Smooth manifolds with prescribed rational cohomology ring
Smooth manifolds with prescribed rational cohomology ring
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具有指定有理上同调环的光滑流形
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
Zhixu Su
中科院分区：
文献类型：
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作者：
J. Fowler;Zhixu Su
The Hirzebruch signature formula provides an obstruction to the following realization question: given a rational Poincaré duality algebra $${\mathcal {A}}$$A, does there exist a manifold M such that $$H^*(M;\mathbb {Q})={\mathcal {A}}$$H∗(M;Q)=A? When $${\mathcal {A}}$$A is the truncated polynomial algebra $$\mathbb {Q}[x]/\langle x^3\rangle $$Q[x]/⟨x3⟩, we prove there exists a realizing closed smooth manifold $$M^n$$Mn only if $$n=8(2^a+2^b)$$n=8(2a+2b). We also eliminate any existence between dimension 32 and 128. For $$n=32$$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 $${\mathcal {A}}=\mathbb {Q}[x]/\langle x^{m+1}\rangle , |x|=8$$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 $$E_8$$E8 manifold has the rational homotopy type of a smooth manifold. The “Appendix” presents a recursive algorithm for efficiently computing the coefficients of the $${\mathcal {L}}$$L-polynomials, which arise in the signature formula.
The Hirzebruch signature formula provides an obstruction to the following realization question: given a rational Poincaré duality algebra $${\mathcal {A}}$$A, does there exist a manifold M such that $$H^*(M;\mathbb {Q})={\mathcal {A}}$$H∗(M;Q)=A? When $${\mathcal {A}}$$A is the truncated polynomial algebra $$\mathbb {Q}[x]/\langle x^3\rangle $$Q[x]/⟨x3⟩, we prove there exists a realizing closed smooth manifold $$M^n$$Mn only if $$n=8(2^a+2^b)$$n=8(2a+2b). We also eliminate any existence between dimension 32 and 128. For $$n=32$$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 $${\mathcal {A}}=\mathbb {Q}[x]/\langle x^{m+1}\rangle , |x|=8$$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 $$E_8$$E8 manifold has the rational homotopy type of a smooth manifold. The “Appendix” presents a recursive algorithm for efficiently computing the coefficients of the $${\mathcal {L}}$$L-polynomials, which arise in the signature formula.