Given a smooth manifold $M$, we can study its rational homotopy type by looking at differential forms. I am wondering if there is a way to represent each $rational$ homotopy class by geometric objects, like submanifolds in $M$? The famous work of Sacks and Unlenbeck says that one can represent the usual $second$ homotopy group by harmonic map. However, it seems it is not clear how to do with rational homotopy groups. Such a theory might be useful to understand the Bott conjecture stating that a compact nonnegatively curved manifold has only finitely many nonvanishing rational homotopy groups.


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