In topology, a branch of mathematics, a nilpotent space, first defined by Emmanuel Dror (1969),[1] is a based topological space X such that
\pi=\pi1(X)
\pi
\pii(X),i\ge2
\pii(X)=
i | |
G | |
1 |
\triangleright
i | |
G | |
2 |
\triangleright...\triangleright
i | |
G | |
ni |
=1
\pi
i | |
G | |
k+1 |
k
Simply connected spaces and simple spaces are (trivial) examples of nilpotent spaces; other examples are connected loop spaces. The homotopy fiber of any map between nilpotent spaces is a disjoint union of nilpotent spaces. Moreover, the null component of the pointed mapping space
\operatorname{Map}*(K,X)
A basic theorem about nilpotent spaces[2] states that any map that induces an integral homology isomorphism between two nilpotent space is a weak homotopy equivalence. For simply connected spaces, this theorem recovers a well-known corollary to the Whitehead and Hurewicz theorems.
Nilpotent spaces are of great interest in rational homotopy theory, because most constructions applicable to simply connected spaces can be extended to nilpotent spaces. The Bousfield–Kan nilpotent completion of a space associates with any connected pointed space X a universal space
\widehat{X}
\widehat{X}
Let X be a nilpotent space and let h be a reduced generalized homology theory, such as K-theory. If h(X)=0, then h vanishes on any Postnikov section of X. This follows from a theorem that states thatany such section is X-cellular.