# Homology of connected sum

From Topospaces

*This article describes the effect of the connected sum operation on the following invariant: homology groups*

## Statement

Suppose and are the connected manifolds of dimension whose connected sum is being taken. Assume both We have:

Case for | Additional condition on | What is known about and ? | Formula for in terms of homology groups of and |
---|---|---|---|

0 | none | both isomorphic to | (because both are connected, so is their connected sum). |

Greater than 0, less than | none | both are finitely generated abelian groups | |

Both manifolds are compact, at least one of them is orientable | both are finitely generated abelian groups, at least one is free abelian | ||

other cases | ? | ? | |

Both are compact and orientable | both are | ||

other cases | ? | ? | |

Greater than | none | both are zero groups | 0 |

### Euler characteristic

The Euler characteristics are related by the following formula when both and are compact connected manifolds: