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Coincidence and self-coincidence of maps between spheres

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Abstract

Let f, g: \({S^{m}\rightarrow S^{2n}}\) be a pair of maps between spheres. We further explore previous results about the coincidence of a pair of maps between spheres. We give special attention to the case of self-coincidence (i.e., (f, f): \({M \rightarrow S^{2n}}\)) in which we consider the question of making the pair homotopy disjoint by a small deformation whenever the pair can be made coincidence free. When M is the sphere \({S^{4n-2}}\), we basically update the results from [C. R. Acad. Sci. Paris Ser. I 342 (2006), 511–513] based on the new results of the Kervaire invariant one problem. When M is either the sphere \({S^{4n-1}}\) or the sphere S 4n, we classify all pairs (\({f_{1}, f_{2}}\)) which are homotopy disjoint; and for maps f such that (f, f) can be deformed to coincidence free, the ones such that (f, f) can be deformed to coincidence free by a small deformation. Finally, we give families of functions \({h_{t} t \in \mathbb{N}}\) such that (\({h_{t},h_{t}}\)) is homotopy disjoint but not by a small deformation.

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References

  1. Barratt, M., Jones, J., Mahowald, M.: The Kervaire invariant problem. Contemp. Math. 19, 9–22 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  2. M. Barratt, J. Jones and M. Mahowald, Relations among Toda brackets and the Kervaire invariant in dimension 62. J. Lond. Math. Soc. (2) 30 (1984), 533–550.

  3. M. Barratt, J. Jones and M. Mahowald, The Kervaire Invariant and the Hopf Invariant. Lecture Notes in Math. 1286, Springer-Verlag, 1987, 135–173.

  4. Borsari, L.D., Goncalves, D.L.: A Van Kampen type theorem for coincidences. Topology Appl. 101, 149–160 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. F. R. Cohen, A course in some aspects of classical homotopy theory. In: Algebraic Topology (Seattle, Wash., 1985), Lecture Notes in Math. 1286, Springer-Verlag, Berlin, 1987, 1–92.

  6. Dimovski, D., Geoghegan, R.: One-parameter fixed point theory. Forum Math. 2, 125–154 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dold, A., Goncalves, D.L.: Self-coincidence of fibre maps. Osaka J. Math. 42, 291–307 (2005)

    MathSciNet  MATH  Google Scholar 

  8. Golasiński, M., Mukai, J.: Gottlieb groups of spheres. Topology 47, 399–430 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. D. L. Gonçalves, Coincidence theory. In: Handbook of Topological Fixed Point Theory (R. F. Brown, ed.), Springer-Verlag, Dordrecht, 2005, 3–42.

  10. D. L. Gonçalves, Coincidence Reidemeister classes on nilmanifolds and nilpotent fibrations. Topology Appl. 83 (1998), 169–186.

  11. D. L. Gonçalves, The coincidence Reidemeister classes of maps on nilmanifolds. Topol. Methods Nonlinear Anal. 12 (1998), 375–386.

  12. D. L. Gonçalves, Coincidence theory for maps from a complex to a manifold. Topology Appl. 92 (1999), 63–77.

  13. D. Gonçalves and J. Jezierski, The Lefschetz coincidence formula on nonorientable manifolds. Fund. Math. 153 (1997), 1–23.

  14. D. Gonçalves and D. Randall, Coincidence of maps from \(S^{2n-1}\)-bundles over \(S^{2n}\) to \(S^{2n}\). Bol. Soc. Mat. Mex. (3) 10 (2004), 181–192.

  15. D. Gonçalves and D. Randall, Self-coincidence of map**s between spheres and the strong Kervaire invariant one problem. C. R. Acad. Sci. Paris Ser. I 342 (2006), 511–513.

  16. D. Gonçalves and P. Wong, Nilmanifolds are Jiang-type spaces for coincidences. Forum Math. 13 (2001), 133–141.

  17. Hatcher, A., Quinn, F.: Bordism invariants of intersections of submanifolds. Trans. Amer. Math. Soc. 200, 327–344 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  18. M. A. Hill, M. J. Hopkins and D. C. Ravenel, On the nonexistence of elements of Kervaire invariant one. Ann. of Math. (2) 184 (2016), 1–262.

