Currently I’m reading the paper of Prof. Peter Teichner On the Signature of four-manifolds with universal covering spin, and I was stuck on the following passage:
The homotopy fibration induces an exact sequence in cohomology
I managed to find a solution to it now, and I thought it could be a good idea to write it here:
So let us start with a definition:
Def 1. is an homotopy fibration sequence if a homotopy is given from the composed map to a constant map $latex z$ and the resulting map from $latex X$ to the homotopy fiber of $latex Y \to Z $ over is a weak homotopy equivalence.
So we start by proving that the sequence we have is really an homotopy fibration.
Claim 1. The sequence is an homotopy fibration sequence.
Proof: Consider the following square:
where we used the fact that the universal cover of the Eilenberg-Maclane space is a model for . The nullhomotopy for is provided by the contractibility of . Now consider this pullback diagram:
where the bended arrow is provided by the null-homotopy chosen before. Notice that that fiber of is really the homotopy fiber of since we can assume that is a fibration. Moreover, since monomorphisms are stable under pullback, the map is really the usual inclusion of fibre. A quick application of the l.e.s. of homotopy groups for a Serre fibration, gives us that
(use the fact that induces iso on first homotopy group). By the above diagram should be clear that the induced map is a weak equivalence. Therefore we concluded the proof of the fact that the sequence is an homotopy fibration.
Prop 1. Let be a fibration with path connected and based. Set . Assume is -connected and is -connected. Then there’s a exact sequence
we will prove this proposition in several steps.
Recall first the following theorem, called the Dual Blakers-Massey Theorem for squares [Munson & Volić, Cubical Homotopy Theory, Thm 4.2.2. page 188]
Theorem 1. Suppose that the following square
is an homotopy pullback square, and that the maps and are respectively -connected for . Then the canonical map is connected.
Claim 2. In the above setting, the map is -connected.
Proof. Consider the following square
we want to apply Theorem 1 to it, therefore we need to show that the square is homotopy pullback (see def. 3.2.4 page 102 in Cubical Homotopy Theory). By definition the homotopy pullback is the subspace of such that is s.t. and , in particular it’s the subspace consisting of points where . If we denote with the subspace of whose paths end in , we have that the homotopy pullback is just . Now it’s well-known that deformation retract to the constant path , therefore, by definition , proving that the square is indeed homotopy pullback. Notice then that the map is -connected since the cone is contractible and[ is -connected by assumption.
Therefore by Theorem 1 we have that the map is connected. Following definition 3.6.3 page 138 of Cubical Homotopy Theory, it’s immediate to see that which concludes the proof.
Proof of Prop 1: Consider the cofibration sequence . Consider the Puppe Sequence for for large enough. Now use the fact that a equivalence gives us isomorphism in cohomology up to [tom Dieck, Algebraic Topology. Theorem 9.5.2 page 236] (so to conclude.
Back to our initial problem: our setting was , and , since . After applying our machinery we have the result. Notice that and therefore using UCT for cohomology it’s even in coefficient.