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

Autores

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

DOI:

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

Palavras-chave:

Poincaré duality, Exotic sphere, Milnor fibration

Resumo

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.

Referências

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. DOI: https://doi.org/10.1112/plms/s1-32.1.277

Alexander J. W. A proof and extension of the Jordan-Brouwer separation theorem. Transactions of the American Mathematical Society. 1922; 23(4):333-49. DOI: https://doi.org/10.1090/S0002-9947-1922-1501206-6

Lefschetz S. Intersections and transformations of complexes and manifolds. Transactions of the American Mathematical Society. 1926; 28(1):1-49. DOI: https://doi.org/10.1090/S0002-9947-1926-1501331-3

Cˇech E. Multiplications on a complex. Annals of Mathematics. 1936: 681-97. DOI: https://doi.org/10.2307/1968483

Whitney H. On products in a complex. Proceedings of the National Academy of Sciences of the United States of America. 1937; 23(5):285. DOI: https://doi.org/10.1073/pnas.23.5.285

Whitney H. On products in a complex. Annals of Mathematics. 1938: 397-432. DOI: https://doi.org/10.2307/1968795

Milnor J. On manifolds homeomorphic to the 7-sphere. Annals of Mathematics. 1956: 399-405. DOI: https://doi.org/10.2307/1969983

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. DOI: https://doi.org/10.2307/1971078

Kervaire M, Milnor J. Groups of homotopy spheres: I. Annals of Mathematics. 1963:504-37. DOI: https://doi.org/10.2307/1970128

Milnor J. Singular points of complex hypersurfaces. 61. Princeton University Press; 1968. DOI: https://doi.org/10.1515/9781400881819

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. DOI: https://doi.org/10.1090/S0002-9947-1915-1501007-5

Alexander JW. Combinatorial analysis situs. Transactions of the American Mathematical Society. 1926; 28(2):301-29. DOI: https://doi.org/10.1090/S0002-9947-1926-1501346-5

Wang G, Xu Z. The triviality of the 61-stem in the stable homotopy groups of pheres. Annals of Mathematics. 2017: 501-80. DOI: https://doi.org/10.4007/annals.2017.186.2.3

Moise EE. Affine structures in 3-manifolds: V. The triangulation theorem and Hauptvermutung. Annals of mathematics. 1952: 96-114. DOI: https://doi.org/10.2307/1969769

Pham F. Formules de Picard-Lefschetz généralisées et ramification des intégrales. Bulletin de la Société Mathématique de France. 1965; 93:333-67. DOI: https://doi.org/10.24033/bsmf.1628

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. DOI: https://doi.org/10.1073/pnas.55.6.1395

Brieskorn E. Beispiele zur differentialtopologie von singularitäten. Inventiones mathematicae. 1966; 2(1):1-14. DOI: https://doi.org/10.1007/BF01403388

Milnor J. Sommes de variétés différentiables et structures différentiables des spheres. Bulletin de la Société Mathématique de France. 1959; 87:439-44. DOI: https://doi.org/10.24033/bsmf.1538

Browder W. The Kervaire invariant of framed manifolds and its generalization. Annals of Mathematics. 1969: 157-86. DOI: https://doi.org/10.2307/1970686

Seade J. On Milnor’s fibration theorem and its offspring after 50 years. Bulletin of the American Mathematical Society. 2019; 56(2):281-348. DOI: https://doi.org/10.1090/bull/1654

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