In terms of Betti numbers
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
In terms of Lefschetz number
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.
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.|
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
There is a corresponding statement for fiber bundles and fibrations. Fill this in later