Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Isotropy groups of homotopy classes of maps


Author: G. Triantafillou
Journal: Trans. Amer. Math. Soc. 334 (1992), 37-48
MSC: Primary 55S37; Secondary 55P62
DOI: https://doi.org/10.1090/S0002-9947-1992-1044966-8
MathSciNet review: 1044966
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \operatorname{aut}(X)$ be the group of homotopy classes of self-homotopy equivalences of a space $ X$ and let $ [f] \in [X,Y]$ be a homotopy class of maps from $ X$ to $ Y$ . The aim of this paper is to prove that under certain nilpotency and finiteness conditions the isotropy group $ \operatorname{aut}{(X)_{[f]}}$ of $ [f]$ under the action of $ \operatorname{aut}(X)$ on $ [X,Y]$ is commensurable to an arithmetic group. Therefore $ \operatorname{aut}{(X)_{[f]}}$ is a finitely presented group by a result of Borel and Harish-Chandra.


References [Enhancements On Off] (What's this?)

  • [B-G] A. K. Bousfield and V. K. A. M. Gugenheim, On $ PL$ de Rham theory and rational homotopy type, Mem. Amer. Math. Soc., vol. 8, no. 179, 1976. MR 0425956 (54:13906)
  • [B-HC] A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962), 485-535. MR 0147566 (26:5081)
  • [B-K] A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Math., vol. 304, Springer-Verlag, 1972. MR 0365573 (51:1825)
  • [B-S] A. Borel and J. P. Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1973), 436-491. MR 0387495 (52:8337)
  • [D-D-K] E. Dror, W. C. Dwyer and D. M. Kan, Self homotopy equivalences of virtually nilpotent spaces, Comment. Math. Helv. 56 (1981), 599-614. MR 656214 (84h:55005)
  • [G-M] P. A. Griffiths and J. W. Morgan, Rational homotopy theory and differential forms, Birkhäuser Verlag, Basel, 1981. MR 641551 (82m:55014)
  • [H] S. Halperin, Lectures on minimal models, Mem. Soc. Math. France 9 (1983). MR 736299 (85i:55009)
  • [H-M-R] P. J. Hilton, G. Mislin and J. Roitberg, Localization of nilpotent groups and spaces, North-Holland Math. Studies, no. 15, North-Holland, 1975.
  • [Hr] S. Hurwitz, The automorphism groups of spaces and fibrations, Pacific J. Math. 96 (1981), 371-388. MR 637978 (83i:55022)
  • [Hu] J. E. Humphreys, Linear algebraic groups, Graduate Texts in Math., vol. 21, Springer-Verlag, 1975. MR 0396773 (53:633)
  • [Q] D. Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205-295. MR 0258031 (41:2678)
  • [R] J. W. Rutter, A homotopy classification of maps into an induced fibre space, Topology 6 (1967), 379-403. MR 0214070 (35:4922)
  • [SI] D. Sullivan, Geometric topology, part I: Localization, periodicity and Galois symmetry, M.I.T. Press, Cambridge, Mass., 1970. MR 0494074 (58:13006a)
  • [S2] -, Infinitesimal computations in topology, Publ. Math. Inst. Hautes Études Sci. 47 (1978), 269-331. MR 0646078 (58:31119)
  • [W] C. W. Wilkerson, Minimal simplicial groups, Topology 15 (1976), 111-130. MR 0402737 (53:6551)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 55S37, 55P62

Retrieve articles in all journals with MSC: 55S37, 55P62


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1044966-8
Keywords: Homotopy self-homotopy equivalence, arithmetic group, minimal model
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society