# Difference between revisions of "Homotopy type of connected sum depends on choice of gluing map"

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(Created page with "==Statement== It is possible to find an example of compact connected orientable manifolds <math>M_1</math> and <math>M_2</math> such that the homotopy type of the [[connecte...") |
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− | To construct an example, we need to find a case where both <math>M_1</math> and <math>M_2</math> are orientable but neither of them has an orientation-reversing self-homeomorphism. One simple choice, by Fact (1), is to set both <math>M_1</math> and <math>M_2</math> as | + | To construct an example, we need to find a case where both <math>M_1</math> and <math>M_2</math> are orientable but neither of them has an orientation-reversing self-homeomorphism. One simple choice, by Fact (1), is to set both <math>M_1</math> and <math>M_2</math> as homeomorphic to the [[complex projective plane]] <math>\mathbb{P}^2(\mathbb{C})</math> which has real dimension 4. |

There are two possible connected sums: | There are two possible connected sums: |

## Revision as of 17:26, 29 July 2011

## Statement

It is possible to find an example of compact connected orientable manifolds and such that the homotopy type of the connected sum is not well defined, i.e., we can get connected sums of different homotopy types depending on the choice of the gluing map.

## Facts used

## Proof

To construct an example, we need to find a case where both and are orientable but neither of them has an orientation-reversing self-homeomorphism. One simple choice, by Fact (1), is to set both and as homeomorphic to the complex projective plane which has real dimension 4.

There are two possible connected sums:

- Connected sum of two complex projective planes with same orientation: This has cohomology ring isomorphic to , where are additive generators of the free abelian group and is the additive generator for .
- Connected sum of two complex projective planes with opposite orientation: This has cohomology ring isomorphic to , where are additive generators of the free abelian group and is the additive generator for .