Connected sum of two complex projective planes with opposite orientation

Definition
This topological space is defined as the connected sum of two copies of the complex projective plane $$\mathbb{P}^2(\mathbb{C})$$, where they are glued with opposite orientations.

Related facts

 * Homotopy type of connected sum depends on choice of gluing map: In particular, this connected sum is of a different homotopy type than the connected sum of two complex projective planes with same orientation.

Homology groups
The homology groups are as follows:

$$H_p(M) = \left\lbrace \begin{array}{rl}\mathbb{Z}, & \qquad p = 0,4 \\ \mathbb{Z} \oplus \mathbb{Z}, & \qquad p = 2 \\ 0, & \qquad \operatorname{otherwise}\\\end{array}\right.$$

Cohomology groups
The cohomology groups are as follows:

$$H^p(M) = \left\lbrace \begin{array}{rl}\mathbb{Z}, & \qquad p = 0,4 \\ \mathbb{Z} \oplus \mathbb{Z}, & \qquad p = 2 \\ 0, & \qquad \operatorname{otherwise}\\\end{array}\right.$$

The cohomology ring is as follows:

$$H^*(M) = \mathbb{Z}[x,y]/(x^2 + y^2, xy, x^3,y^3)$$

Homotopy groups
The fundamental group is the trivial group.