# The Groove of Math

## Filling the Details #2: Representing 2-homology classes of a 4-manifold.

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 $2$-homology class of a smooth 4-manifold is generated by a surface

Let $M = M^n$ and $A = A^p$ be compact oriented manifolds with smooth embedding $i : M \rightarrow A$. Let $k = p-n$. Show that the Poincare duality isomorphism  $\frown \mu_A : H^k(A) \rightarrow H_n(A)$ maps the cohomology class $u^{'}|A$ dual to $M$ to the homology class $(-1)^{nk} i_{*} (\mu_M)$. Assume moreover that the normal bundle $\nu^k$ is oriented so that $\tau_M \oplus \nu^k$ is orientation preserving isomorphic to $\tau_A|M$.