Poincaré duality

Target entity: Poincaré duality
Context triple: [Henri Poincaré, notableWork, Poincaré duality]

• A. Atiyah–Singer index theorem
  The Atiyah–Singer index theorem is a fundamental result in mathematics that links the analytical properties of elliptic differential operators to topological invariants of manifolds, unifying analysis, topology, and geometry.
• B. Whitney approximation theorem
  The Whitney approximation theorem is a fundamental result in differential topology stating that any continuous function between smooth manifolds can be uniformly approximated by smooth functions.
• C. Hodge theory
  Hodge theory is a branch of mathematics that studies the relationship between differential forms, cohomology, and complex geometry, particularly on complex manifolds.
• D. Lefschetz operator
  The Lefschetz operator is a linear operator in Kähler geometry that acts on differential forms by wedging with the Kähler form, playing a central role in the Hard Lefschetz theorem and Hodge theory.
• E. Gauss–Bonnet theorem (early form)
  The Gauss–Bonnet theorem (early form) is an early version of the fundamental result in differential geometry that links the total curvature of a surface to its topological characteristics, originally developed by Carl Friedrich Gauss.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Poincaré duality
Target entity description: Poincaré duality is a fundamental theorem in algebraic topology that relates the homology and cohomology groups of an oriented closed manifold in complementary dimensions.

• A. Atiyah–Singer index theorem
  The Atiyah–Singer index theorem is a fundamental result in mathematics that links the analytical properties of elliptic differential operators to topological invariants of manifolds, unifying analysis, topology, and geometry.
• B. Whitney approximation theorem
  The Whitney approximation theorem is a fundamental result in differential topology stating that any continuous function between smooth manifolds can be uniformly approximated by smooth functions.
• C. Hodge theory
  Hodge theory is a branch of mathematics that studies the relationship between differential forms, cohomology, and complex geometry, particularly on complex manifolds.
• D. Lefschetz operator
  The Lefschetz operator is a linear operator in Kähler geometry that acts on differential forms by wedging with the Kähler form, playing a central role in the Hard Lefschetz theorem and Hodge theory.
• E. Gauss–Bonnet theorem (early form)
  The Gauss–Bonnet theorem (early form) is an early version of the fundamental result in differential geometry that links the total curvature of a surface to its topological characteristics, originally developed by Carl Friedrich Gauss.
• F. None of above. chosen