Connected sum of two complex projective planes with opposite orientation

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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

Algebraic topology

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.