Smooth manifolds with prescribed rational cohomology ring

Abstract

The Hirzebruch signature formula provides an obstruction to the following realization question: given a rational Poincare duality algebra \({\mathcal {A}}\), does there exist a manifold M such that \(H^*(M;\mathbb {Q})={\mathcal {A}}\)? When \({\mathcal {A}}\) is the truncated polynomial algebra \(\mathbb {Q}[x]/\langle x^3\rangle \), we prove there exists a realizing closed smooth manifold \(M^n\) only if \(n=8(2^a+2^b)\). We also eliminate any existence between dimension 32 and 128. For \(n=32\), we show that such a realizing manifold does not admit a Spin structure, and therefore is not 2-connected. In the case that \({\mathcal {A}}=\mathbb {Q}[x]/\langle x^{m+1}\rangle , |x|=8\), we apply the rational surgery realization theorem to conclude that a rational octonionic projective space exists for m odd. Similar technique is applied to study if the Milnor \(E_8\) manifold has the rational homotopy type of a smooth manifold. The “Appendix” presents a recursive algorithm for efficiently computing the coefficients of the \({\mathcal {L}}\)-polynomials, which arise in the signature formula.