[Verse 1]
Picture modules lined in row, arrows flowing left to right
Each differential maps you down, composition must be tight
When you chase two arrows through, something magical appears
Zero always waits for you, that's the rule that engineers
[Chorus]
Chain complex, chain complex
Boundary of boundary is zero
D-squared equals nothing left
That's the rule of every hero
Kernel over image, that's your homology
Measuring the holes you see
Chain complex, chain complex
[Verse 2]
Start with C-sub-n up high, differential takes you down
To the next module below, in this algebraic town
Image sits inside the kernel, when the sequence behaves well
But the quotient tells the story, of the holes that you can't quell
[Chorus]
Chain complex, chain complex
Boundary of boundary is zero
D-squared equals nothing left
That's the rule of every hero
Kernel over image, that's your homology
Measuring the holes you see
Chain complex, chain complex
[Bridge]
When exactness fails to hold
Homology takes control
Every element that's trapped
In the kernel's careful grasp
But escapes the image net
Shows the gaps that we detect
[Verse 3]
Like a telescope extending, modules stretch toward infinity
Each differential dancing, with perfect symmetry
If your homology vanishes, then exact is what you've got
But when H-sub-n survives, then a hole marks the spot
[Chorus]
Chain complex, chain complex
Boundary of boundary is zero
D-squared equals nothing left
That's the rule of every hero
Kernel over image, that's your homology
Measuring the holes you see
Chain complex, chain complex
[Outro]
From topology to algebra
The same patterns always flow
Chain complexes hold the secrets
Of the holes we need to know