Tautological rings of fake quaternionic spaces

• Nils Prigge ^[1]
  1. [1] Stockholm University

     Stockholm University

     Suecia

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 31, Nº. 1, 2025
• Idioma: inglés
• DOI: 10.1007/s00029-024-00993-9
• Enlaces
  • Texto completo
• Resumen

  • The tautological ring R∗(M) of a smooth manifold M is the ring of characteristic classes generated by the Miller–Morita–Mumford classes, and is often more accessible than the ring of all characteristic classes of smooth M-bundles. In this paper, we introduce a new method to obtain upper bounds on the Krull dimension of R∗(M) for manifolds homotopy equivalent to a fixed, simply connected Poincaré duality space by using recent progress in rational homotopy theory and the family signature theorem. In particular, we show that the Krull dimension of the tautological ring vanishes for almost all manifolds homotopy equivalent to HP2.

• Referencias bibliográficas
  • 1. Atiyah, M., Hirzebruch, F.: Spin-manifolds and group actions. In Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham),...
  • 2. Atiyah, M.F.: The signature of fibre-bundles. In: Global Analysis (Papers in Honor of K. Kodaira), pp. 73–84. Univ. Tokyo Press, Tokyo...
  • 3. Baues, H.J.: On the group of homotopy equivalences of a manifold. Trans. Amer. Math. Soc. 348(12), 4737–4773 (1996)
  • 4. Berglund, A.: Rational homotopy theory of mapping spaces via Lie theory for L∞-algebras. Homol. Homotopy Appl. 17(2), 343–369 (2015)
  • 5. Berglund, A.: Rational models for automorphisms of fiber bundles. Doc. Math. 25, 239–265 (2020)
  • 6. Berglund, A.: Characteristic classes for families of bundles. Selecta Math. 28(3), 51 (2022)
  • 7. Berglund, A., Madsen, I.: Rational homotopy theory of automorphisms of manifolds. Acta Math. 224(1), 67–185 (2020)
  • 8. Borel, A., Hirzebruch, F.: Characteristic classes and homogeneous spaces. I. Amer. J. Math. 80, 458– 538 (1958)
  • 9. Eells, J., Kuiper, N.: Manifolds which are like projective planes. Inst. Hautes Études Sci. Publ. Math. 14, 5–46 (1962)
  • 10. Eisenbud, D.: Commutative algebra, vol. 150 of Graduate Texts in Mathematics. Springer-Verlag, New York. With a view toward algebraic...
  • 11. Félix, Y., Halperin, S., Thomas, J.-C.: Rational homotopy theory. Graduate Texts in Mathematics, Springer-Verlag, New York (2001)
  • 12. Galatius, S., Grigoriev, I., Randal-Williams, O.: Tautological rings for high-dimensional manifolds. Compos. Math. 153(4), 851–866 (2017)
  • 13. Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/
  • 14. Grigoriev, I.: Relations among characteristic classes of manifold bundles. Geom. Topol. 21(4), 2015– 2048 (2017)
  • 15. Hebestreit, F., Land, M., Lück, W., Randal-Williams, O.: A vanishing theorem for tautological classes of aspherical manifolds. Geom. Topol....
  • 16. Hirzebruch, F.: Topological methods in algebraic geometry. Die Grundlehren der mathematischen Wissenschaften, Band 131. Springer-Verlag...
  • 17. Hsiang, W.: A note on free differentiable actions of S1 and S3 on homotopy spheres. Ann. Math. 2(83), 266–272 (1966)
  • 18. Kemper, G.: A course in commutative algebra. Graduate Texts in Mathematics, Springer, Heidelberg (2011)
  • 19. Kramer, L., Stolz, S.: A diffeomorphism classification of manifolds which are like projective planes. J. Differential Geom. 77(2), 177–188...
  • 20. May, J.P.: Classifying spaces and fibrations. Mem. Amer. Math. Soc., 1(1, 155):xiii+98, (1975)
  • 21. Mumford, D.: Towards an enumerative geometry of the moduli space of curves. In Arithmetic and geometry, Vol. II, vol. 36 of Progr. Math.,...
  • 22. Prigge, N.: Tautological Rings of Fibrations. page arXiv:1905.02676, May (2019)
  • 23. Prigge, N.: On tautological classes of fibre bundles and self-embedding calculus. University of Cambridge, Cambridge (2020)
  • 24. Randal-Williams, O.: Some phenomena in tautological rings of manifolds. Selecta Math. 24(4), 3835– 3873 (2018)
  • 25. Randal-Williams, O.: The family signature theorem. page arXiv:2204.11696, April (2022)
  • 26. Stoll, R.: The stable cohomology of self-equivalences of connected sums of products of spheres. page arXiv:2203.15650, March (2022)
  • 27. Tanré, D.: Homotopie rationnelle: modèles de Chen, Quillen. Lecture Notes in Mathematics, Sullivan, Springer-Verlag, Berlin (1983)
  • 28. Wall, C.T.C.: Poincaré complexes. I. Ann. Math. 2(86), 213–245 (1967)