nLab long exact sequence of homotopy groups



Homotopy theory

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



Paths and cylinders

Homotopy groups

Basic facts




For YZY \to Z a morphism of pointed ∞-groupoids and XYX \to Y its homotopy fiber, there is a long exact sequence of homotopy groups

π n+1(Z)π n(X)π n(Y)π n(Z)π n1(X). \cdots \to \pi_{n+1}(Z) \to \pi_n(X) \to \pi_n(Y) \to \pi_n(Z) \to \pi_{n-1}(X) \to \cdots \,.

In terms of presentations this means:

for YZY \to Z a fibration in the classical model structure on topological spaces or in the classical model structure on simplicial sets, and for XYX \to Y the ordinary fiber of topological spaces or simplicial sets, respectively, we have such a long exact sequence.

For background and details see fibration sequence.


The observation of long exact sequences of homotopy groups for homotopy fiber sequences originates (according to Switzer 75, p. 35) in

  • M. G. Barratt, Track groups I, II. Proc. London Math. Soc. 5, 71-106, 285-329 (1955).

The first exhaustive study of these is due to

  • Dieter Puppe, Homotopiemengen und ihre induzierten Abbildungen I, Math. Z. 69, 299-344 (1958).

whence the terminology Puppe sequences.

Textbook accounts:

See also:

In the generality of categorical homotopy groups in an ( , 1 ) (\infty,1) -topos:

Last revised on January 4, 2024 at 17:11:31. See the history of this page for a list of all contributions to it.