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On cobordism of manifolds with corners


Author: Gerd Laures
Journal: Trans. Amer. Math. Soc. 352 (2000), 5667-5688
MSC (2000): Primary 55N22, 55T15; Secondary 55Q10, 55N34, 57R20
DOI: https://doi.org/10.1090/S0002-9947-00-02676-3
Published electronically: August 21, 2000
MathSciNet review: 1781277
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Abstract: This work sets up a cobordism theory for manifolds with corners and gives an identification with the homotopy of a certain limit of Thom spectra. It thereby creates a geometrical interpretation of Adams-Novikov resolutions and lays the foundation for investigating the chromatic status of the elements so realized. As an application, Lie groups together with their left invariant framings are calculated by regarding them as corners of manifolds with interesting Chern numbers. The work also shows how elliptic cohomology can provide useful invariants for manifolds of codimension 2.


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  • [Ada74] J.F. Adams, Stable homotopy and generalized homology, The University of Chicago Press, 1974. MR 53:6534
  • [AS74] M.F. Atiyah and L. Smith, Compact Lie groups and the stable homotopy of spheres, Topology 13 (1974), 135-142. MR 49:8013
  • [Bak97] A. Baker, Hecke operations and the Adams $e_2$-term based on elliptic cohomology, Canad. Math. Bull. 42 (1999), 129-138. MR 2000c:55008
  • [BH58] A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces, I-III, Am. J. Math. 80-82 (1958), 458-538, 315-382,491-504. MR 21:1586; MR 22:988; MR 22:11413.
  • [Bry90] J.L. Brylinski, Representations of loop groups, Dirac operators on loop space, and modular forms, Topology 29 (1990), 461-480. MR 91j:58151
  • [Cer61] J. Cerf, Topologie de certains espaces de plongements, Bull. Soc. Math. France 89 (1961), 227-380. MR 25:3543
  • [CF66] P.E. Conner and E.E. Floyd, The relation of cobordism to $K$-theories, LNM 28, Springer, Berlin, 1966. MR 35:7344
  • [Dol78] A. Dold, Geometric cobordism and the fixed point transfer, Algebraic Topology, Lecture Notes in Mathematics, vol. 673, Springer Verlag, 1978, pp. 32-87. MR 80g:57052
  • [Dou61] A. Douady, Variétés à bord anguleux et voisinages tubulaires, théorèmes d'isotopie et de recollement, Séminaire Henri Cartan 1961/62, Exposé 1, Secrétariat Math., Paris, 1961/62. MR 28:3435
  • [Dye69] E. Dyer, Cohomology theories, W.A. Benjamin, New York Amsterdam, 1969. MR 42:3780
  • [Elm88] A.D. Elmendorf, The Grassmannian geometry of spectra, Journal of Pure and Applied Algebra 54 (1988), 37-94. MR 90c:55007
  • [Fra92] J. Franke, On the construction of elliptic cohomology, Math. Nachr. 158 (1992), 43-65. MR 94h:55007
  • [HBJ92] F. Hirzebruch, T. Berger, and R. Jung, Manifolds and modular forms, Aspects of Mathematics, vol. E20, Vieweg, Braunschweig, 1992. MR 94d:57001
  • [HS96] M. Hovey and H. Sadofsky, Invertible spectra in the $E(n)$-local stable homotopy category, J. London Math. Soc. (2) 60 (1999), 284-302. CMP 2000:04
  • [Jän68] K. Jänich, On the classification of $O(n)$-manifolds, Math. Annalen 176 (1968), 53-76. MR 37:2261
  • [Kna78] K. Knapp, Rank and Adams filtration of a Lie group, Topology 17 (1978), 41-52. MR 57:10703
  • [Lau99] G. Laures, The topological $q$-expansion principle, Topology 38 (1999), 387-425. MR 2000c:55009
  • [Lil73] J. Lillig, A union theorem for cofibrations, Arch. Math. 24 (1973), 410-415. MR 48:12512
  • [LMS80] L.G. Lewis, J.P. May, and M. Steinberger, Equivariant stable homotopy theory, Lecture Notes in Mathematics, vol. 1213, Springer Verlag, 1986. MR 88e:55002
  • [MR77] H.R. Miller and D.C. Ravenel, Morava stabilizer algebras and the localization of Novikov's $E_2$-term, Duke Math. Journal 44 (1977), 433-447. MR 56:16613
  • [MRW77] H.R. Miller, D.C. Ravenel, and W.S. Wilson, Periodic phenomena in the Adams-Novikov spectral sequence, Annals of Math. 106 (1977), 469-516. MR 56:16626
  • [Oss82] E. Ossa, Lie-groups as framed manifolds, Topology 21 (1982), 315-323. MR 83e:55009
  • [Qui71] D. Quillen, Elementary proofs of some results of cobordism theory using Steenrod operations, Advances in Mathematics 7 (1971), 29-56. MR 44:7566
  • [Rav86] D.C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Academic Press, Orlando, 1986. MR 87j:55003
  • [Ray79] N. Ray, Invariants of reframed manifolds, Proc. London Math. Soc. 39 (1979), 253-275. MR 81a:55024
  • [RS95] P.S. Landweber D.C. Ravenel and R.E. Stong, Periodic cohomology theories defined by elliptic curves, The Cech Centennial: A Conference on Homotopy Theory (Providence) (M. Cenkl and H. Miller, eds.), Contemp. Math., vol. 181, Amer. Math. Soc., 1995, pp. 317-337. MR 96i:55009
  • [Seg87] G. Segal, Elliptic cohomology, Séminaire Bourbaki 1987/88, Exposé 695, Astérisque, no. 161-162, Soc. Math. France, Paris, 1988, pp. 187-201. MR 91b:55005
  • [Ste76] B. Steer, Orbits and the homotopy class of a compactification of a classical map, Topology 15 (1976), 383-393. MR 56:3864
  • [Sto68] R. E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, 1968. MR 40:2108
  • [Str68] A. Strøm, Note on cofibrations. II, Math. Scand. 22 (1968), 130-142. MR 39:4846
  • [Tho54] R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17-86. MR 15:890a
  • [Wit86] E. Witten, The index of the Dirac operator in loop space, Elliptic Curves and Modular Forms in Algebraic Topology (Berlin and New York), Lecture Notes in Math., no. 1326, Springer, 1986, pp. 161-181. MR 41a:57021
  • [Woo76] R.M.W. Wood, Framing the exceptional Lie group $G_2$, Topology 15 (1976), 303-320. MR 58:7665

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Additional Information

Gerd Laures
Affiliation: Fachbereich Mathematik, Johannes Gutenberg Universität Mainz, D-55099 Mainz, Germany
Address at time of publication: Mathematisches Institut der Universität Heidelberg, Im Neuenheimer Feld 288, D-69120 Heidelberg, Germany
Email: gerd@laures.de

DOI: https://doi.org/10.1090/S0002-9947-00-02676-3
Keywords: Cobordism theory, manifolds with corners, Lie groups, Adams-Novikov spectral sequence, elliptic cohomology
Received by editor(s): June 24, 1998
Published electronically: August 21, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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