nLab
nilpotent topological space

Idea

A connected topological space $x$ 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 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.

Related concepts

• nilpotent element

• nilpotent Lie algebra, nilpotent L-infinity algebra

• nilpotent group

References

• Wikipedia, Nilpotent space

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)