[Verse 1]
In manifolds where boundaries disappear
A secret symmetry becomes crystal clear
Each k-dimensional hole finds its twin
At n minus k, the mirror begins
The fundamental class holds the key
To unlock this strange geometry
[Chorus]
Poincaré duality, fold it in the middle
High reflects the low like solving a riddle
Cap product maps the cohomology
Self-mirroring topology
Know the first half, second half's determined
Perfect pairing, nothing left unlearned
[Verse 2]
On surfaces where n equals two
H-zero pairs with H-two
Connected components meet the top class
While H-one loops reflect in looking glass
Betti numbers cut in half reveal
The constraint that makes the structure real
[Chorus]
Poincaré duality, fold it in the middle
High reflects the low like solving a riddle
Cap product maps the cohomology
Self-mirroring topology
Know the first half, second half's determined
Perfect pairing, nothing left unlearned
[Bridge]
Mayer-Vietoris shows the way
Simple pieces, then we glue
Orientation gives the play
Isomorphism pushes through
From intersection theory's birth
To Lefschetz and Serre's worth
[Verse 3]
Closed and oriented, that's the rule
For this topological jewel
The fundamental class in H-n
Makes the pairing work again
What transfers out to algebraic ground
Where duality patterns can be found
[Chorus]
Poincaré duality, fold it in the middle
High reflects the low like solving a riddle
Cap product maps the cohomology
Self-mirroring topology
Know the first half, second half's determined
Perfect pairing, nothing left unlearned
[Outro]
Manifolds constrained by their reflection
Geometric self-perfection
In the middle dimensions where they meet
The duality is complete