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Verschlingung von Fixpunktmengen in Darstellungsformen. I

Part of the Lecture Notes in Mathematics book series (2766,volume 1172)

Abstract

Let G=H0 × H1 be a product of two cyclic groups Hi of odd order. We show that there exist smooth actions of G on the standard sphere X=Sn(0)+n(1)+1 with the following properties:

  1. i)

    The isotropy groups are 1, H0, and H1.

  2. ii)

    The fixed point set XH i is a standard sphere Sn(i).

  3. iii)

    The linking number k of the fixed point sets can be any integer in the kernel of the Swan homomorphisms sG:ℤ/|G|*→Ko(ℤG).

For certain values of thelinking number X does not have the equivariant homotopy type of a representation sphere.

Keywords

  • Standard Sphere
  • Equivariant Homotopy Type

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Literatur

  1. Bak, A.: Odd dimension surgery groups of odd torsion groups vanish. Topology 14, 367–374 (1975).

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. Bredon, G. E.: Introduction to compact transformation groups. New York, Academic Press 1972.

    MATH  Google Scholar 

  3. Browder, W.: Surgery on simply-connected manifolds. Berlin-Heidelberg-New York, Springer 1972.

    CrossRef  MATH  Google Scholar 

  4. tom Dieck, T.: Homotopiedarstellungen endlicher Gruppen: Dimensionsfunktionen. Invent. math. 67, 231–252 (1982).

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. tom Dieck, T. und Petrie, T.: Homotopy representations of finite groups. Publ. math. I.H.E.S. 56, 129–169 (1983).

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. Dotzel, R. M. und Hamrick, G.: p-group actions on homology spheres. Invent. math. 62, 437–442 (1981).

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Gottlieb, D. H.: Poincaré duality and fibrations. Proc. Amer. Math. Soc. 76, 148–150 (1979).

    MathSciNet  MATH  Google Scholar 

  8. Hirzebruch, F. und Mayer, K. H.: O(n)-Mannigfaltigkeiten exotische Sphären und Singularitäten. LNM 57. Berlin-Heidelberg-NewYork, Springer 1968.

    CrossRef  MATH  Google Scholar 

  9. Löffler, P.: Über rationale Homologiesphären. Math. Ann. 249, 141–152 (1980).

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. Ribenboim, P.: 13 lectures on Fermat's last theorem. Berlin-Heidelberg-New York, Springer 1979.

    CrossRef  MATH  Google Scholar 

  11. Taylor, M. J.: Locally free classgroups of groups of prime power order. J. of Algebra 50, 463–487 (1978).

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. Wall, C. T. C.: classification of Hermitian forms VI Group rings. Ann. of Math. 103, 1–80 (1976).

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. Madsen, I. und M. Raußen: Smooth and locally linear G-homotopy representations. Aarhus Universitet, Preprint Series 1984/85, No. 22.

    Google Scholar 

  14. Pedersen, E. K.: These Proceedings.

    Google Scholar 

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© 1985 Springer-Verlag

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tom Dieck, T., Löffler, P. (1985). Verschlingung von Fixpunktmengen in Darstellungsformen. I. In: Smith, L. (eds) Algebraic Topology Göttingen 1984. Lecture Notes in Mathematics, vol 1172. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0074431

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  • DOI: https://doi.org/10.1007/BFb0074431

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-16061-8

  • Online ISBN: 978-3-540-39745-8

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