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Almost complex manifolds with total Betti number three

Department of Mathematics, Stony Brook University, 100 Nicolls Road, 11794 Stony Brook, USA

E-mail Address: jiahao.hu@stonybrook.edu

https://doi.org/10.1142/S1793525323500164Cited by:0 (Source: Crossref)

Abstract

In this paper, we show the minimal total Betti number of a closed almost complex manifold of dimension $2 n \geq 8$ is four, thus confirming a conjecture of Sullivan except for dimension $6$ . Along the way, we prove the only simply connected closed complex manifold having total Betti number three is the complex projective plane.

Keywords:

AMSC: 32Q60, 53C15

References

1. M. Albanese and A. Milivojević , On the minimal sum of Betti numbers of an almost complex manifold, Differ. Geom. Appl. 62 (2019) 101–108. Crossref, Google Scholar
2. M. H. Freedman , The topology of four-dimensional manifolds, J. Differ. Geom. 17 (1982) 357–453. Crossref, Google Scholar
3. F. Hirzebruch, Topological Methods in Algebraic Geometry, Classics in Mathematics (Springer-Verlag, 1995), Translated from the German and Appendix One by R. L. E. Schwarzenberger, With a preface to the third English edition by the author and Schwarzenberger, Appendix Two by A. Borel, Reprint of the 1978 edition. Google Scholar
4. F. Hirzebruch, T. Berger and R. Jung , Manifolds and Modular Forms, Aspects of Mathematics, Vol. 20 (Friedr. Vieweg & Sohn, 1992), With appendices by Nils-Peter Skoruppa and by Paul Baum. Crossref, Google Scholar
5. J. Milnor , On the cobordism ring $Ω^{*}$ and a complex analogue. I, Amer. J. Math. 82 (1960) 505–521. Crossref, Google Scholar
6. G. Prasad and S.-K. Yeung , Fake projective planes, Invent. Math. 168 (2007) 321–370. Crossref, Google Scholar
7. Z. Su, Almost complex manifold with betti number $b_{i} = 0$ except $i = 0, n / 2, n$ , preprint (2020), arXiv:2001.02313. Google Scholar
8. S. T. Yau , Calabi’s conjecture and some new results in algebraic geometry, Proc. Natl. Acad. Sci. USA 74 (1977) 1798–1799. Crossref, Google Scholar