On dimensions supporting a rational projective plane

Lee Kennard, Zhixu Su

Research output: Contribution to journalArticle

Abstract

A rational projective plane ((Formula presented.)) is a simply connected, smooth, closed manifold (Formula presented.) such that (Formula presented.). 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 paper, 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 (Formula presented.). We then confirm the existence of a (Formula presented.) in two new dimensions and prove several non-existence results using factorization of the numerators of the 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.

Original languageEnglish (US)
Pages (from-to)1-21
Number of pages21
JournalJournal of Topology and Analysis
DOIs
StateAccepted/In press - Jan 1 2017
Externally publishedYes

Fingerprint

Projective plane
Quadratic residue
Bernoulli numbers
Numerator
Integrality
Projective Space
Surgery
Nonexistence
Existence Results
Open Problems
Resolve
Factorization
Simplify
Signature
Classify
Necessary Conditions
Closed
Sufficient Conditions
Theorem

Keywords

  • characteristic classes
  • Rational projective planes
  • rational surgery realization

ASJC Scopus subject areas

  • Analysis
  • Geometry and Topology

Cite this

On dimensions supporting a rational projective plane. / Kennard, Lee; Su, Zhixu.

In: Journal of Topology and Analysis, 01.01.2017, p. 1-21.

Research output: Contribution to journalArticle

@article{f01d93f32ef0432098866a685b12c45c,
title = "On dimensions supporting a rational projective plane",
abstract = "A rational projective plane ((Formula presented.)) is a simply connected, smooth, closed manifold (Formula presented.) such that (Formula presented.). 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 paper, 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 (Formula presented.). We then confirm the existence of a (Formula presented.) in two new dimensions and prove several non-existence results using factorization of the numerators of the 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.",
keywords = "characteristic classes, Rational projective planes, rational surgery realization",
author = "Lee Kennard and Zhixu Su",
year = "2017",
month = "1",
day = "1",
doi = "10.1142/S1793525319500237",
language = "English (US)",
pages = "1--21",
journal = "Journal of Topology and Analysis",
issn = "1793-5253",
publisher = "World Scientific Publishing Co. Pte Ltd",

}

TY - JOUR

T1 - On dimensions supporting a rational projective plane

AU - Kennard, Lee

AU - Su, Zhixu

PY - 2017/1/1

Y1 - 2017/1/1

N2 - A rational projective plane ((Formula presented.)) is a simply connected, smooth, closed manifold (Formula presented.) such that (Formula presented.). 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 paper, 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 (Formula presented.). We then confirm the existence of a (Formula presented.) in two new dimensions and prove several non-existence results using factorization of the numerators of the 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.

AB - A rational projective plane ((Formula presented.)) is a simply connected, smooth, closed manifold (Formula presented.) such that (Formula presented.). 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 paper, 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 (Formula presented.). We then confirm the existence of a (Formula presented.) in two new dimensions and prove several non-existence results using factorization of the numerators of the 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.

KW - characteristic classes

KW - Rational projective planes

KW - rational surgery realization

UR - http://www.scopus.com/inward/record.url?scp=85032803495&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85032803495&partnerID=8YFLogxK

U2 - 10.1142/S1793525319500237

DO - 10.1142/S1793525319500237

M3 - Article

AN - SCOPUS:85032803495

SP - 1

EP - 21

JO - Journal of Topology and Analysis

JF - Journal of Topology and Analysis

SN - 1793-5253

ER -