Poincar´e duality and the existence of exotic structures on n-spheres

Maico Felipe Silva Ribeiro; Leandro Nery de Oliveira; Thiago Filipe da Silva

doi:10.14244/lajm.v2i01.25

Authors

Maico Ribeiro Universidade Federal do Espirito Santo
Leandro Oliveira Universidade Federal de São Carlos
Thiago da Silva Universidade Federal do Espírito Santo

DOI:

https://doi.org/10.14244/lajm.v2i01.25

Keywords:

Poincaré duality, Exotic sphere, Milnor fibration

Abstract

Poincar´e duality is a remarkable result in Algebraic Topology. It guarantees the existence of an isomorphism between the homology and cohomology groups of manifolds. We present a survey of the most general version of this result and its most important variations such as the Lefschetz duality and the Alexander duality. We consider an important application of these results in the study of the existence of exotic structures on n-spheres.

References

Gergonne J. Annales de matématiques pures et appliquées. T. 16.(1825- 1826)(Annales de Gergonne). In: Annales de mathématiques pures et appliquées. vol. 16. Imprimerie de P. Durand Belle Nimes, 1826 Format: 404 p.; 1826. p. 1825-6.

Poincaré H. Sur la généralisation d’un théoreme d’Euler relatif aux polyedres. Comptes Rendus de Séances de l’Academie des Sciences. 1893; 117:144.

Poincaré H. Analysis situs. Gauthier-Villars; 1895.

Poincaré H. Second complément à l'analysis situs. Proceedings of the London Mathematical Society. 1900; 1(1):277-308.

Alexander J. W. A proof and extension of the Jordan-Brouwer separation theorem. Transactions of the American Mathematical Society. 1922; 23(4):333-49.

Lefschetz S. Intersections and transformations of complexes and manifolds. Transactions of the American Mathematical Society. 1926; 28(1):1-49.

Cˇech E. Multiplications on a complex. Annals of Mathematics. 1936: 681-97.

Whitney H. On products in a complex. Proceedings of the National Academy of Sciences of the United States of America. 1937; 23(5):285.

Whitney H. On products in a complex. Annals of Mathematics. 1938: 397-432.

Milnor J. On manifolds homeomorphic to the 7-sphere. Annals of Mathematics. 1956: 399-405.

Jr JE, Kuiper NH. An invariant for certain smooth manifolds. COLUMBIA UNIV NEW YORK; 1963.

Gromoll D, Meyer W. An exotic sphere with nonnegative sectional curvature. Annals of Mathematics. 1974:401-6.

Kervaire M, Milnor J. Groups of homotopy spheres: I. Annals of Mathematics. 1963:504-37.

Milnor J. Singular points of complex hypersurfaces. 61. Princeton University Press; 1968.

Hatcher A. Algebraic Topology. Cambridge University Press; 2001.

Hirzebruch F. Singularities and exotic spheres. Séminaire N Bourbaki. 1968; 10(314):13-32.

Alexander JW. A proof of the invariance of certain constants of analysis situs. Transactions of the American Mathematical Society. 1915; 16(2):148-54.

Alexander JW. Combinatorial analysis situs. Transactions of the American Mathematical Society. 1926; 28(2):301-29.

Wang G, Xu Z. The triviality of the 61-stem in the stable homotopy groups of pheres. Annals of Mathematics. 2017: 501-80.

Moise EE. Affine structures in 3-manifolds: V. The triangulation theorem and Hauptvermutung. Annals of mathematics. 1952: 96-114.

Pham F. Formules de Picard-Lefschetz généralisées et ramification des intégrales. Bulletin de la Société Mathématique de France. 1965; 93:333-67.

Brieskorn EV. Examples of singular normal complex spaces which are topological manifolds. Proceedings of the National Academy of Sciences of the United States of America. 1966; 55(6):1395.

Brieskorn E. Beispiele zur differentialtopologie von singularitäten. Inventiones mathematicae. 1966; 2(1):1-14.

Milnor J. Sommes de variétés différentiables et structures différentiables des spheres. Bulletin de la Société Mathématique de France. 1959; 87:439-44.

Browder W. The Kervaire invariant of framed manifolds and its generalization. Annals of Mathematics. 1969: 157-86.

Seade J. On Milnor’s fibration theorem and its offspring after 50 years. Bulletin of the American Mathematical Society. 2019; 56(2):281-348.

Griffiths P, Harris J. Principles of algebraic geometry. John Wiley & Sons; 2014.