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Homotopy investigation of classifying spaces of cobordisms of singular maps

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Abstract

The classifying spaces of cobordisms of singular maps have two fairly different constructions. We expose a homotopy theoretical connection between them. As a corollary we show that the classifying spaces in some cases have a simple product structure.

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References

  1. V. I. Arnold, Normal forms of functions near degenerate critical points, the Weyl groups \(A_k\), \(D_k\), \(E_k\), and Lagrangian singularities, Funkts. Anal. Prilozh., 6 (1974), 3–25 (in Russian); translation in Funct. Anal. Appl., 6 (1972), 254– 272.

  2. Barratt, M.G., Eccles, P.J.: \(\Gamma ^+\)-structures. I, Topology 13, 25–45 (1974)

    Article  MATH  Google Scholar 

  3. F. R. Cohen, T. J. Lada and J. P. May, The Homology of Iterated Loop Spaces, Lecture Notes in Math., 533, Springer (Berlin–Heidelberg, 1976)

  4. A. Hatcher, Algebraic Topology, pi.math.cornell.edu/ hatcher/AT/ATch4.4.pdf

  5. Jänich, K.: Symmetry properties of singularities of \(C^\infty \) functions. Math. Ann. 238, 147–156 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Mather, Stability of \(C^\infty \)-mappings. I, Ann. Math., 89 (1969), 89–104; II, Ann. Math., 87 (1968), 254–291; III, Publ. Math., Inst. Hautes Étud. Sci., 35 (1969), 127–156; IV, Publ. Math., Inst. Hautes Étud. Sci., 37 (1970), 223–248; V, Adv. Math., 4 (1970), 301–336; VI, in: Proceedings of Liverpool Singularities Symposium. I, Lect. Notes in Math., 192, Springer (Berlin, 1971), pp. 207–253

  7. J. P. May, The Geometry of Iterated Loop Spaces, Lecture Notes in Math. 271, Springer (Berlin, 1972)

  8. B. Morin, Formes canoniques des singularités d'une application différentiable, C. R. Acad. Sci. Paris, 260 (1965), 5662–5665, 6503–6506.

  9. R. Rimányi, Doctoral thesis, www.cs.elte.hu/phd_th/rimanyi.dvi

  10. Rimányi, R., Szűcs, A.: Pontrjagin-Thom-type construction for maps with singularities. Topology 37, 1177–1191 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. E. H. Spanier, Algebraic Topology, Springer-Verlag (New York, 1966)

  12. A. Szűcs, Cobordism of maps with simplest singularities, in: Topology Symposium (Siegen, 1979), Lecture Notes in Math., 788, Springer (Berlin, 1980)

  13. Szűcs, A.: Cobordism of singular maps. Geom. Topol. 12, 2379–2452 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. T. Terpai, Fibration of classifying spaces in the cobordism theory of singular maps, Tr. Mat. Inst. Steklova, Osobennosti i Prilozheniya, 267 (2009), 280–287 (in Russian); translation in Proc. Steklov Inst. Math., 267 (2009), 270–277

  15. Wall, C.T.C.: A second note on symmetry of singularities. Bull. London Math. Soc. 12, 347–354 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  16. Wells, R.: Cobordism groups of immersions. Topology 5, 281–294 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  17. H. Whitney, The singularities of a smooth \(n\)-manifold in \((2n-1)\)-space, Ann. Math. (2), 45 (1944), 247–293

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Correspondence to A. Szűcs.

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Both authors were supported by the National Research, Development and Innovation Office NKFIH (OTKA) Grant K 112735; the second author was supported by the National Research, Development and Innovation Office NKFIH (OTKA) Grant K 120697.

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Szűcs, A., Terpai, T. Homotopy investigation of classifying spaces of cobordisms of singular maps. Acta Math. Hungar. 157, 489–502 (2019). https://doi.org/10.1007/s10474-019-00926-5

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  • DOI: https://doi.org/10.1007/s10474-019-00926-5

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