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

• 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.

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.