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Smooth manifolds with prescribed rational cohomology ring Su, Zhixu


Given a rational Poincare duality algebra A, is there a manifold M whose rational cohomology ring realizes A? The Hirzebruch signature formula provides an obstruction to the existence of such manifold. In the case of A=Q[x]/<x^3>, a realizing smooth manifold M^n (called a rational projective plane) could only exist in dimensions n=8(2^a+2^b). Sullivan's rational surgery realization theorem provides the necessary and sufficient condition to the existence; the problem boiled down to finding possible Pontryagin numbers satisfying a set of integrality conditions, which can be reduced to a Diophantine equation in our case. Similar techniques can be applied to study the realization of rational Octonionic projective spaces. This is joint work with Jim Fowler.

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