nLab
long exact sequence of homotopy groups

Contents

Idea

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 ordinary model structure on topological spaces or in the 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.

References

Revised on October 30, 2014 19:14:35 by Urs Schreiber (185.26.182.33)