Emily Riehl; def. 14.4.9 in: Categorical Homotopy Theory, New Mathematical Monographs 24, Cambridge University Press (2014) [pdf, doi:10.1017/CBO9781107261457]

See also:

• Wikipedia, Nilpotent space

O the rational homotopy theory of nilpotent topological spaces:

• 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]