# Euler characteristic

## Contents

## Definition

### In terms of Betti numbers

The **Euler characteristic** of a space with finitely generated homology , denoted , is defined as a signed sum of its Betti numbers, viz.:

where is the Betti number, i.e., the rank of the torsion-free part of the homology group of .

Note that by assumption, all the are finite and only finitely many of them are nonzero.

*The Euler characteristic can take any integer value, including zero, positive, and negative integers.*

### In terms of Poincare polynomial

The **Euler characteristic** of a space with finitely generated homology , denoted , is defined as the value of its Poincare polynomial at the number .

### In terms of Lefschetz number

The **Euler characteristic** of a space with finitely generated homology is the Lefschetz number (also called Lefschetz trace) of the identity map from to itself.

### For a CW-complex with finitely many cells

For a CW-complex with finitely many cells, the Euler characteristic can be defined as:

where ranges over all cells and isthe dimension of the cell. In particular, if there are cells of dimension for each , this becomes:

By assumption, all the are finite and only finitely many of them are nonzero.

Note that corresponding CW-space is a space with finitely generated homology and the Euler characteristic of that topological space equals the Euler characteristic of the CW-complex. *However, it is possible for a CW-complex with infinitely many cells (either infinitely many cells at a given dimension, or arbitrarily large dimensions) that still has finitely generated homology.* In that case, the above definition of Euler characteristic of a CW-complex will not apply, but we can still compute the Euler characteristic as a topological space.

## Particular cases

Two particular numerical values of the Euler characteristic are of significance -- and :

### Euler characteristics of some typical spaces

Space | Value of Euler characteristic (may depend on a parameter if the space is not specific but describes a family with a parameter) | Justification |
---|---|---|

contractible space | 1 | All homology groups except zeroth one are zero |

acyclic space | 1 | (similar to above) |

finite discrete space with points | The zeroth homology is free of rank . | |

circle | 0 | Compatible with explanation for spheres, also with explanation for connected Lie groups |

underlying topological space of a nontrivial compact connected Lie group | 0 | Euler characteristic of nontrivial compact connected Lie group is zero |

sphere | , which is 2 if is even, and 0 if is odd. | See homology of spheres |

torus | 0 | compatible with statement for Lie groups, also from fact about circles and Euler characteristic of product is product of Euler characteristics |

compact orientable genus surface | follows from homology of compact orientable surfaces | |

real projective space , odd | 0 | compatible with fact that Euler characteristic of covering space is degree of covering times Euler characteristic of base, and the fact that the double cover has Euler characteristic zero. |

real projective space , even | 1 | compatible with fact that Euler characteristic of covering space is degree of covering times Euler characteristic of base, and the fact that the double cover has Euler characteristic 2. |

## Facts

### Effect of operations

Operation | Arity | Brief description | Effect on Euler characteristics | Proof/explanation |
---|---|---|---|---|

disjoint union | 2 | is a set-theoretic disjoint union and open subsets of this are those whose intersection with each piece is open in that. | Sum of Euler characteristics: | Euler characteristic of disjoint union is sum of Euler characteristics |

disjoint union | finite number | or is a set-theoretic disjoint union and open subsets of this are those whose intersection with each piece is open in that. | Sum of Euler characteristics: | Euler characteristic of disjoint union is sum of Euler characteristics |

wedge sum | 2 | , defined in the context of a basepoint choice for both spaces, is the quotient of the disjoint union by identification of the two basepoints | ||

product of topological spaces | 2 | , the Cartesian product, equipped with the product topology | Euler characteristic of product is product of Euler characteristics | |

product of topological spaces | finite number | , the Cartesian product, equipped with the product topology | Euler characteristic of product is product of Euler characteristics |

### Covering spaces and fibrations

We have Euler characteristic of covering space is degree of covering times Euler characteristic of base.

There is a corresponding statement for fiber bundles and fibrations. *Fill this in later*