nilpotent topological space




topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


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topological homotopy theory



A connected topological space xx is called nilpotent if

  1. its fundamental group π 1(X)\pi_1(X) is a nilpotent group;

  2. the action of π 1(X)\pi_1(X) on the higher homotopy groups is nilpotent in that the sequence N 0,nπ n(X)N_{0,n} \coloneqq \pi_n(X), N k+1,n{gnn|nN k,n,gπ 1(X)}N_{k+1,n} \coloneqq \{g n - n | n \in N_{k,n}, g \in \pi_1(X)\} terminates.


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.


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)

Last revised on June 17, 2019 at 05:21:29. See the history of this page for a list of all contributions to it.