On dimensions supporting a rational projective plane
| Autor: | Zhixu Su, Lee Kennard |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
Pure mathematics
Closed manifold Open problem 010102 general mathematics Geometric Topology (math.GT) 01 natural sciences Manifold Characteristic class Mathematics - Geometric Topology Factorization 0103 physical sciences FOS: Mathematics 57R20 (Primary) 57R65 57R67 57R15 (Secondary) 010307 mathematical physics Geometry and Topology Projective plane 0101 mathematics Signature (topology) Bernoulli number Analysis Mathematics |
| Zdroj: | Journal of Topology and Analysis. 11:535-555 |
| ISSN: | 1793-7167 1793-5253 |
| Popis: | A rational projective plane ($\mathbb{QP}^2$) is a simply connected, smooth, closed manifold $M$ such that $H^*(M;\mathbb{Q}) \cong \mathbb{Q}[\alpha]/\langle \alpha^3 \rangle$. An open problem is to classify the dimensions at which such a manifold exists. The Barge-Sullivan rational surgery realization theorem provides necessary and sufficient conditions that include the Hattori-Stong integrality conditions on the Pontryagin numbers. In this article, we simplify these conditions and combine them with the signature equation to give a single quadratic residue equation that determines whether a given dimension supports a $\mathbb{QP}^2$. We then confirm existence of a $\mathbb{QP}^2$ in two new dimensions and prove several non-existence results using factorizations of numerators of divided Bernoulli numbers. We also resolve the existence question in the Spin case, and we discuss existence results for the more general class of rational projective spaces. Comment: to appear in J. Topol. Anal |
| Databáze: | OpenAIRE |
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