Definition of Homology groups
Definitions 1 2
- Let’s denote by $n \in \mathbb{N}_{0}$. A chain of Abelian groups $C_{n}$ and homomorphisms $\partial_{n} : C_{n} \longrightarrow C_{n-1}$ $$ \cdots \longrightarrow C_{n+1} \overset{\partial_{n+1}}{\longrightarrow} C_{n} \overset{\partial_{n}}{\longrightarrow} C_{n-1} \longrightarrow \cdots \longrightarrow C_{1} \overset{\partial_{1}}{\longrightarrow} C_{0} \overset{\partial_{0}}{\longrightarrow} 0 $$ that satisfies $$ \partial_{n} \circ \partial_{n+1} = 0 $$ for all $n$ is called a Chain Complex.
- The quotient group $H_{n} := \ker \partial_{n} / \operatorname{Im} \partial_{n+1}$ is called the $n$-th Homology group of $\mathcal{C}$.
- The homomorphism $\partial_{n} : C_{n} \longrightarrow C_{n-1}$ is called a Boundary or Differential operator.
- An element of $Z_{n} := \ker \partial_{n}$ is called a $n$-Cycle, and an element of $B_{n} := \operatorname{Im} \partial_{n+1}$ is called a $n$-Boundary.
- A group $0$ is a magma as defined in $\left\{ 0 \right\}$. Essentially, it is an algebraic structure that is empty.
- The homomorphism $\partial^{2} = 0$ is a zeromorphism.
- $\operatorname{Im}$ is an image.
- $\ker$ is a kernel.
Explanation
It’s normal to feel puzzled. The definitions introduced are very strictly algebraic statements, so it’s recommended to quickly move on to simplicial homology for an intuitive understanding. (Although that isn’t particularly easy either) It can be difficult to grasp the algebraic terms for boundaries and differentials without looking at them geometrically.
Generalizability
Actually, it’s known that the index set in a Chain Complex can not only be expanded to negative numbers beyond $\mathbb{N}_{0} = \left\{ 0, 1, 2, \cdots \right\}$ but also to real numbers. However, after $0$ as a reference point, moving towards negative entails a significant fade in topological or geometric meaning.
Existence of Homology groups
Let’s assume $U, V, W$ is a vector space, and $T_{1} : U \to V$, $T_{2} : V \to W$ are linear transformations. Then, the following holds:
$$ T_{2}T_{1} = 0 \iff \operatorname{Im} (T_{1}) \subset \ker (T_{2}) $$
The condition of a chain complex, $\partial_{n} \circ \partial_{n+1} = 0$, is commonly abbreviated as $\partial^{2} = 0$. No matter what $\operatorname{Im} \partial_{n+1}$ is, after taking $\partial_{n}$, it means that $0$ is generously defined to completely encompass $\operatorname{Im} \partial_{n+1}$, implying $\operatorname{Im} \partial_{n+1} \subset \ker \partial_{n}$.
$\partial^{2} = 0$ leading to $\ker \partial_{n} / \operatorname{Im} \partial_{n+1}$ might seem out of the blue, but historically, there was substantial research on algebraic structures that partition kernels into images as in $\ker f / \operatorname{Im} g$, and $\partial^{2} = 0$ was included in the definition more for its elegant expression than its intuitive meaning.
Boundary and Differential
The term for a $n$-cycle $Z_{n}$, “Zyklus,” comes from German.
$\partial_{n}$, when viewed as the boundary of a simplex, naturally fits its naming, and the term differential, as defined in $$ \lim_{h \to 0} {{ f(x + h) - f(x) } \over { h }} $$ like $$ \partial \left[ v_{0} , v_{1} \right] = \left[ v_{1} \right] - \left[ v_{0} \right] $$, is understandably derived from the mathematical form of Difference. However, one cannot comprehend this from the bare definition of homology groups alone. These explanations only become plausible after the specific definition of $\partial_{n}$ is given, and the universal applicability is understood. For now, it’s best to overlook the exact terminology and move forward.
Infamy
Homology is surprisingly well-known to the general public. While they might not remember the term “homology,” it has gained a cult-like popularity through stories on Twitter, becoming known as something even Seoul National University students can’t easily explain.
