long exact sequence of homotopy groups in nLab

Contents

Contents

Idea

For Y→ZY \to Z a morphism of pointed ∞-groupoids and X→YX \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 n−1(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 Y→ZY \to Z a fibration in the classical model structure on topological spaces or in the classical model structure on simplicial sets, and for X→YX \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.

• long exact sequence in homology

  • long exact sequence in chain homology

  • long exact sequence in generalized homology

• 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:

• Norman Steenrod, Thm. 17.4 in: The topology of fibre bundles, Princeton Mathematical Series 14, Princeton Univ. Press, 1951 (jstor:j.ctt1bpm9t5)

• Robert Switzer, around 2.59 in: Algebraic Topology - Homotopy and Homology, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975 (doi:10.1007/978-3-642-61923-6)

• Stanley Kochmann, Corollary 3.2.7 in: Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

• Tammo tom Dieck, Thm. 6.1.2 in: Algebraic topology, European Mathematical Society, Zürich (2008) (doi:10.4171/048, pdf)

See also:

• Wikipedia, Long exact sequence of a fibration

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

• Jacob Lurie, Rem. 6.5.1.5 of: Higher Topos Theory, Annals of Mathematics Studies 170, Princeton University Press 2009 (pup:8957, pdf)