Contents

# Contents

## 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.

## 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