Isotropy groups of homotopy classes of maps

Triantafillou, G.

doi:10.1090/S0002-9947-1992-1044966-8

Isotropy groups of homotopy classes of maps
HTML articles powered by AMS MathViewer

by G. Triantafillou PDF

Trans. Amer. Math. Soc. 334 (1992), 37-48 Request permission

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

A. K. Bousfield and V. K. A. M. Gugenheim, On $\textrm {PL}$ de Rham theory and rational homotopy type, Mem. Amer. Math. Soc. 8 (1976), no. 179, ix+94. MR 425956, DOI 10.1090/memo/0179
Armand Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962), 485–535. MR 147566, DOI 10.2307/1970210
A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR 0365573
A. Borel and J.-P. Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1973), 436–491. MR 387495, DOI 10.1007/BF02566134
E. Dror, W. G. Dwyer, and D. M. Kan, Self-homotopy equivalences of virtually nilpotent spaces, Comment. Math. Helv. 56 (1981), no. 4, 599–614. MR 656214, DOI 10.1007/BF02566229
Phillip A. Griffiths and John W. Morgan, Rational homotopy theory and differential forms, Progress in Mathematics, vol. 16, Birkhäuser, Boston, Mass., 1981. MR 641551
S. Halperin, Lectures on minimal models, Mém. Soc. Math. France (N.S.) 9-10 (1983), 261. MR 736299

Localization of nilpotent groups and spaces

Sara Hurvitz, The automorphism groups of spaces and fibrations, Pacific J. Math. 96 (1981), no. 2, 371–388. MR 637978
James E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975. MR 0396773
Daniel Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205–295. MR 258031, DOI 10.2307/1970725
John W. Rutter, A homotopy classification of maps into an induced fibre space, Topology 6 (1967), 379–403. MR 214070, DOI 10.1016/0040-9383(67)90025-0
Dennis Sullivan, Geometric topology. Part I, Massachusetts Institute of Technology, Cambridge, Mass., 1971. Localization, periodicity, and Galois symmetry; Revised version. MR 0494074
Dennis Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269–331 (1978). MR 646078
Clarence W. Wilkerson, Applications of minimal simplicial groups, Topology 15 (1976), no. 2, 111–130. MR 402737, DOI 10.1016/0040-9383(76)90001-X