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A survey of bordism and cobordism

Published online by Cambridge University Press:  24 October 2008

Peter S. Landweber
Peter S. Landweber
Affiliation:
Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA
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Extract

In the paper ‘Bordism and Cobordism’, which appeared in vol. 57 (1961) of this journal [5], Michael Atiyah introduced and began the study of bordism and cobordism theory. The present article will trace developments in this area since this beginning.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 100 , Issue 2 , September 1986 , pp. 207 - 223
Copyright
Copyright © Cambridge Philosophical Society 1986

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References

REFERENCES

[1] Adams, J. F.. Stable Homotopy and Generalised Homology. (University of Chicago Press, 1974).Google Scholar
[2] Anderson, D. W., Brown, E. H. Jr. and Peterson, F. P.. SU cobordim, KO-characteristic numbers, and the Kervaire invariant. Ann. of Math. 83 (1966), 54–87.CrossRefGoogle Scholar
[3] Anderson, D. W., Brown, E. H. Jr. and Peterson, F. P.. The structure of the Spin cobordism ring. Ann. of Math. 86 (1967), 271–298.CrossRefGoogle Scholar
[4] Araki, S.. Typical formal groups in complex cobordism and K-theory. Lecture Notes Math., Kyoto University, no. 3 (Kinokuniya Book Store, 1973).Google Scholar
[5] Atiyah, M. F.. Bordism and cobordism. Proc. Cambridge Philos. Soc. 57 (1961), 200–208.CrossRefGoogle Scholar
[6] Atiyah, M. F.. Vector bundles and the Künneth formula. Topology 1 (1962), 245–248.CrossRefGoogle Scholar
[7] Atiyah, M. F.. Power operations in K-theory. Quart. J. Math. Oxford (2) 17 (1966), 165–193.CrossRefGoogle Scholar
[8] Atiyah, M. F.. K-theory (Benjamin, 1967).Google Scholar
[9] Atiyah, M. F., Bott, R. and Shapiro, A.. Clifford modules. Topology 3, suppl. 1 (1964), 3–38.CrossRefGoogle Scholar
[10] Atiyah, M. F. and Hirzebruch, F.. Riemann-Roch theorems for differentiable manifolds. Bull. Amer. Math. Soc. 65 (1959), 276–281.CrossRefGoogle Scholar
[11] Atiyah, M. F. and Hirzebruch, F.. Vector bundles and homogeneous spaces. In Proc. Symp. Pure Math., vol. III. (Amer. Math. Soc., 1961), pp. 7–38.CrossRefGoogle Scholar
[12] Atiyah, M. F. and Hirzebruch, F.. Cohomologie Operationen und charakteristische Klassen Math. Zeit. 77 (1961), 149–187.CrossRefGoogle Scholar
[13] Atiyah, M. F. and Hirzebruch, F.. Spin-manifolds and group actions. In Essays on Topology and Related Topics (Springer-Verlag, 1970), pp. 18–28.CrossRefGoogle Scholar
[14] Atiyah, M. F. and Singer, I. M.. The index of elliptic operators on compact manifolds. Bull. Amer. Math. Soc. 69 (1963), 422–433.CrossRefGoogle Scholar
[15] Atiyah, M. F. and Singer, I. M.. The index of elliptic operators: III. Ann. of Math. 87 (1968), 546–604.CrossRefGoogle Scholar
[16] Atiyah, M. F. and Singer, I. M.. Index theory for skew-adjoint Fredholm operators. Publ. Math. I.H.E.S. 37 (1969), 5–26.CrossRefGoogle Scholar
[17] Baas, N. A.. On bordism theory of manifolds with singularity. Math. Scand. 33 (1973), 279–302.CrossRefGoogle Scholar
[18] Borel, A. and Hirzebruch, F.. Characteristic classes and homogeneous spaces, I, II. Amer. J. Math. 80 (1958), 458–538CrossRefGoogle Scholar
Borel, A. and Hirzebruch, F.. Characteristic classes and homogeneous spaces, I, II. Amer. J. Math. 81 (1959), 315–382.CrossRefGoogle Scholar
[19] Bousfield, A. K.. The localization of spectra with respect to homology. Topology 18 (1979), 257–281.CrossRefGoogle Scholar
[20] Brown, E. H. Jr. and Peterson, F. P.. A spectrum whose p cohomology is the algebra of reduced pth powers. Topology 5 (1966), 149–154.CrossRefGoogle Scholar
[21] Buchstaber, V. M.. Cobordism in problems of algebraic topology. J. Soviet Math. 7 (1977), 629–653. [Translated from Itogi Nauki i Tekhniki (Algebra Topologiya, Geometriya) 13 (1975), 231–271.]CrossRefGoogle Scholar
[22] Chern, S. S., Hirzebruch, F. and Serre, J. P.. On the index of a flbered manifold. Proc. Amer. Math. Soc. 8 (1957), 587–596.CrossRefGoogle Scholar
[23] Conner, P. E. and Floyd, E. E.. Differentiable Periodic Maps. Ergeb. Math, Band 33 (Springer-Verlag, 1964).Google Scholar
[24] Conner, P. B. and Floyd, E. E.. Torsion in SU-bordism. Mem. Amer. Math. Soc. no. 60, 1966.CrossRefGoogle Scholar
[25[ Conner, P. E. and Floyd, E. E.. The relation of cobordism to K-theories. Lecture Notes in Math., vol. 28 (Springer-Verlag, 1966).CrossRefGoogle Scholar
[26] Conner, P. E. and Smith, L.. On the complex bordism of finite complexes. Publ. Math. I.H.E.S. 37 (1969), 117–221.CrossRefGoogle Scholar
[27] Dieck, T. Tom. Steenrod Operationen in Kobordismen-Theorien. Math. Z. 107 (1968), 380–401.CrossRefGoogle Scholar
[28] Dold, A.. Erzeugende der Thomschen Algebra . Math. Z. 65 (1956), 25–35.CrossRefGoogle Scholar
[29] Eilenberg, S. and Steenrod, N.. Foundations of Algebraic Topology. (Princeton University Press, 1952.)CrossRefGoogle Scholar
[30] Floyd, E. B.. Stiefel–Whitney numbers of quaternionic and related manifolds. Trans. Amer. Math. Soc. 155 (1971), 77–94.CrossRefGoogle Scholar
[31] Fröhlich, A.. Formal groups. Lecture Notes in Math., vol. 74 (Springer-Verlag, 1968).CrossRefGoogle Scholar
[32] Gromov, M. and Lawson, H. B. Jr. Spin and scalar curvature in the presence of a fundamental group. Ann. of Math. 111 (1980), 209–230.CrossRefGoogle Scholar
[33] Gromov, M. and Lawson, H. B. Jr. The classification of simply connected manifolds of positive scalar curvature. Ann. of Math. 111 (1980), 423–434.CrossRefGoogle Scholar
[34] Gromov, M. and Lawson, H. B. Jr. Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Publ. Math. I.H.E.S. 58 (1983), 83–196.CrossRefGoogle Scholar
[35] Hattori, A.. Integral characteristic numbers for weakly almost complex manifolds. Topology 5 (1966), 259–280.CrossRefGoogle Scholar
[36] Hazewinkel, M.. Constructing formal groups I, II. J. Pure Appl. Algebra 9 (1977), 131–150 and 151–162.CrossRefGoogle Scholar
[37] Hazewinkel, M.. Formal Groups and Applications. (Academic Press, 1978.)Google Scholar
[38] Hirzebruch, F.. Topological Methods in Algebraic Geometry. (Springer-Verlag, 1966.)Google Scholar
[39] Hitchin, N.. Harmonic spinors. Adv. in Math. 14 (1974), 1–55.CrossRefGoogle Scholar
[40] Johnson, D. C. and Wilson, W. S.. Projective dimension and Brown–Peterson homology. Topology 12 (1973), 327–353.CrossRefGoogle Scholar
[41] Johnson, D. C. and Wilson, W. S.. BP-operations and Morava's extraordinary K-theories. Math. Zeit. 144 (1975), 55–75.CrossRefGoogle Scholar
[42] Kochman, S. O.. The symplectic cobordism I, II, III. Mem. Amer. Math. Soc. No. 228 (1980), No. 271 (1982) and (to appear).CrossRefGoogle Scholar
[43] Landweber, P. S.. Künneth formulas for bordism theories. Trans. Amer. Math. Soc. 121 (1966), 242–256.Google Scholar
[44] Landweber, P. S.. Cobordism operations and Hopf algebras. Trans. Amer. Math. Soc. 129 (1967), 94–110.CrossRefGoogle Scholar
[45] Landweber, P. S.. Homological properties of comodules over MU *(MU) and BP *(BP). Amer. J. Math. 98 (1976), 591–610.CrossRefGoogle Scholar
[46[ Landweber, P. S. and Stong, R. E.. Circle actions on Spin manifolds and characteristic numbers (to appear).Google Scholar
[47] Lashof, R.. Poincaré duality and cobordism. Trans. Amer. Math. Soc. 109 (1963), 257–277.Google Scholar
[48] Lawson, H. B. Jr. and Michelsohn, M. L.. Spin Geometry (to appear).Google Scholar
[49] Lawson, H. B. Jr. and Yau, S. T.. Scalar curvature, non-abelian group actions and the degree of symmetry of exotic spheres. Comm. Math. Helv. 49 (1974), 232–244.CrossRefGoogle Scholar
[50] Lichnerowicz, A.. Spineurs harmonique. C. B. Acad. Sci. Paris, Sér. A–B, 257 (1963), 7–9.Google Scholar
