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The Exotic World of Milnor's Spheres
El exótico mundo de las esferas de Milnor
DOI:
https://doi.org/10.15446/recolma.v57n1.112373Parole chiave:
sphere, signature, characteristic classes (en)esfera, signatura, clases características (es)
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In his celebrated article of 1956, John Milnor established the existence of smooth structures on the 7-dimensional sphere that differs from the usual one. These so-called "exotic" structures have been of great interest ever since. The purpose of this article is to give a clear exposition of the different tools that Milnor used in order to provide an almost self-contained construction of exotic structures on the 7-dimensional sphere and then to show that they are not diffeomorphic to the standard sphere.
En su célebre artículo de 1956, John Milnor estableció la existencia de estructuras suaves en la esfera de dimensión 7 que difieren de la estructura usual. Éstas son llamadas estructuras "exóticas" las cuáles desde entonces fueron de gran interés. El propósito del presente artículo es dar una clara exposición de las diferentes herramientas que Milnor usó para ofrecer una construcción de las esferas exóticas en dimensión 7 y verificar que no son difeomorfas a la esfera usual.
Riferimenti bibliografici
A. Fomenko and D. Fuchs, Homotopical Topology, Second edition. Graduate Texts in Mathematics, 273. Springer, 2016. DOI: https://doi.org/10.1007/978-3-319-23488-5
M. H. Freedman, The topology of four-dimensional manifolds, J. Differential Geometry 17 (1982), 357-453. DOI: https://doi.org/10.4310/jdg/1214437136
M. Kervaire and J. W. Milnor, Groups of homotopy spheres I, Ann. of Math. 77 (1963), no. 2, 504-537. DOI: https://doi.org/10.2307/1970128
I. G. Macdonald, Symmetric functions and Hall polynomials, Second edition, Oxford University Press, 2015.
S. MacLane, Categorties for the working mathmatician, vol. 5, Graduates Texts in Mathematics, Springer-Verlag, 1971. DOI: https://doi.org/10.1007/978-1-4612-9839-7
R. McEnroe, Milnor's construction of exotic 7-spheres, (2015), Chicago REU.
J. W. Milnor, On manifolds homeomorphic to the 7-sphere, Ann. of Math. 64 (1956), no. 2, 399-405. DOI: https://doi.org/10.2307/1969983
J. W. Milnor, Morse theory, Princeton University Press, Annals of Mathematics Studies, 1963.
J. W. Milnor and J. D. Stasheff, On manifolds homeomorphic to the 7-sphere, Princeton University Press, Annals of Mathematics Studies, 1974.
E. Riehl, Category theory in context, Cambridge University Press, 2014.
J. C. Santos, When is a group homomorphism a covering homomorphism?, Extracta Mathematicae (2007).
J. Stallings, The piecewise-linear structure of euclidean space, Proc. Cambridge Philos. Soc. 58 (1962), 481-488. DOI: https://doi.org/10.1017/S0305004100036756
N. Steenrod, The topology of fibre bundles, Princeton University Press, Princeton, 1999.
R. Thom, Quelques propriétés globales des variétés differentiables, Comment. Math. Helv. 28 (1954), 17-86. DOI: https://doi.org/10.1007/BF02566923
G. Tiozzo, Differentiable structures on the 7-sphere, (2009), Yale University.
