homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
A connected topological space $X$ (or rather its homotopy type) is called nilpotent if
its fundamental group $\pi_1(X)$ is a nilpotent group;
the action of $\pi_1(X)$ on the higher homotopy groups is a nilpotent module in that the sequence $N_{0,n} \coloneqq \pi_n(X)$, $N_{k+1,n} \coloneqq \{g n - n | n \in N_{k,n}, g \in \pi_1(X)\}$ terminates.
Directly from the definition we have that:
and more generally
As a special case of this
and thus
every loop space is nilpotent
(since all its connected components are homotopy equivalent to the unit component, which is a connected H-space).
(See May-Ponto 12, p. 49 (77 of 542))
(See Hilton 82, Section 3).
Nilpotency is involved in sufficient conditions for many important constructions in (stable) homotopy theory, see for instance at
Peter Hilton, Nilpotency in group theory and topology, Publicacions de la Secció de Matemàtiques Vol. 26, No. 3 (1982), pp. 47-78 (jstor:43741908)
Peter May, Kate Ponto, More Concise Algebraic Topology, University of Chicago Press (2012) (pdf)
Emily Riehl, def. 14.4.9 in: Categorical Homotopy Theory, New Mathematical Monographs 24, Cambridge University Press 2014 (pdf, doi:10.1017/CBO9781107261457)
See also
See also
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:
Last revised on September 15, 2020 at 11:03:58. See the history of this page for a list of all contributions to it.