Nilpotent Space
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In topology, a branch of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a nilpotent space, first defined by Emmanuel Dror (1969), is a based topological space ''X'' such that * the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
\pi = \pi_1 (X) is a nilpotent group; * \pi acts nilpotently on the
higher homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotop ...
\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_ = 1 such that the induced action of \pi on the quotient group G^i_k/G^i_ is trivial for all k.
Simply connected space In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
s and
simple space In algebraic topology, a branch of mathematics, a simple space is a connected topological space that has a homotopy type of a CW complex and whose fundamental group is abelian and acts trivially on the homotopy and homology of the universal coverin ...
s 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_*(K,X), where ''K'' is a pointed, finite-dimensional CW complex and ''X'' is any pointed space, is a nilpotent space. The odd-dimensional real projective spaces are nilpotent spaces, while the projective plane is not. A basic theorem about nilpotent spaces 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 through which any map of ''X'' to a nilpotent space ''N'' factors uniquely up to a contractible space of choices. Often, however, \widehat itself is not nilpotent but only an inverse limit of a tower of nilpotent spaces. This tower, as a pro-space, always models the homology type of the given pointed space ''X''. Nilpotent spaces admit a good arithmetic localization theory in the sense of Bousfield and Kan cited above, and the unstable Adams spectral sequence strongly converges for any such space. 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 that any such section is ''X''-cellular.


References

Topological spaces {{topology-stub