Simplicial homology groups are a powerful tool for understanding the topological features of spaces. They use chain complexes and boundary operators to capture information about connected components, holes, and voids in simplicial complexes.

By computing homology groups and their ranks (Betti numbers), we can quantify and compare the topological structures of different spaces. This approach provides a systematic way to analyze and classify complex geometric objects.

Chain Complexes and Boundary Operators

Definition and Notation

• A chain complex is a sequence of abelian groups and homomorphisms between them, denoted as $(C_n, \partial_n)$, where $C_n$ is the n-th chain group and $\partial_n$ is the n-th boundary operator
• The boundary operator $\partial_n$ maps elements from $C_n$ to $C_{n-1}$, satisfying the condition $\partial_{n-1} \circ \partial_n = 0$ for all $n$
• The n-th chain group $C_n$ is the free abelian group generated by the n-simplices of a simplicial complex
• Example: In a simplicial complex with vertices $\{a, b, c\}$ and edges $\{[a, b], [b, c]\}$, $C_0 = \langle a, b, c \rangle$ and $C_1 = \langle [a, b], [b, c] \rangle$

Boundary Operator Definition and Properties

• The boundary operator $\partial_n$ is defined on the generators of $C_n$ (the n-simplices) and extended linearly to all elements of $C_n$
• For an n-simplex $[v_0, v_1, \ldots, v_n]$, the boundary operator is defined as $\partial_n([v_0, v_1, \ldots, v_n]) = \sum_{i=0}^n (-1)^i [v_0, \ldots, v_{i-1}, v_{i+1}, \ldots, v_n]$
• Example: For a 2-simplex $[a, b, c]$, $\partial_2([a, b, c]) = [b, c] - [a, c] + [a, b]$
• The boundary of a boundary is always zero: $\partial_{n-1} \circ \partial_n = 0$ for all $n$
• Example: $\partial_1(\partial_2([a, b, c])) = \partial_1([b, c] - [a, c] + [a, b]) = (c - b) - (c - a) + (b - a) = 0$

Simplicial Homology Groups

Definition and Computation

• The n-th homology group $H_n$ is defined as the quotient group $\text{Ker}(\partial_n) / \text{Im}(\partial_{n+1})$, where $\text{Ker}(\partial_n)$ is the kernel of the n-th boundary operator and $\text{Im}(\partial_{n+1})$ is the image of the $(n+1)$-th boundary operator
• Elements of $\text{Ker}(\partial_n)$ are called n-cycles, and elements of $\text{Im}(\partial_{n+1})$ are called n-boundaries
• To compute the n-th homology group, first find a basis for the n-cycles (elements in $\text{Ker}(\partial_n)$) and a basis for the n-boundaries (elements in $\text{Im}(\partial_{n+1})$)
• Express the n-boundaries in terms of the basis for the n-cycles to determine the quotient group $H_n = \text{Ker}(\partial_n) / \text{Im}(\partial_{n+1})$

Betti Numbers and Homological Features

• The rank of the n-th homology group $H_n$ is called the n-th Betti number, denoted as $\beta_n$, which counts the number of "n-dimensional holes" in the simplicial complex
• Example: In a simplicial complex representing a circle, $\beta_0 = 1$ (one connected component) and $\beta_1 = 1$ (one 1-dimensional hole)
• Betti numbers provide a way to quantify the topological features of a simplicial complex
• Example: In a simplicial complex representing a torus, $\beta_0 = 1$ (one connected component), $\beta_1 = 2$ (two 1-dimensional holes), and $\beta_2 = 1$ (one 2-dimensional void)

Geometric Interpretation of Homology

Low-Dimensional Homology Groups

• The 0-th homology group $H_0$ represents the connected components of the simplicial complex
• The 1-st homology group $H_1$ represents the "1-dimensional holes" or non-contractible loops in the simplicial complex
• The 2-nd homology group $H_2$ represents the "2-dimensional voids" or non-contractible cavities in the simplicial complex
• Example: In a simplicial complex representing a sphere, $H_0 \cong \mathbb{Z}$ (one connected component), $H_1 \cong 0$ (no 1-dimensional holes), and $H_2 \cong \mathbb{Z}$ (one 2-dimensional void)

Higher-Dimensional Homology Groups

• Higher-dimensional homology groups $H_n$ ($n > 2$) represent higher-dimensional "holes" or non-contractible subspaces in the simplicial complex
• Example: In a simplicial complex representing a 3-torus, $H_3 \cong \mathbb{Z}$ (one 3-dimensional void)
• Higher-dimensional homology groups capture more intricate topological features of the simplicial complex
• The geometric interpretation of homology groups provides insight into the shape and connectivity of the underlying space

Functoriality of Simplicial Homology

Simplicial Maps and Induced Homomorphisms

• A simplicial map $f$ between two simplicial complexes $K$ and $L$ is a function that maps vertices of $K$ to vertices of $L$, preserving the simplicial structure (i.e., if $[v_0, \ldots, v_n]$ is a simplex in $K$, then $[f(v_0), \ldots, f(v_n)]$ is a simplex in $L$)
• A simplicial map $f$ induces a chain map $f_\#$ between the chain complexes of $K$ and $L$, which is a sequence of homomorphisms $f_n: C_n(K) \to C_n(L)$ that commute with the boundary operators (i.e., $\partial_n \circ f_n = f_{n-1} \circ \partial_n$)
• The induced chain map $f_\#$ further induces a homomorphism $f_: H_n(K) \to H_n(L)$ between the homology groups of $K$ and $L$ for each dimension $n$

Functoriality Property and Its Consequences

• The induced homomorphism f_*$ satisfies the functoriality property: if $g: L \to M$ is another simplicial map, then $(g \circ f)_* = g_* \circ f_*
• Functoriality allows for the study of simplicial complexes and their homology groups through simplicial maps, enabling the comparison of topological properties between different spaces
• Example: If $f: K \to L$ and $g: L \to M$ are simplicial maps inducing isomorphisms f_*: H_n(K) \to H_n(L)$ and $g_*: H_n(L) \to H_n(M) for all $n$, then $K$, $L$, and $M$ have the same homology groups and are thus homologically equivalent
• Functoriality is a powerful tool for studying the relationships between simplicial complexes and their homological invariants, allowing for the classification of spaces up to homological equivalence