Simplicial Homology - Key Properties

The homology groups defined from the chain complex capture topological invariants that detect holes of various dimensions.

Definition

Let $K$ be a simplicial complex with chain complex $(C_\bullet(K), \partial)$ . Define:

$n$ -cycles: $Z_n(K) = \ker(\partial_n) = \{c \in C_n(K) : \partial_n c = 0\}$
$n$ -boundaries: $B_n(K) = \text{im}(\partial_{n+1}) = \{\partial_{n+1}c : c \in C_{n+1}(K)\}$

The $n$ -th homology group is the quotient: $H_n(K) = Z_n(K) / B_n(K)$

Since $\partial^2 = 0$ , we have $B_n(K) \subseteq Z_n(K)$ , so the quotient is well-defined. Elements of $H_n(K)$ are equivalence classes $[c]$ of cycles modulo boundaries.

Theorem

Homology groups are topological invariants: if $|K| \cong |L|$ (homeomorphic), then $H_n(K) \cong H_n(L)$ for all $n$ . Moreover, homology is a homotopy invariant for simplicial maps.

Example

For a point $K = \{v\}$ :

$H_0(K) = \mathbb{Z}$ (one connected component)
$H_n(K) = 0$ for $n > 0$ (no higher-dimensional holes)

Theorem

(Euler Characteristic) For a finite simplicial complex $K$ , the Euler characteristic is: $\chi(K) = \sum_{n=0}^\infty (-1)^n \text{rank}(H_n(K)) = \sum_{n=0}^\infty (-1)^n c_n$ where $c_n$ is the number of $n$ -simplices in $K$ .

Example

For the circle $S^1$ :

$H_0(S^1) = \mathbb{Z}$ (connected)
$H_1(S^1) = \mathbb{Z}$ (one 1-dimensional hole)
$H_n(S^1) = 0$ for $n \geq 2$

The generator of $H_1(S^1)$ represents a loop around the circle.

Definition

A simplicial map $f : K \to L$ is a map sending vertices of $K$ to vertices of $L$ such that if $\{v_0, \ldots, v_n\}$ spans a simplex in $K$ , then $\{f(v_0), \ldots, f(v_n)\}$ spans a simplex in $L$ (possibly with repetitions).

Theorem

A simplicial map $f : K \to L$ induces homomorphisms $f_* : H_n(K) \to H_n(L)$ for all $n$ , and this is functorial: $(g \circ f)_* = g_* \circ f_*$ and $(\text{id})_* = \text{id}$ .

Remark

Homology transforms the category of simplicial complexes into the category of graded abelian groups, preserving composition and identities. This functoriality makes homology a powerful computational tool.

The reduced homology $\tilde{H}_n$ (obtained by replacing $H_0$ with its augmentation) has better formal properties, particularly for wedge sums.