nilpotent homotopy type



Homotopy theory

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topological homotopy theory



A topological space/simplicial set/homotopy type/infinity-groupoid XX is nilpotent if

  1. its fundamental group is a nilpotent group

  2. the canonical action of π 1(X)\pi_1(X) on all the higher homotopy groups π n2(X)\pi_{n \geq 2}(X) is nilpotent.

The first condition means that π 1(X)\pi_1(X) is isomorphic to an iterated central extension of abelian groups.

The second condition means that each π n2(X)\pi_{n \geq 2}(X) admits a sequence of subgroups

*=G kG 1=π n(X) \ast = G_k \hookrightarrow \cdots \hookrightarrow G_1 = \pi_n(X)

such that for all ii

  1. G i+1G iG_{i+1} \hookrightarrow G_i is a normal subgroup;

  2. the quotient G i/G i+1G_i/G_{i+1} is an abelian group;

  3. each G iG_i is closed under the action of π 1(X)\pi_1(X);

  4. the induced action on G i/G i+1G_i/G_{i+1} is trivial.

This implies that given any element aπ n2(X)a \in \pi_{n \geq 2}(X), then after acting on it at most kk times with elements from π 1(X)\pi_1(X) the result is zero.


Nilpotency is involved in sufficient conditions for many important constructions in (stable) homotopy theory, see for instance at


Review includes

  • Emily Riehl, def. 14.4.9 in Categorical homotopy theory, new mathematical monographs 24, Cambridge University Press 2014 (published version)

See also

Last revised on April 19, 2018 at 03:11:26. See the history of this page for a list of all contributions to it.