Question

This must be an elementary question: could somebody tell me a reference for the Mayer-Vietoris homotopy groups sequence of a pull-back of a fibration?

I'm working in the category of pointed simplicial sets. So I've a pull-back of a (Kan) fibration of pointed simplicial sets, and I've read that in this situation you have an associated Mayer-Vietoris sequence relating the homotopy groups of the simplicial sets of the pull-back that looks like the classical Mayer-Vietoris sequence for the singular homology of a pair of open sets covering a topological space.

I've been searching in May's "Simplicial objects in Algebraic Topology" and Goerss-Jardine's "Simplicial Homotopy Theory", but I couldn't find it.

Answer 1

I don't know of a reference, but here is a quick argument. Suppose we want to compute the homotopy pullback P = X ×^h_Z Y of two maps f : X → Z and g : Y → Z of pointed simplicial sets. Assume for convenience that everything is fibrant. There is a fibration Z^Δ[1] → Z^∂Δ[1] = Z × Z with fiber ΩZ. Now P is the pullback of the diagram X × Y → Z × Z ← Z^Δ[1]. In particular, P → X × Y is also a fibration with fiber ΩZ, and the Mayer-Vietoris sequence follows from the long exact sequence of homotopy groups of this fibration.