Let $M$ be a compact manifold. What is your favorite proof of finite generatedness of the singular cohomology $H^*(M,\mathbf{Z})$ of $M$ that avoids invoking the isomorphism with cellular cohomology, but only uses compactness of $M$ and existence of good covers?
Could you please provide references?
Thanks!
This is essentially a trivial consequence of the existence of Mayer-Vietoris sequences. More generally, if $M$ is a space that has a finite good cover (meaning an open cover such that all the intersections of sets in the cover are either empty or contractible), then the cohomology of $M$ is finitely generated. The proof is a very straightforward induction on the number of sets in the good cover: if $U_1,\dots,U_n$ is a good cover of $M$, let $N=U_1\cup\dots\cup U_{n-1}$. Then $N$ and $N\cap U_n$ both have good covers by fewer than $n$ sets ($U_1,\dots,U_{n-1}$ for $N$ and their intersections with $U_n$ for $N\cap U_n$), so by induction they have finitely generated cohomology. Since $U_n$ is contractible, the Mayer-Vietoris sequence for covering $M$ by $N$ and $U_n$ gives that the cohomology of $M$ is finitely generated.