# Difference between revisions of "Connected sum of two complex projective planes with same orientation"

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The cohomology ring is as follows: | The cohomology ring is as follows: | ||

− | <math>H^*(M) = \mathbb{Z}[x,y]/(x^2 - y^2, x^3,y^3)</math> | + | <math>H^*(M) = \mathbb{Z}[x,y]/(x^2 - y^2, xy, x^3,y^3)</math> |

===Homotopy groups=== | ===Homotopy groups=== | ||

The [[fundamental group]] is the trivial group. | The [[fundamental group]] is the trivial group. |

## Latest revision as of 16:40, 13 December 2016

This article is about a particular topological space (uniquely determined up to homeomorphism)|View a complete list of particular topological spaces

## Contents

## Definition

This topological space is defined as the connected sum of two copies of the complex projective plane , where they are glued with the same orientation.

## 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 opposite 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.