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 .