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Smooth manifolds with prescribed rational cohomology ring
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Journal Article
Smooth manifolds with prescribed rational cohomology ring
2016
Overview
The Hirzebruch signature formula provides an obstruction to the following realization question: given a rational Poincaré duality algebra A , does there exist a manifold M such that H∗(M;Q)=A ? When A is the truncated polynomial algebra Q[x]/⟨x3⟩ , we prove there exists a realizing closed smooth manifold Mn only if n=8(2a+2b) . 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 A=Q[x]/⟨xm+1⟩,|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 E8 manifold has the rational homotopy type of a smooth manifold. The “Appendix” presents a recursive algorithm for efficiently computing the coefficients of the L -polynomials, which arise in the signature formula.
Publisher
Springer Nature B.V
Subject
