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

$$\cdots \to {\pi }_{n+1}(Z)\to {\pi }_n(X)\to {\pi }_n(Y)\to {\pi }_n(Z)\to {\pi }_{n-1}(X)\to \cdots .$$\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