Abstract
We study the homology groups of certain 2-connected 7-manifolds admitting quasi-regular Sasaki–Einstein metrics, among them, we found 52 new examples of Sasaki–Einstein rational homology 7-spheres, extending the list given by Boyer et al. (Ann Inst Fourier 52(5):1569–1584, 2002). As a consequence, we exhibit new families of positive Sasakian homotopy 9-spheres given as cyclic branched covers, determine their diffeomorphism types and find out which elements do not admit extremal Sasaki metrics. We also improve previous results given by Boyer (Note Mat 28:63–105, 2008) showing new examples of Sasaki–Einstein 2-connected 7-manifolds homeomorphic to connected sums of \(S^3\times S^4\). Actually, we show that manifolds of the form \(\#k\left( S^{3} \times S^{4}\right) \) admit Sasaki–Einstein metrics for 22 different values of k. All these links arise as Thom–Sebastiani sums of chain type singularities and cycle type singularities where Orlik’s conjecture holds due to a recent result by Hertling and Mase (J Algebra Number Theory 16(4):955–1024, 2022).
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Acknowledgements
The authors would like to thank the anonymous referees for a careful reading and suggestions. The first author would like to thank Charles Boyer for encouragement in writing this article.
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Part of this article was prepared with the financial support from Pontificia Universidad Católica del Perú through Project VRI-DFI PI0655.
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J.C. wrote the manuscript. J.L. made contributions to the interpretation of data and wrote the programs in MATLAB used to determine the torsion of the homology groups.
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Appendix
Appendix
- (a):
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The Johnson–Kollár list of hypersurfaces in weighted projective 4-space admitting Kähler–Einstein orbifold metrics is available at https://web.math.princeton.edu/~jmjohnso/delpezzo/KEandTiger.txt. This list includes the weight vectors followed by data on whether or not it is known if the orbifold is Kähler–Einstein.
- (b):
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Table I, cited in Theorem 3.6 is given here https://github.com/Jcuadrosvalle/TABLES
- (c):
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Table II and Table III, where we exhibit the third homology group of links of hypersurface singularities that are neither rational homology 7-spheres nor homeomorphic to connected sums of \(S^3\times S^4,\) are available at https://github.com/Jcuadrosvalle/TABLES
- (d):
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Four codes in MATLAB are available at https://github.com/Jcuadrosvalle/Codes-in-Matlab. Codes1, 2 and 3 determine whether or not the singularities are of chain type, cycle type or an iterated Thom–Sebastiani sum of chain type and cycle type singularities, Code4 computes the third homology groups of the corresponding links.
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Cuadros Valle, J., Lope Vicente, J. Sasaki–Einstein 7-manifolds and Orlik’s conjecture. Ann Glob Anal Geom 65, 3 (2024). https://doi.org/10.1007/s10455-023-09930-z
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DOI: https://doi.org/10.1007/s10455-023-09930-z