| Abstrakt: |
The Hirzebruch signature formula provides an obstruction to the following realization question: given a rational Poincaré 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. [ABSTRACT FROM AUTHOR] |
Full text from SpringerLink