⇤Math diary
May 11, 2024: 30
Here’s a crazy fact I just learned: For a compact Kahler manifold, the cohomology ring determines the manifold up to homotopy. This is called the formality theorem, and is from the paper Real homotopy theory of Kähler manifolds, by Pierre Deligne, Phillip Griffiths, John Morgan & Dennis Sullivan. More precisely, any morphism between kahler manifolds that preserves the cohomology ring must also preserve homotopy type.
A similar theorem is whiteheads theorem, which states that any map between CW complexes that preserves homotopy groups must be a homotopy equivelence. The Homotopy groups are a much stronger invariant than the cohomology ring. If we assume a manifold is compact kahler, then the extreme power of hodge theory restricts things to the point that the cohomology is enough.