TY - JOUR
T1 - On dimensions supporting a rational projective plane
AU - Kennard, Lee
AU - Su, Zhixu
N1 - Funding Information:
We also want to thank Yang Su and Jim Davis for communication about this problem and Sam Wagstaff for discussions on Bernoulli numbers that made possible the proof of Theorem B. Finally we are grateful to the referee for carefully reading and making suggestions to improve the paper. The first author was supported by NSF Grant DMS 1622541.
Publisher Copyright:
© 2019 World Scientific Publishing Company.
PY - 2019/9/1
Y1 - 2019/9/1
N2 - A rational projective plane (2) is a simply connected, smooth, closed manifold M such that H - (M;)[α](α3). 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 2. We then confirm the existence of a 2 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 (2) is a simply connected, smooth, closed manifold M such that H - (M;)[α](α3). 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 2. We then confirm the existence of a 2 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 - Rational projective planes
KW - characteristic classes
KW - rational surgery realization
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U2 - 10.1142/S1793525319500237
DO - 10.1142/S1793525319500237
M3 - Article
AN - SCOPUS:85032803495
SN - 1793-5253
VL - 11
SP - 535
EP - 555
JO - Journal of Topology and Analysis
JF - Journal of Topology and Analysis
IS - 3
ER -