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On the characterization of rational homotopy types and Chern classes of closed almost complex manifolds

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Milivojević,  Aleksandar
Max Planck Institute for Mathematics, Max Planck Society;

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Milivojević, A. (2022). On the characterization of rational homotopy types and Chern classes of closed almost complex manifolds. Complex Manifolds, 9(1), 138-169. doi:10.1515/coma-2021-0133.


Cite as: https://hdl.handle.net/21.11116/0000-000A-75C0-9
Abstract
We give an exposition of Sullivan’s theorem on realizing rational homotopy types by closed smooth manifolds, including a discussion of the necessary rational homotopy and surgery theory, adapted to the realization problem for almost complex manifolds: namely, we give a characterization of the possible simply connected rational homotopy types, along with a choice of rational Chern classes and fundamental class, realized by simply connected closed almost complex manifolds in real dimensions six and greater. As a consequence, beyond demonstrating that rational homotopy types of closed almost complex manifolds are plenty, we observe that the realizability of a simply connected rational homotopy type by a simply connected closed almost complex manifold depends only on its cohomology ring. We conclude with some computations and examples.