Nilpotent Space
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In
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, a branch of
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a nilpotent space, first defined by Emmanuel Dror Farjoun (1969), is a
based Brandon Christopher McCartney (born August 17, 1989), known professionally as Lil B and Lil B The BasedGod, is an American rapper. He began his career as a member of the Berkeley, California-based hip hop group the Pack in 2005, who signed wi ...
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
''X'' such that * the
fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...
\pi = \pi_1 (X) is a
nilpotent group In mathematics, specifically group theory, a nilpotent group ''G'' is a group that has an upper central series that terminates with ''G''. Equivalently, it has a central series of finite length or its lower central series terminates with . I ...
; * \pi acts nilpotently on the higher homotopy groups \pi_i (X), i \ge 2, i.e., there is a
central series In mathematics, especially in the fields of group theory and Lie theory, a central series is a kind of normal series of subgroups or Lie subalgebras, expressing the idea that the commutator is nearly trivial. For groups, the existence of a central ...
\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 into any other such path while preserving the two endpoi ...
s and simple spaces are (trivial) examples of nilpotent spaces; other examples are connected
loop space In topology, a branch of mathematics, the loop space Ω''X'' of a pointed topological space ''X'' is the space of (based) loops in ''X'', i.e. continuous pointed maps from the pointed circle ''S''1 to ''X'', equipped with the compact-open topolog ...
s. The
homotopy fiber In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber)Joseph J. Rotman, ''An Introduction to Algebraic Topology'' (1988) Springer-Verlag ''(See Chapter 11 for construction.)'' is part of a construction ...
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 In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called ''cells'') of different dimensions in specific ways. It generali ...
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 Witold Hurewicz (June 29, 1904 – September 6, 1956) was a Polish mathematician who worked in topology. Early life and education Witold Hurewicz was born in Łódź, at the time one of the main Polish industrial hubs with economy focused on th ...
theorems. Nilpotent spaces are of great interest in
rational homotopy theory In mathematics and specifically in topology, rational homotopy theory is a simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored. It was founded by and . This simplification of homoto ...
, 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 In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within t ...
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 In mathematics, the Adams spectral sequence is a spectral sequence introduced by which computes the stable homotopy groups of topological spaces. Like all spectral sequences, it is a computational tool; it relates homology theory to what is now c ...
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 In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometr ...
. 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