Smooth manifolds with prescribed rational cohomology ring

TL;DR

This paper investigates the conditions under which certain rational cohomology rings can be realized by smooth manifolds, revealing number theoretic constraints and providing algorithms for signature computations.

Contribution

It establishes new dimension restrictions for the realization of rational cohomology algebras and introduces an efficient recursive algorithm for L-polynomial coefficients.

Findings

Certain rational cohomology rings cannot be realized in specific dimensions.

The minimal dimension for realization of some algebras is identified as 128.

An algorithm for computing L-polynomial coefficients is presented.

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.