Euler Characteristics and Mayer-Vietoris Sequence

Euler Characteristic Revisited

Recall the rank of an abelian group:

Every finite abelian group is isomorphic to a direct product (sum) of cyclic groups:
such that divides for all , and .
More generally, every finitely generated abelian group is isomorphic to

where is called the rank of , divides for all , and is called the torsion subgroup of .

Link to original

e.g. .

Homological Euler Characteristic

The Euler characteristic of a finite CW complex can be computed from its homology groups:

is a topological invariant and does not depend on the specific CW structure chosen for .

The Euler characteristic

Proof Let , , and . These groups are related by two short exact sequences for each :From a lemma on short exact sequences of finitely generated abelian groups, we know that ranks are additive. Thus:

Now, we compute : The last two sums cancel each other out via a change of index. Therefore, we are left with the desired result: $$\chi(X) = \sum_n (-1)^n \text{rank}(H_n(X)) $$$\square$

e.g.

• Sphere: .
• Real Projective Space: .
• Klein Bottle: .
• Genus Surface: .
• Complex Projective Space: .
  

Mayer-Vietoris Sequence

Mayer-Vietoris Sequence

Suppose a space is the union of the interiors of two subspaces and , i.e., . Then there is a long exact sequence in homology:

Proof This sequence arises from the short exact sequence of chain complexes: Here represents chains in , and by the Excision Theorem, its homology is .

e.g. The suspension of a space is given by , where and are two cones on joined at their base.
Let and . Then and . Both and are contractible, so their reduced homology groups are zero, i.e., for all . The reduced Mayer-Vietoris sequence contains the segment: Since the cone homology groups are zero, this simplifies to:This shows that the connecting homomorphism is an isomorphism: