Abstract
There is a rick mathematical structure attached to the cobordism invariants of manifolds. In the cases described by the index theorem, a generalized cohomology theory is used to express the global properties of locally defined analytic objects. Hirzebruch’s theory of multiplicative sequences gives an expression for these invariants in terms of characteristic classes, and brings to focus their remarkable arithmetic properties. Quillen’s theory of formal groups and complex oriented cohomology theories illuminates the generalized cohomology theories themselves.
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References
M. F. Atiyah and F. Hirzebruch, Riemann-Roch theorems for differentiable manifolds, Bull. Amer. Math. Soc. 65 (1959), 276–281.
M. F. Atiyah and I. M. Singer, The index of elliptic operators on compact manifolds, Bull. Amer. Math. Soc. 69 (1963), 422–433.
M. F. Atiyah and I. M. Singer, The index of elliptic operators: V, Ann. of Math. 93 (1971), 139–149.
A. Borel and J.-P. Serre, Le theoreme de Riemann-Roch (d’apres Grothendieck), Bull. Soc. Math. France 86 (1958), 97–136.
L. Breen, Fonctions theta et theoreme du cube, Lecture Notes in Math. 980, SpringerVerlag, Berlin and New York, 1983.
P. Deligne, Courbes elliptiques: formulaire (d’apres J. Tate), Modular Functions of One Variable III, Lecture Notes in Math. 476, Springer-Verlag, Berlin and New York, 1975, pp. 53–73.
P. Deligne and D. Mumford, Irreducibility of the space of curves of given genus, Inst. Hautes Etudes Sci. Publ. Math. 36 (1969), 75–109.
P. Deligne and M. Rapoport, Les schémas de modules de courbes elliptiques, Modular Functions of One Variable II, Lecture Notes in Math. 349, Springer-Verlag, Berlin and New York, 1973, pp. 143–316.
J. Franke, On the construction of elliptic cohomology, Max Planck Institut preprint, 1991.
F. Hirzebruch, Topological Methods in Algebraic Geometry, Springer-Verlag, New York, 1956.
M. J. Hopkins and M. Mahowald, The spectrum eo2, to appear.
P. S. Landweber, Elliptic cohomology and modular forms, Elliptic Curves and Modular Forms in Algebraic Topology (New York) (P. S. Landweber, ed.), Lecture Notes in Math. 1326, Springer-Verlag, 1988, pp. 55–68.
P. S. Landweber, D. C. Ravenel, and R. E. Stong, Periodic cohomology theories defined by elliptic curves, To appear.
J. W. Milnor, On the cobordism ring 12* and a complex analogue, Part I, Amer. J. Math. 82 (1960), 505–521.
D. Mumford, Picard groups of moduli problems, Arithmetic Algebraic Geometry (O. F. G. Schilling, ed.), Harper & Row, New York, 1965.
S. P. Novikov, Some problems in the topology of manifolds connected with the theory of Thom spaces, Soviet Mathematics Dokl. 1 (1960), 717–720.
S. Ochanine, Sur les genres multiplicatifs definis par des integrals elliptiques, Topology 26 (1987), 143–151.
D. G. Quillen, On the formal group laws of unoriented and complex cobordism theory, Bull. Amer. Math. Soc. 75 (1969), 1293–1298.
G. Segal, Elliptic cohomology, Seminaire Bourbaki 1987/88, Astérisque, vol. 161–162, Societe Mathematique de France, Fev. 1988, pp. 187–201.
R. Thom, Quelques propriétés globales des variétés differentiables, Comm. Math. Helv. 28 (1954), 17–86.
C. T. C. Wall, Determination of the cobordism ring, Ann. Math. (2) 72 (1960), 292–311.
E. Witten, Elliptic genera and quantum field theory, Comm. Math. Phys. 109 (1987), 525–536.
E. Witten, The index of the Dirac operator in loop space, Elliptic Curves and Modular Forms in Algebraic Topology (New York) (P. S. Landweber, ed.), Lecture Notes in Math. 1326, Springer-Verlag, Berlin and New York, 1988, pp. 161–181.
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© 1995 Birkhäser Verlag, Basel, Switzerland
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Hopkins, M.J. (1995). Topological Modular Forms, the Witten Genus, and the Theorem of the Cube. In: Chatterji, S.D. (eds) Proceedings of the International Congress of Mathematicians. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-9078-6_49
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DOI: https://doi.org/10.1007/978-3-0348-9078-6_49
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