[51] Madsen, I. and Milgram, R. J.. The Classifying Spaces for Surgery and Cobordism of Manifolds. Annals of Math. Studies no. 92 (Princeton University Press, 1979).Google Scholar
[52] Milnor, J. W.. On the cobordism ring Ω* and a complex analogue. Amer. J. Math. 82 (1960), 505–521.CrossRefGoogle Scholar
[53] Milnor, J. W.. Remarks concerning Spin manifolds. In Differential Geometry and Combinatorial Topology (Princeton University Press, 1965), pp. 55–62.CrossRefGoogle Scholar
[54] Milnor, J. W. and Stasheff, J. D.. Characteristic Classes. Annals of Math. Studies no. 76 (Princeton University Press, 1974).CrossRefGoogle Scholar
[55] Miyazaki, T.. Simply connected Spin manifolds and positive scalar curvature. Proc. Amer. Math. Soc. 93 (1985), 730–734.CrossRefGoogle Scholar
[56] Morava, J.. Extraordinary K-theories. Columbia University preprint (1971).Google Scholar
[57] Novikov, S. P.. Homotopy properties of Thom complexes. Mat. Sb. 57 (1962), 407–442 (Russian).Google Scholar
[58] Novikov, S. P.. The methods of algebraic topology from the point of view of cobordism theories. Izvestiya Akad. Nauk SSSR, Ser. Mat. 31 (1967), 885–951Google Scholar
Novikov, S. P.. The methods of algebraic topology from the point of view of cobordism theories Math. USSR Izvestiya (1967), 827–913 (translation).CrossRefGoogle Scholar
[59] Ochanine, S.. Sur le genres multiplicatifs définis par des intégrales elliptiques (to appear).Google Scholar
[60] Palais, R. S.. Seminar on the Atiyah–Singer Index Theorem. Annals of Math. Studies no. 57 (Princeton University Press, 1965).Google Scholar
[61] Quillen, D. G.. On the formal group laws of unoriented and complex cobordism theory. Bull. Amer. Math. Soc. 75 (1969), 1293–1298.CrossRefGoogle Scholar
[62] Quillen, D. G.. Elementary proofs of some results of cobordism theory using Steenrod operations. Adv. in Math. 7 (1971), 29–56.CrossRefGoogle Scholar
[63] Ravenel, D. C.. A novice's guide to the Adams–Novikov spectral sequence. In Geometric Applications of Homotopy Theory II, Lecture Notes in Math. no. 658 (Springer-Verlag, 1978), pp. 404–475.CrossRefGoogle Scholar
[64] Ravenel, D. C.. Localization with respect to certain periodic homology theories. Amer. J. Math. 106 (1984), 351–414.CrossRefGoogle Scholar
[65] Ravenel, D. C.. Complex Cobordism and Stable Homotopy Groups of Spheres. (Academic Press, 1986.)Google Scholar
[66] Roush, F. W.. On the image of symplectic cobordism in unoriented cobordism. Proc. Amer. Math. Soc. 38 (1973), 647–652.CrossRefGoogle Scholar
[67] Smith, L.. On the finite generation of . J. Math. Mech. 18 (1969), 1017–1024.Google Scholar
[68] Stong, R. E.. Relations among characteristic numbers – I, II. Topology 4 (1985), 267–281CrossRefGoogle Scholar
Stong, R. E.. Relations among characteristic numbers – I, II Topology 5 (1966), 133–148.CrossRefGoogle Scholar
[69] Stong, R. E.. Notes on Cobordism Theory (Princeton University Press, 1968).Google Scholar
[70] Thom, R.. Quelques propriétés globales des variétés differentiables. Comm. Math. Helv. 28 (1954), 17–86.CrossRefGoogle Scholar
[71] Toda, H. and Kozima, K.. The symplectic Lazard ring. J. Math. Kyoto Univ. 22 (1982), 131–153.Google Scholar
[72] Wall, C. T. C.. Determination of the cobordism ring. Ann. of Math. 72 (1960), 292–311.CrossRefGoogle Scholar
[73] Whitehead, G. W.. Generalized homology theories. Trans. Amer. Math. Soc. 102 (1962), 227–283.CrossRefGoogle Scholar
[74] Wilson, W. S.. Brown–Peterson homology, an introduction and sampler. Regional Conf. Series in Math. 48 (Amer. Math. Soc., 1980).Google Scholar
[75] Witten, E.. Fermion quantum numbers in Kaluza–Klein theory. In Shelter lsland II: Proceedings of the 1983 Shelter Island Conference on Quantum Field Theory and the Fundamental Problems of Physics (MIT Press, 1985), pp. 227–277.Google Scholar