Mayer-Vietoris homotopy groups sequence of a pull-back of a fibration

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.

Answer 2

Here is a lower-tech version of Reid's answer. In general, if one has a Serre fibration p: E->B, and a map f: X->B, then there is a Mayer-Vietoris type sequence that comes from weaving together the long exact sequences in homotopy associated to p and to f*(p): f*(E)->X. These exact sequences agree on every third term, because the fibers of f*(p) and p are homeomorphic. The weaving process I'm talking about is an exercise in Hatcher's book (Exercise 38 in Section 2.2).

The relation between this exact sequence and the usual Mayer-Vietoris sequence in cohomology is explained here: Mathematically mature way to think about Mayer–Vietoris

Answer 3

A low level Mayer-Vietoris sequence for a pull-back of a fibration of groupoids is in (R. BROWN, P.R. HEATH and H. KAMPS), ``Groupoids and the Mayer-Vietoris sequence'', {\em J. Pure Appl. Alg.} 30 (1983)

and you will also find a version for coverings of groupoids in `Topology and Groupoids', R. Brown (available on amazon.com).

I've set as an exercise in my new coauthored book `Nonabelian algebraic topology' (see my web pages) to get a Mayer-Vietoris sequence for a pullback of a fibration of crossed complexes.