Homology: Why Boundary of Boundary = 0 ?

Homology and co-homology are the Top 10 Toughest Math in the world (1st & 8th topics in the list, of which 3rd, 4th, 9th and 10th topics received the Fields Medals). Like most Math concepts, which were discovered few decades or centuries ago, now become useful in scientific / industrial / computer applications never thought before by their discoverers.

Examples: Prime numbers, Chinese Remainder Theorem 韩信点兵 / Fermat’s Little Theorem / Gauss Modular Arithmetic / Elliptic Curve in Cryptography; Homology in Big Data Analytics for Epidemic Medicine, Pharmaceutical Drugs, Consumer Behavior Study, Stock Market, Economy, etc.

This Homology Fundamental Equation puzzles most people. WHY ?
\boxed {{\delta}^2 = 0 { ?}}

It is analogous to the Vector Algebra:
Let the boundary of {A, B} =
\delta (A,B) = \overrightarrow{AB } 

\overrightarrow{AB } + \overrightarrow {BA} =\overrightarrow{AB } - \overrightarrow {AB} = \vec 0

Source: http://mathoverflow.net/questions/640/what-is-cohomology-and-how-does-a-beginner-gain-intuition-about-it

Note: (co)-homology: (上) 同调

Euclid Geometry & Homology:

References: (Videos)

1. Isabell Darcy Lecture: cohomology

2. 同调代数 Homological Algebra

References: (Books)

4. “From Calculus to co-homology” (eBook download)

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