Euler characteristic

In terms of Betti numbers
The Euler characteristic of a space with finitely generated homology $$X$$, denoted $$\chi(X)$$, is defined as a signed sum of its defining ingredient::Betti numbers, viz.:

$$\chi(X) = \sum_{k=0}^\infty (-1)^k b_k(X)$$

where $$b_k$$ is the $$k^{th}$$ Betti number, i.e., the rank of the torsion-free part of the $$k^{th}$$ homology group of $$X$$.

Note that by assumption, all the $$b_k(X)$$ 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 $$X$$, denoted $$\chi(X)$$, is defined as the value of its defining ingredient::Poincare polynomial at the number $$-1$$.

In terms of Lefschetz number
The Euler characteristic of a space with finitely generated homology $$X$$ is the defining ingredient::Lefschetz number (also called Lefschetz trace) of the identity map from $$X$$ 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:

$$\sum_i (-1)^{d_i}$$

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

$$\sum_j (-1)^j c_j$$

By assumption, all the $$c_j$$ 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 -- $$0$$ and $$1$$:


 * Space with zero Euler characteristic
 * Space with Euler characteristic one

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.