Manifold whose rational cohomology ring is same like the rational cohomology ring of complex projective plane.

Question

There are many examples of rational homology spheres whose cohomology ring are same like usual spheres. Is there are some manifolds whose rational cohomology ring same like complex projective spaces(in particular complex projective plane).

Answer 1

Let $M$ be an $n$-dimensional rational homology sphere, i.e. a closed $n$-dimensional manifold with $H_k(M; \mathbb{Q}) \cong H_k(S^{2n}; \mathbb{Q})$ for every $k$. Then for any closed, orientable $n$-dimensional manifold $N$, the manifolds $M\# N$ and $N$ have the same rational cohomology ring. In particular, $M\#\mathbb{CP}^n$ and $\mathbb{CP}^n$ have the same rational cohomology ring for any $2n$-dimensional rational homology sphere $M$.

An explicit example of an even dimensional rational homology sphere is the grassmanian $\operatorname{Gr}_2(\mathbb{R}^4)$ which has dimension four and integral homology groups $\mathbb{Z}, \mathbb{Z}_2, \mathbb{Z}_2, 0, \mathbb{Z}$ (see this question). So $\operatorname{Gr}_2(\mathbb{R}^4)\#\mathbb{CP}^2$ has the same rational cohomology ring as $\mathbb{CP}^2$.

Answer 2

In addition to Michael's answer, you may be interested to know some other examples.

We'll start with simply connected homogeneous examples, which have been completely classified.

First, the Grassmannian of oriented planes in $\mathbb{R}^{2n+1}$ is a simply connected rational $\mathbb{C}P^n$. However, it's integral cohomology ring contains $2$-torsion, so it is not homotopy equivalent to $\mathbb{C}P^n$

Viewing the Grassmannian as the homogeneous space $SO(2n+1)/(SO(2n-1)\times S^1)$ leads us to our second example: One can view this as $SO(2n+1)/SO(2n-1)$ with a circle acting freely. A circle can act freely in a different way, giving the biquotient $S^1\backslash SO(2n+1)/SO(2n-1)$. It was shown in

V.Kapovitch-W.Ziller, Biquotients with singly generated rational cohomology, Geom. Dedicata 104 (2004), 149-160.

that the biquotient and the Grassmannian have isomorphic integral cohomology rings, but their Pontryagin classes are different when $n\geq 2$.

The last homogeneous example is a homogeneous space of the form $G_2/U(2)$, where $U(2)$ is a subgroup of the natural $SO(4)$ in $G_2$, but it is not a subgroup of the natural $SU(3)$. (If one uses the $U(2)$ in $SU(3)\cap SO(4)$, then $G_2/U(2)$ is diffeomorphic to one of the Grassmannian examples given above.)

None of these examples is homotopy equivalent to $\mathbb{C}P^n$. But there are non-homogeneous spaces which have the homotopy type of $\mathbb{C}P^n$, but are not homeomorphic to it. I don't know much about these, but they are called fake projective spaces.