# Connected sum of two complex projective planes with opposite orientation

From Topospaces

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## Contents

## Definition

This topological space is defined as the connected sum of two copies of the complex projective plane , 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:

### Cohomology groups

The cohomology groups are as follows:

The cohomology ring is as follows:

### Homotopy groups

The fundamental group is the trivial group.