This talk is primarily expository and aimed at graduate students and newcomers to homotopical algebra. We will introduce the language of bidifferential complexes--graded vector spaces equipped with both a differential and a codifferential--and use it to discuss deformation retractions and an algebraic form of the Hodge decomposition.

We will then examine in more detail the classical setting of differential forms on a Riemannian manifold, focusing on the homotopy transfer of the wedge product, which yields a complete invariant of its rational homotopy type.

In a subsequent talk, I plan to move beyond the Riemannian case and consider the Lorentzian setting, where physical information is also encoded in the transferred structure.

A compact 1-connected 4-dimensional CW complex has its homotopy type determined by its cohomology ring together with the Bockstein homomorphisms and the Pontryagin squares. This result has many applications, including to the cohomology rigidity problem. Following the work of J. H. C. Whitehead, we will see that morphisms between such cohomology rings can be lifted to maps between CW complexes.