nLab
long exact sequence of homotopy groups

Idea

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

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

In terms of presentations this means:

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

For background and details see fibration sequence.

Related concepts

• fiber sequence

• Gysin sequence

• Serre long exact sequence

• Given a tower of homotopy fibers such as a Whitehead tower or Adams resolution, the long exact sequences of homotopy groups for each stage combine to yield an exact couple. The corresponding spectral sequence is the Adams spectral sequence.

References

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 include

• Robert Switzer, around 2.59 of Algebraic Topology - Homotopy and Homology, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.

• Stanley Kochmann, corollary 3.2.7 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

See also

• Wikipedia, Long exact sequence of a fibration