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Every finite group is the group of self-homotopy equivalences of an elliptic space

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Acta Mathematica

Abstract

We prove that every finite group G can be realized as the group of self-homotopy equivalences of infinitely many elliptic spaces X. To construct those spaces we introduce a new technique which leads, for example, to the existence of infinitely many inflexible manifolds. Further applications to representation theory will appear in a separate paper.

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References

  1. Amann, M., Mapping degrees of self-maps of simply-connected manifolds. Preprint, 2011. arXiv:1109.0960 [math.AT].

  2. Arkowitz, M., The group of self-homotopy equivalences—a survey, in Groups of Self- Equivalences and Related Topics (Montreal, PQ, 1988), Lecture Notes in Math., 1425, pp. 170–203. Springer, Berlin–Heidelberg, 1990.

  3. Arkowitz, M., Problems on self-homotopy equivalences, in Groups of Homotopy Self-Equivalences and Related Topics (Gargnano, 1999), Contemp. Math., 274, pp. 309–315. Amer. Math. Soc., Providence, RI, 2001.

  4. Arkowitz M., Lupton G.: Rational obstruction theory and rational homotopy sets. Math. Z. 235, 525–539 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Barge J.: Structures diffërentiables sur les types d’homotopie rationnelle simplement connexes. Ann. Sci. École Norm. Sup. 9, 469–501 (1976)

    MathSciNet  MATH  Google Scholar 

  6. Benkhalifa M.: Rational self-homotopy equivalences and Whitehead exact sequence. J. Homotopy Relat. Struct. 4, 111–121 (2009)

    MathSciNet  MATH  Google Scholar 

  7. Benkhalifa M.: Realizability of the group of rational self-homotopy equivalences. J. Homotopy Relat. Struct. 5, 361–372 (2010)

    MathSciNet  MATH  Google Scholar 

  8. Bollobás, B., Modern Graph Theory. Graduate Texts in Mathematics, 184. Springer, New York, 1998.

  9. Copeland A. H. Jr., Shar A. O.: Images and pre-images of localization maps. Pacific J. Math. 57, 349–358 (1975)

    Article  MathSciNet  Google Scholar 

  10. Costoya, C. & Viruel, A., Faithful actions on commutative differential graded algebras and the group isomorphism problem. To appear in Q. J. Math.

  11. Crowley, D. & Lӧh, C., Functorial semi-norms on singular homology and (in)flexible manifolds. Preprint, 2011. arXiv:1103.4139 [math.GT].

  12. Federinov J., Felix Y.: Realization of 2-solvable nilpotent groups as groups of classes of homotopy self-equivalences. Topology Appl. 154, 2425–2433 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Félix, Y., La dichotomie elliptique-hyperbolique en homotopie rationnelle. Astérisque, 176 (1989).

  14. Félix, Y., Problems on mapping spaces and related subjects, in Homotopy Theory of Function Spaces and Related Topics, Contemp. Math., 519, pp. 217–230. Amer. Math. Soc., Providence, RI, 2010.

  15. Félix, Y., Halperin, S. & Thomas, J. C., Rational Homotopy Theory. Graduate Texts in Mathematics, 205. Springer, New York, 2001.

  16. Frucht R.: Herstellung von Graphen mit vorgegebener abstrakter Gruppe. Compos. Math. 6, 239–250 (1939)

    MathSciNet  Google Scholar 

  17. Frucht R.: Graphs of degree three with a given abstract group. Canadian J. Math., 1, 365–378 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gromov, M., Metric Structures for Riemannian and Non-Riemannian Spaces. Progress in Mathematics, 152. Birkhäuser, Boston, MA, 1999.

  19. Halperin S.: Finiteness in the minimal models of Sullivan. Trans. Amer. Math. Soc. 230, 173–199 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kahn D. W.: Realization problems for the group of homotopy classes of self-equivalences. Math. Ann. 220, 37–46 (1976)

    Article  MathSciNet  Google Scholar 

  21. Kahn, D. W., Some research problems on homotopy-self-equivalences, in Groups of Self-Equivalences and Related Topics (Montreal, PQ, 1988), Lecture Notes in Math., 1425, pp. 204–207. Springer, Berlin–Heidelberg, 1990.

  22. Maruyama K.: Finite complexes whose self-homotopy equivalence groups realize the infinite cyclic group. Canad. Math. Bull. 37, 534–536 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  23. Nešetřil, J., Homomorphisms of derivative graphs. Discrete Math., 1 (1971/72), 257–268.

  24. Oka S.: Finite complexes whose self-homotopy equivalences form cyclic groups. Mem. Fac. Sci. Kyushu Univ. Ser. A 34, 171–181 (1980)

    MathSciNet  MATH  Google Scholar 

  25. Puppe, V., Simply connected 6-dimensional manifolds with little symmetry and algebras with small tangent space, in Prospects in Topology (Princeton, NJ, 1994), Ann. of Math. Stud., 138, pp. 283–302. Princeton Univ. Press, Princeton, NJ, 1995.

  26. Rutter, J.W., Spaces of Homotopy Self-Equivalences. Lecture Notes in Mathematics, 1662. Springer, Berlin–Heidelberg, 1997.

  27. Sullivan D.: Infinitesimal computations in topology. Inst. Hautes Études Sci. Publ. Math. 47, 269–331 (1977)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Departamento de Computación, Álxebra, Universidade da Coruña, Campus de Elviña, ES-15071, A Coruña, Spain

    Cristina Costoya

  2. Departamento de Álgebra, Geometría y Topología, Universidad de Málaga, Campus de Teatinos, ES-29071, Málaga, Spain

    Antonio Viruel

Authors
  1. Cristina Costoya
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  2. Antonio Viruel
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Corresponding author

Correspondence to Antonio Viruel.

Additional information

The first author was partially supported by Ministerio de Ciencia e Innovacióon (European FEDER support included), grant MTM2009-14464-C02-01. The second author was partially supported by Ministerio de Ciencia e Innovacióon (European FEDER support included), grant MTM2010-18089, and JA grants FQM-213 and P07-FQM-2863. Both authors were partially supported by Xunta de Galicia grant EM2013/16.

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Costoya, C., Viruel, A. Every finite group is the group of self-homotopy equivalences of an elliptic space. Acta Math 213, 49–62 (2014). https://doi.org/10.1007/s11511-014-0115-4

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  • DOI: https://doi.org/10.1007/s11511-014-0115-4

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