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.:

$\chi (X)=\sum _{k=0}^{\mathrm{\infty }}(-1{)}^k{b}_k(X)$

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.

Particular cases

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)
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 $\mathbb{R}{\mathbb{P}}^n$, $n$ 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 $\mathbb{R}{\mathbb{P}}^n$, $n$ 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	$X\bigsqcup Y$ 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: $\chi (X\bigsqcup Y)\chi (X)+\chi (Y)$	Euler characteristic of disjoint union is sum of Euler characteristics
disjoint union	finite number $n$	$\underset{i=1}{\overset{n}{⨆}}{X}_i$ or $X_1\bigsqcup X_2\bigsqcup \dots \bigsqcup X_n$ 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: $\chi (\underset{i=1}{\overset{n}{⨆}}{X}_i)=\sum _{i=1}^n\chi (X_i)$
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

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