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

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.

## 3 Answers

I don't know of a reference, but here is a quick argument. Suppose we want to compute the homotopy pullback P = X ×hZ 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 ZΔ → Z∂Δ = Z × Z with fiber ΩZ. Now P is the pullback of the diagram X × Y → Z × Z ← ZΔ. 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.

• I would add that this is Eckmann-Hilton dual to the argument that gets the usual Mayer-Vietoris sequence from the long exact sequence of a cofibration. – Eric Wofsey Oct 30 '09 at 14:15

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.

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