Let $M$ be a closed, smooth $4$-manifold with integral cohomology ring isomorphic to that of $\mathbb{CP}^2$, is it diffeomorphic to it?



4-dimensional cohomology $\mathbb{CP}^2$'s
No. If $\Sigma$ is any homology 4-sphere with non-trivial fundamental group, $\mathbb{CP}^2 \# \Sigma$ is a homology $\mathbb{CP}^2$ with non-trivial fundamental group. (Here $\#$ denotes connected sum.) There are many examples of homology 4-spheres: for instance, Kervaire produced many examples in his paper *Smooth homology spheres and their fundamental groups*.\*\*


Whether there is any simply connected 4-manifold with the same homology as $\mathbb{CP}^2$ but that is not diffeomorphic to it, is an important open question, which is somewhat related to the 4-dimensional smooth Poincaré conjecture.


\*\* The main statement is about homology $n$-spheres for $n$ strictly larger than 4, but the remark at the end of the introduction mentions that the constructive part of the proof works for $n = 4$ as well.