  19. Hilton, P.: An Introduction to Homotopy Theory. Cambridge University Press, Cambridge (1953)

    Book  MATH  Google Scholar 

  20. James, I.M.: Products on spheres. Mathematika 6, 1–13 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jezierski, J.: One codimensional Wecken type theorems. Forum Math. 5, 421–439 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kervaire, M.: Some nonstable homotopy groups of Lie groups. Illinois J. Math. 4, 161–169 (1960)

    MathSciNet  MATH  Google Scholar 

  23. S. O. Kochman and M. E. Mahowald, On the computation of the stable stems. In: The Čech Centennial: A Conference on Homotopy Theory, Contemp. Math. 181, Amer. Math. Soc., Providence, 1995, 299–316.

  24. Koschorke, U.: Selfcoincidences in higher codimensions. J. Reine Angew. Math. 576, 1–10 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Koschorke, U.: Nielsen coincidence theory in arbitrary codimensions. J. Reine Angew. Math. 598, 211–236 (2006)

    MathSciNet  MATH  Google Scholar 

  26. Koschorke, U.: Nonstabilized Nielsen coincidence invariants and Hopf-Ganea homomorphisms. Geom. Topol. 10, 619–665 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. U. Koschorke,Geometric and homotopy theoretic methods in Nielsen coincidence theory. Fixed Point Theory Appl. (2006), Article ID 84093, 15 pages.

  28. Koschorke, U.: Selfcoincidences and roots in Nielsen theory. J. Fixed Point Theory Appl. 2, 241–259 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. U. Koschorke, Minimizing coincidence numbers of maps into projective spaces. In: The Zieschang Gedenkschrift, Geom. Topol. Monogr. 14, Geom. Topol. Publ., Coventry, 2008, 373–391.

  30. Koschorke, U., Randall, D.: Kervaire invariants and selfcoincidences. Geom. Topol. 17, 621–638 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lam, K.Y., Randall, D.: Block bundle obstruction to Kervaire invariant one. Contemp. Math. 407, 163–171 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Morisugi, K., Mukai, J.: Lifting to mod 2 Moore spaces. J. Math. Soc. Japan 52, 515–533 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  33. J. Mukai, Private communication, 2015.

  34. Y. Nomura, Some homotopy groups of real Stiefel manifolds in the metastable range. III. Sci. Rep. College Gen. Ed. Osaka Univ. 28 (1979), 1–26.

  35. Y. Nomura, Some homotopy groups of real Stiefel manifolds in the metastable range. V. Sci. Rep. College Gen. Ed. Osaka Univ. 29 (1981), 159–183.

  36. Oda, N.: Unstable homotopy groups of spheres. Bull. Inst. Adv. Res. Fukuoka Univ. 44, 49–152 (1979)

    MathSciNet  MATH  Google Scholar 

  37. D. Randall, Embedding Homotopy Spheres and the Kervaire Invariant. Contemp. Math. 239, Amer. Math. Soc., Providence, RI, 1999, 239–243.

  38. J. P. Serre, Groupes d’homotopie et classes des groupes abélians. Ann. of Math. (2) 58 (1953), 258–294.

  39. Thomeier, S.: Einige Ergebnisse: über Homotopiegruppen von Sphären. Math. Ann. 164, 225–250 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  40. H. Toda, Composition Methods in Homotopy Groups of Spheres. Ann. of Math. Stud. 49, Princeton University Press, Princeton, NJ, 1962.

  41. G. Whitehead, Elements of Homotopy Theory. Grad. Texts in Math., Springer-Verlag, New York, 1978.

  42. Wong, P.: Reidemeister number, Hirsch rank, coincidences on polycyclic groups and solvmanifolds. J. Reine Angew. Math. 524, 185–204 (2000)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Departamento de Matemática, IME-USP, Caixa Postal 6628, Ag. Cidade de São Paulo, CEP 05314-970, São Paulo, SP, Brazil

    Daciberg L. Gonçalves

  2. Department of Mathematical Sciences, Loyola University, 6363, St Charles Avenue, New Orleans, LA, 70118, USA

    Duane Randall

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  1. Daciberg L. Gonçalves
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Gonçalves, D.L., Randall, D. Coincidence and self-coincidence of maps between spheres. J. Fixed Point Theory Appl. 19, 1011–1040 (2017). https://doi.org/10.1007/s11784-016-0376-y

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