Nilpotent space

In topology, a branch of mathematics, a nilpotent space^[1] is a based topological space X such that

• the fundamental group $\pi = \pi_1 X$ is a nilpotent group;
• $\pi$ acts nilpotently on higher homotopy groups $\pi_i X, i \ge 2$, i.e. there is a central series $\pi_i X = G^i_1 \triangleright G^i_2 \triangleright \dots \triangleright G^i_{n_i} = 1$ such that the induced action of $\pi$ on the quotient $G^i_k/G^i_{k+1}$ is trivial for all $k$.

Simply connected spaces and simple spaces are (trivial) examples of nilpotent spaces. Nilpotent spaces are of great interest in rational homotopy theory, because most constructions applicable to simply connected spaces can be extended to nilpotent spaces.

References

1. Bousfield, A. K.; Kan, D. M. (1987), Homotopy Limits, Completions and Localizations, Lecture Notes in Mathematics 304, Springer, p. 59, ISBN 9783540061052.