Abstract
We give an exposition of Sullivan’s theorem on realizing rational homotopy types by closed smooth manifolds, including a discussion of the necessary rational homotopy and surgery theory, adapted to the realization problem for almost complex manifolds: namely, we give a characterization of the possible simply connected rational homotopy types, along with a choice of rational Chern classes and fundamental class, realized by simply connected closed almost complex manifolds in real dimensions six and greater. As a consequence, beyond demonstrating that rational homotopy types of closed almost complex manifolds are plenty, we observe that the realizability of a simply connected rational homotopy type by a simply connected closed almost complex manifold depends only on its cohomology ring. We conclude with some computations and examples.
References
[1] Albanese, M. and Milivojević, A., 2019. On the minimal sum of Betti numbers of an almost complex manifold. Differential Geometry and its Applications, 62, pp.101-108.10.1016/j.difgeo.2018.10.002Search in Google Scholar
[2] Barge, J., 1976. Structures différentiables sur les types d’homotopie rationnelle simplement connexes. Annales scientifiques de l’École Normale Supérieure (Vol. 9, No. 4, pp. 469-501).10.24033/asens.1315Search in Google Scholar
[3] Browder, W., 1962, August. Homotopy type of differentiable manifolds. In Proceedings of the Aarhus Symposium (pp. 42-46).Search in Google Scholar
[4] Browder, W., 1972. Surgery on Simply Connected Manifolds. Ergeb. der Math, (2).10.1007/978-3-642-50020-6Search in Google Scholar
[5] Chern, S.S., 1946. Characteristic classes of Hermitian manifolds. Annals of Mathematics, pp.85-121.10.2307/1969037Search in Google Scholar
[6] Deligne, P., Griffiths, P., Morgan, J. and Sullivan, D., 1975. Real homotopy theory of Kähler manifolds. Inventiones mathematicae, 29(3), pp.245-274.10.1007/BF01389853Search in Google Scholar
[7] Müller, S. and Geiges, H., 2000. Almost complex structures on 8-manifolds. L’Enseignement Mathématique, 46(1/2), pp.95-108.Search in Google Scholar
[8] Griffiths, P. and Morgan, J.W., 1981. Rational homotopy theory and differential forms (Vol. 16). Boston: Birkhäuser.Search in Google Scholar
[9] Hilbert, D., 1902. Mathematical problems. Bulletin of the American Mathematical Society, 8(10), pp.437-479.10.1090/S0002-9904-1902-00923-3Search in Google Scholar
[10] Hirzebruch, F., 1960. Komplexe mannigfaltigkeiten. In International Congress of Mathematicians (pp. 119-136). Cambridge University Press.Search in Google Scholar
[11] Hirzebruch, F., 1987. Gesammelte Abhandlungen-Collected Papers I: 1951-1962 (Vol. 2). Springer.10.1007/978-3-642-61711-9Search in Google Scholar
[12] Hu, J., 2021. Almost complex manifolds with total Betti number three. arXiv preprint arXiv:2108.06067.Search in Google Scholar
[13] Kadeishvili, T., 2008. Cohomology C∞-algebra and Rational Homotopy Type. arXiv preprint arXiv:0811.1655.10.4064/bc85-0-16Search in Google Scholar
[14] Kahn, P.J., 1969. Obstructions to extending almost X-structures. Illinois Journal of Mathematics, 13(2), pp.336-357.10.1215/ijm/1334250797Search in Google Scholar
[15] Kervaire, M.A. and Milnor, J.W., 1963. Groups of homotopy spheres: I. Annals of Mathematics, pp.504-537.10.2307/1970128Search in Google Scholar
[16] Markl, M., 1990. The rigidity of Poincaré duality algebras and classification of homotopy types of manifolds. In Théorie de L’homotopie: Colloque CNRS-NSF-SMF Au CIRM Du 11 Au 15 Juillet 1988 (Vol. 191, p. 221). Société Mathématique de France.Search in Google Scholar
[17] Massey, W.S., 1962. Non-existence of almost-complex structures on quaternionic projective spaces. Pacific Journal of Mathematics, 12(4), pp.1379-1384.10.2140/pjm.1962.12.1379Search in Google Scholar
[18] Lectures in Topology, The University of Michigan, Conference of 1940.Search in Google Scholar
[19] Macaulay2, a software system for research in algebraic geometry. available at https://www.unimelb-macaulay2.cloud.edu.au/home, package “SymmetricPolynomials” by Alexandra Seceleanu, available at https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packagesSearch in Google Scholar
[20] Milivojević, A., 2021. On the characterization of rational homotopy types and Chern classes of closed almost complex manifolds (Doctoral dissertation, State University of New York at Stony Brook).10.1515/coma-2021-0133Search in Google Scholar
[21] Milnor, J.W. and Husemoller, D., 1973. Symmetric bilinear forms (Vol. 73). New York: Springer-Verlag.10.1007/978-3-642-88330-9Search in Google Scholar
