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On the homotopy groups of the self-equivalences of linear spheres

Published online by Cambridge University Press:  26 October 2015

Assaf Libman
Assaf Libman*
Affiliation:
Institute of Mathematics, University of Aberdeen, Fraser Noble Building, Aberdeen AB24 3UE, UK ([email protected])
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Abstract

Let S(V) be a complex linear sphere of a finite group G. Let S(V)*n denote the n-fold join of S(V) with itself and let aut G(S(V)*) denote the space of G-equivariant self-homotopy equivalences of S(V)*n. We show that for any k ≥ 1 there exists M > 0 that depends only on V such that |πk autG(S(V)*n)|≤ M for all n ≫0.

Keywords

homotopy groupsself-equivalencesequivariant spherescomplex representationsPrimary 55Q5255P9155N91
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 2 , May 2016 , pp. 445 - 462
Copyright
Copyright © Edinburgh Mathematical Society 2016 

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References

1. Becker, J. C. and Schultz, R. E., Spaces of equivariant self-equivalences of spheres, Bull. Am. Math. Soc. 79 (1973), 158–162.Google Scholar
2. Bousfield, A. K. and Kan, D. M., Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Volume 304 (Springer, 1972).Google Scholar
3. Bredon, G. E., Equivariant cohomology theories, Lecture Notes in Mathematics, Volume 34 (Springer, 1967).Google Scholar
4. Dwyer, W. G., Hirschhorn, P. S., Kan, D. M. and Smith, J. H., Homotopy limit functors on model categories and homotopical categories, Mathematical Surveys and Monographs, Volume 113 (American Mathematical Society, Providence, RI, 2004).Google Scholar
5. Hauschild, H., Äquivariante homotopie, I, Arch. Math. 29(2) (1977), 158–165.CrossRefGoogle Scholar
6. Illman, S., Smooth equivariant triangulations of G-manifolds for G a finite group, Math. Annalen 233(3) (1978), 199–220.CrossRefGoogle Scholar
7. Klaus, M., Constructing free actions of p-groups on products of spheres, Alg. Geom. Topol. 11(5) (2011), 3065–3084.Google Scholar
8. Møller, J. M., On equivariant function spaces, Pac. J. Math. 142(1) (1990), 103–119.Google Scholar
9. Munkres, J. R., Elements of algebraic topology (Addison-Wesley, 1984).Google Scholar
10. Rutter, J. W., Spaces of homotopy self-equivalences: a survey, Lecture Notes in Mathematics, Volume 1662 (Springer, 1997).Google Scholar
11. Schultz, R., Homotopy decompositions of equivariant function spaces, I, Math. Z. 131 (1973), 49–75.CrossRefGoogle Scholar
12. Schultz, R., Homotopy decompositions of equivariant function spaces, II, Math. Z. 132 (1973), 69–90.Google Scholar
13. Spanier, E. H., Algebraic topology (McGraw-Hill, 1966).Google Scholar
14. Ünlü, Ö. and Yalçin, E., Constructing homologically trivial actions on products of spheres, Indiana Univ. Math. J. 62(3) (2013), 927–945.Google Scholar
15. Weibel, C. A., An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, Volume 38 (Cambridge University Press, 1994).Google Scholar
16. Whitehead, G. W., Elements of homotopy theory, Graduate Texts in Mathematics, Volume 61 (Springer, 1978).Google Scholar