For the second article under the FtD tag, I’ve chosen a nice proposition which bothered me some times ago. It’s intuitively easy to believe but I found a scarce amount of details spelled out in the proofs one can find on the classical references. So here we have the following result:
Proposition: Every -homology class of a smooth 4-manifold is generated by a surface
The classical proof goes as follows
Proof: (from Kirby’s Topology of -Manifolds Theorem 1.1 page 20) There is an isomorphism:
so letting being the Poincaré dual of a chosen , there is an homotopy class of maps corresponding to . By cellular approximation, we can homotopy (a representative of) in order to obtain a map . In fact the 4-skeleton of is and cellular approximation tells you that the image of lies there. Make smoothly transverse to . Consider , this will be an oriented surface representing .
yes but… why That’s the real question, why represents our homology class
we need the following facts:
Claim 1: Let be a submanifold and let the homology class it defines in . Then in the Thom class of the normal bundle is dual to .
Proof: Have a look here 🙂.
Back to our original exercise. First of all we need to fix some notations: Let us denote with the Thom Class associated to the normal bundle of the inclusion . Similarly, let be the Thom class associated to the normal bundle of . With we will mention the Poincaré Dual of (both in the case of homology and cohomology).
So let be our homology class, as we said before , where is a map as above and is the fundamental class associated to ( or in our case, doesn’t change anything by cellular approximation). Using the fact that (see the addendum below), we have:
By Claim 1 and naturality of the Thom classes (the normal bundle over is the pullback via of the one over )
Using PD again,
Using Claim 1 again,
Which was exactly what we needed to show.
ADDENDUM: Let us recall briefly the definition of the fundamental class. Let be a polarised Eilenberg-MacLane space of type (i.e. we fixed an isomorphism from the homotopy group to . Using UCT (with map ) there is a unique class , called the fundamental class s.t. the composite
is the chosen . (it depends on the chosen iso!). In fact is connected, and abelian, the Hurewicz map is an isomorphism and therefore
Is a group homomorphism (an iso actually). Since UCT in degree is an iso, we define .
In our case, , and since , the map is a generator of it, and UCT maps generator to generator. So up to a sign, we can assume .