# Euler characteristic

## Definition

### 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 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 Poincare polynomial at the number $-1$.

### In terms of Lefschetz number

The Euler characteristic of a space with finitely generated homology $X$ is the 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 $1$:

### 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 $n$ points $n$ The zeroth homology is free of rank $n$.
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 $S^n$ $1 + (-1)^n$, which is 2 if $n$ is even, and 0 if $n$ is odd. See homology of spheres
torus $T^n$ 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 $g$ surface $2 - 2g$ follows from homology of compact orientable surfaces
real projective space $\R\mathbb{P}^n$, $n$ 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 $S^n$ has Euler characteristic zero.
real projective space $\R\mathbb{P}^n$, $n$ 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 $S^n$ has Euler characteristic 2.

## Facts

### Effect of operations

Operation Arity Brief description Effect on Euler characteristics Proof/explanation
disjoint union 2 $X \sqcup Y$ 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: $\chi(X \sqcup Y) = \chi(X) + \chi(Y)$ Euler characteristic of disjoint union is sum of Euler characteristics
disjoint union finite number $n$ $\bigsqcup_{i=1}^n X_i$ or $X_1 \sqcup X_2 \sqcup \dots \sqcup X_n$ 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: $\chi(\bigsqcup_{i=1}^n X_i) = \sum_{i=1}^n \chi(X_i)$ Euler characteristic of disjoint union is sum of Euler characteristics
wedge sum 2 $X \vee Y$, defined in the context of a basepoint choice for both spaces, is the quotient of the disjoint union by identification of the two basepoints $\chi(X \vee Y) = \chi(X) + \chi(Y) - 1$
product of topological spaces 2 $X \times Y$, the Cartesian product, equipped with the product topology $\chi(X \times Y) = \chi(X)\chi(Y)$ Euler characteristic of product is product of Euler characteristics
product of topological spaces finite number $n$ $X_1 \times X_2 \times \dots \times X_n$, the Cartesian product, equipped with the product topology $\chi(X_1 \times X_2 \times \dots \times X_n)$ 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