Euler characteristic

From Topospaces
Jump to: navigation, search

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

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