Dear all, sorry for not having posted anything so far but my master thesis (stable classification of certain families of four manifolds) and PhD interviews have been keeping me very busy 🙂
This is a quick exercise which I was asked to solve by a friend of mine and since I found it funny I decided to spell some more details about it in order to show the importance of the Bockstein exact sequence. This could be a typical exam exercise in “Topology 2” here in Bonn, so dear fellows students, read and comment if anything is unclear!
The following is exercise 7, page 366 from Bredon’s Topology and Geometry:
Show that there can be no -manifold with
Clearly one has to reason by absurd, so let us assume that there exists indeed such manifold . The absurd will be given by the Bockstein homomorphism.
First of all, a quick application of UCT gives us the following:
Let us denote with a generator of . Let a generator of .
Claim 1 We can choose , where is the Bockstein homomorphism associated to the s.e.s.
Proof: It’s enough to prove that is an isomorphism. Let us have a look at the relevant piece of the exact sequences for
exactness gives us that is an isomorphism and naturality w.r.t. coefficients gives us that is an isomorphism as claimed.
Claim 2 .
Proof: Start by noticing that , since we are working with field coefficients. Therefore we can consider (i.e. it exists) an element as the dual of . By Poincaré Duality we have that for or (but not !!), . Therefore the pairing:
proves that .
In order to finish the exercise, let us compute , recall that by assumption :
where we used the fact that the Bockstein for our s.e.s. is a derivation (you can find it, as an exercise on Bredon’s book. You can prove it by hand via diagram chasing of the definition).
This is a clear absurd, which concludes our proof.
Note that the exercise can be generalized to -dim Poincaré complexes with the same homology groups but , with any odd prime. Think about it, so you can see where we used what, which is always important to know.