## Finiteness properties of self-equivalence groups of rational $$co$$-$$H$$- spaces.(English)Zbl 0762.55011

Given a nilpotent complex $$X$$ which is rationally a bouquet of spheres, a co-$$H_ 0$$-space, the author investigates finiteness properties involving the group $$E(X)$$ of the self-homotopy equivalences of $$X$$ and the canonical subgroups $$E_ \pi(X)$$ and $$E_ H(X)$$, self-equivalences inducing the identity in homotopy and homology respectively. Besides, if $$X$$ is 1-connected, explicit formulae for the rank of $$E_ H(X)$$ and $$E(X)$$ in terms of the rational homotopy and homology of $$X$$ are displayed. The author also shows through a list of examples the necessity of $$X$$ being finite and that $$E_ \pi(X)$$ is not always finite for non co-$$H_ 0$$-spaces.
Reviewer: F.Gomez (Malaga)

### MSC:

 55P62 Rational homotopy theory 55P10 Homotopy equivalences in algebraic topology 55P45 $$H$$-spaces and duals
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### References:

 [1] M. Arkowitz and C. R. Curjel.: Groups of homotopy classes. (Lecture Notes in Math., 4) Springer-Verlag 1966 [2] M. Arkowitz and C. R. Curjel.: The Hurewicz homomorphism and finite homotopy invariants. Trans. Amer. Math. Soc.110, 538–551 (1964) · Zbl 0132.19203 [3] W. D. Barcus and M. G. Barratt.: On the homotopy classification of a fixed map. Trans. Amer. Math. Soc.88, 57–74 (1958) · Zbl 0095.16801 [4] E. Dror and A. Zabrodsky.: Unipotency and nilpotency in homotopy equivalences. Topology18, 187–197 (1979) · Zbl 0417.55008 [5] P. Hilton, G. Mislin and J. Roitberg.: Localization of nilpotent groups and spaces. (Mathematics Studies, 15) North Holland 1975 · Zbl 0323.55016 [6] K. Maruyama.: Localization of a certain subgroup of self-homotopy equivalences. Pacific J. Math.136 293–301 (1989) · Zbl 0673.55006 [7] K. Maruyama.: Localization of self-homotopy equivalences inducing the identity on homology. Math. Proc. Camb. Phil. Soc.108 291–297 (1990) · Zbl 0718.55006 [8] J. Mukai.: Stable homotopy of some elementary complexes. Mem. Fac. Sci., Kyushu University Ser. A20 266–282 (1966) · Zbl 0187.20302 [9] S. Oka, N. Sawashita and M. Sugawara.: On the group of self-homotopy equivalences of a mapping cone. Hiroshima Math. J.4 9–28 (1974) · Zbl 0284.55013 [10] R. Piccinini (Ed).: Groups of self-equivalences and related topics. (Lecture Notes in Math., 1425) Springer-Verlag 1988 [11] K. Tsukiyama.: Self-homotopy-equivalences of a space with two nonvanishing homotopy groups. Proc. Amer. Math. Soc.79, 134–138 (1980) · Zbl 0389.55014 [12] C. W. Wilkerson.: Application of minimal simplicial groups. Topology15, 111–130 (1976) · Zbl 0345.55011 [13] A. Zabrodsky.: Endomorphisms in the homotopy category. Cont. Math. Amer. Math. Soc.44, 227–277 (1985) · Zbl 0583.55007 [14] A. Zabrodsky.: Hopf spaces. (Mathematics Studies, 22) North Holland 1976
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