Smooth manifolds with prescribed rational cohomology ring

Abstract

The Hirzebruch signature formula provides an obstruction to the following realization question: given a rational Poincar\'e duality algebra $\mathcal{A}$, does there exist a smooth manifold $M$ such that $H^*(M;\mathbb{Q})=\mathcal{A}$? This problem is especially interesting for rational truncated polynomial algebras whose corresponding integral algebra is not realizable. For example, there are number theoretic constraints on the dimension $n$ in which there exists a closed smooth manifold $M^n$ with $H^*(M^n;\mathbb{Q})= \mathbb{Q}[x]/\langle x^3\rangle$. We limit the possible existence dimension to $n=8(2^a+2^b)$. For $n = 32$, such manifolds are not two-connected. We show that the next smallest possible existence dimension is $n=128$. As there exists no integral $\mathbb{O}P^m$ for $m>2$, the realization of the truncated polynomial algebra $\mathbb{Q}[x]/\langle x^{m+1}\rangle, |x|=8$ is studied. Similar considerations provide examples of topological manifolds which do not have the rational homotopy type of a smooth closed manifold. The appendix presents a recursive algorithm for efficiently computing the coefficients of the L-polynomials which arise in the signature formula.