[22] Milnor, J. and Stasheff, J.D., 2016. Characteristic Classes.(AM-76), Volume 76. Princeton university press.Search in Google Scholar
[23] Miller, T.J., 1979. On the formality of K − 1 connected compact manifolds of dimension less than or equal to 4K − 2. Illinois Journal of Mathematics, 23(2), pp.253-258.10.1215/ijm/1256048237Search in Google Scholar
[24] Novikov, S.P., 1964. Homotopy equivalent smooth manifolds, I. Appendices 1, 2. Izv. Akad. Nauk SSSR. Ser. Mat, 28(2).Search in Google Scholar
[25] Oprea, J.F., review of “Real and rational homotopy theory” by Brown, Edgar H. and Szczarba, R. H., zbMATH Zbl 0865.55009Search in Google Scholar
[26] Panov, T., 2011. Complex bordism. Bulletin of the Manifold Atlas, 30, p.39.Search in Google Scholar
[27] Pontryagin, L., 1938. A classification of continuous transformations of a complex into a sphere. In Dokl. Akad. Nauk SSSR (Vol. 19, pp. 361-363).Search in Google Scholar
[28] Pontryagin, L.S., 1942. Characteristic cycles of smooth manifolds. In Dokl. Akad. Nauk SSSR (Vol. 35, No. 2, pp. 35-39).Search in Google Scholar
[29] Quillen, D.G., 1968. Some remarks on etale homotopy theory and a conjecture of Adams. Topology, 7(2), pp.111-116.10.1016/0040-9383(68)90017-7Search in Google Scholar
[30] Quillen, D., 1969. Rational homotopy theory. Annals of Mathematics, pp.205-295.10.2307/1970725Search in Google Scholar
[31] Scorpan, A., 2005. The wild world of 4–manifolds. American Mathematical Soc..Search in Google Scholar
[32] Smale, S., 1962. On the structure of manifolds. American Journal of Mathematics, 84(3), pp.387-399.10.2307/2372978Search in Google Scholar
[33] Spivak, M., 1964. On spaces satisfying Poincare duality (Doctoral dissertation, Princeton University).Search in Google Scholar
[34] Stiefel, E., 1935. Richtungsfelder und Fernparallelismus in n–dimensionalen Mannigfaltigkeiten (Doctoral dissertation, ETH Zürich).10.1007/BF01199559Search in Google Scholar
[35] Stong, R.E., 1965. Relations among characteristic numbers—I. Topology, 4(3), pp.267-281.10.1016/0040-9383(65)90011-XSearch in Google Scholar
[36] Stong, R.E., 1966. Relations among characteristic numbers—II. Topology, 5(2), pp.133-148.10.1016/0040-9383(66)90014-0Search in Google Scholar
[37] Stong, R.E., 2015. Notes on cobordism theory. Princeton University Press.Search in Google Scholar
[38] Su, Z., 2009. Rational homotopy type of manifolds. PhD thesis, Indiana University.Search in Google Scholar
[39] Su, Z., 2014. Rational analogs of projective planes. Algebraic & Geometric Topology, 14(1), pp.421-438.10.2140/agt.2014.14.421Search in Google Scholar
[40] Su, Z., 2021. Almost complex manifold with Betti number bi = 0 except i = 0, n/2, n, preprint.Search in Google Scholar
[41] Sullivan, D., 1965. Triangulating homotopy equivalences (Doctoral dissertation, Princeton University).Search in Google Scholar
[42] Sullivan, D., 1970. Geometric Topology. Part I. Localization, periodicity and Galois symmetry. Notes, MIT.Search in Google Scholar
[43] Sullivan, D., 1974. Genetics of homotopy theory and the Adams conjecture. Annals of Mathematics, pp.1-79.10.2307/1970841Search in Google Scholar
[44] Sullivan, D., 1977. Infinitesimal computations in topology. Publications Mathématiques de l’IHÉS, 47, pp.269-331.10.1007/BF02684341Search in Google Scholar
[45] Taylor, L. and Williams, B., 1979. Local surgery: foundations and applications. In Algebraic Topology Aarhus 1978 (pp. 673-695). Springer, Berlin, Heidelberg.10.1007/BFb0088110Search in Google Scholar
[46] Thom, R., 1954. Quelques propriétés globales des variétés différentiables. Commentarii Mathematici Helvetici, 28(1), pp.17-86.10.1007/BF02566923Search in Google Scholar
[47] Wallace, A.H., 1960. Modifications and cobounding manifolds. Canadian Journal of Mathematics, 12, pp.503-528.10.4153/CJM-1960-045-7Search in Google Scholar
[48] Whitney, H., 1935. Sphere-spaces. Proceedings of the National Academy of Sciences of the United States of America, 21(7), p.464.10.1073/pnas.21.7.464Search in Google Scholar PubMed PubMed Central
[49] Whitney, H., 1936. Differentiable manifolds. Annals of Mathematics, pp.645-680.10.2307/1968482Search in Google Scholar
[50] Whitney, H., 1944. The self-intersections of a smooth n–manifold in 2n–space. Annals of Math, 45(220-446), p.180.10.2307/1969265Search in Google Scholar
[51] Whitney, H., 1957. Geometric integration theory.10.1515/9781400877577Search in Google Scholar
[52] Wu, W., 1952. Sur les classes caractéristiques des structures fibrées sphériques, Publ. Inst. Math. Univ. Strasbourg, (11).Search in Google Scholar
© 2022 Aleksandar Milivojević, published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.