Chain Complexes and Homology - Applications

Homology theory provides powerful computational and theoretical tools with applications throughout algebra and topology.

Theorem1.4Euler-Poincaré Formula

Let $C_\bullet$ be a chain complex of finite-dimensional vector spaces with finitely many non-zero terms. Then:

$\chi(C_\bullet) = \sum_{n} (-1)^n \dim(C_n) = \sum_{n} (-1)^n \dim(H_n(C_\bullet))$

where $\chi(C_\bullet)$ is the Euler characteristic.

Proof

For each $n$ , the short exact sequence $0 \to Z_n \to C_n \to B_{n-1} \to 0$ gives: $\dim(C_n) = \dim(Z_n) + \dim(B_{n-1})$

Similarly, $0 \to B_n \to Z_n \to H_n \to 0$ gives: $\dim(Z_n) = \dim(B_n) + \dim(H_n)$

Combining these and summing alternately: $\sum (-1)^n \dim(C_n) = \sum (-1)^n (\dim(B_n) + \dim(H_n) + \dim(B_{n-1}))$

The boundary terms telescope to zero, leaving $\sum (-1)^n \dim(H_n)$ .

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ExampleEuler Characteristic of Surfaces

For a closed orientable surface $\Sigma_g$ of genus $g$ , the simplicial chain complex satisfies:

$\dim(H_0) = 1$ (one connected component)
$\dim(H_1) = 2g$ (fundamental group)
$\dim(H_2) = 1$ (orientation)

The Euler characteristic is $\chi(\Sigma_g) = 1 - 2g + 1 = 2 - 2g$ .

Theorem1.5Universal Coefficient Theorem (Homology)

Let $C_\bullet$ be a chain complex of free abelian groups and $G$ any abelian group. Then there is a natural short exact sequence:

$0 \to H_n(C_\bullet) \otimes G \to H_n(C_\bullet \otimes G) \to \text{Tor}_1(H_{n-1}(C_\bullet), G) \to 0$

This sequence splits (non-naturally).

Remark

The Universal Coefficient Theorem relates homology with different coefficient groups. It shows that homology with coefficients in $G$ can be computed from homology with integer coefficients, up to a correction term involving the Tor functor.

ExampleHomology with Field Coefficients

When $G = \mathbb{F}$ is a field, $\text{Tor}_1(H_{n-1}(C_\bullet), \mathbb{F}) = 0$ since fields are flat. Thus: $H_n(C_\bullet; \mathbb{F}) \cong H_n(C_\bullet; \mathbb{Z}) \otimes \mathbb{F}$

This means homology with field coefficients simply tensorizes the integer homology with the field.

Theorem1.6Künneth Formula

Let $C_\bullet$ and $D_\bullet$ be chain complexes of free abelian groups. Then:

$H_n(C_\bullet \otimes D_\bullet) \cong \bigoplus_{p+q=n} (H_p(C_\bullet) \otimes H_q(D_\bullet)) \oplus \bigoplus_{p+q=n-1} \text{Tor}_1(H_p(C_\bullet), H_q(D_\bullet))$

Remark

The Künneth formula computes the homology of a tensor product of chain complexes from the homologies of the factors. It's essential for computing homology of product spaces.