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Self-homotopy equivalences of product spaces

Published online by Cambridge University Press:  14 November 2011

Petar Pavešić
Petar Pavešić
Affiliation:
Department of Mathematics, University of Ljubljana, Jadranska 19, Ljubljana, Slovenia (petar.pavesic@uni-lj.si)
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Extract

The group of self-homotopy equivalences Aut(X ć Y) is represented as a product of two subgroups under the assumption that the self-equivalences of X ć Y can be diagonalized. Moreover, an analogous result holds for the subgroup Aut#(X ć Y) of self-equivalences, which induce identity automorphisms on homotopy groups. Other methods for the computation of Aut(X ć Y) are studied, especially when the spaces involved have an H- or coH-structure, and several examples are considered, among others, some non-simply connected H-spaces of rank 2.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 1 , 1999 , pp. 181 - 197
Copyright
Copyright © Royal Society of Edinburgh 1999

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References

1Amberg, B., Pranciosi, S. and De Giovanni, F.. Products of groups (Oxford: Clarendon, 1992).Google Scholar
2Arkowitz, M.. The group of self-homotopy equivalences—a survey. In Group of selfequivalences and related topics (ed. Piccinini, R. A.), pp. 170–203. Lecture Notes in Mathematics 1425 (Springer, 1990).CrossRefGoogle Scholar
3Arkowitz, M. and Lupton, G.. On finiteness of self-homotopy equivalences. In Contemporary mathematics (ed. Cenkl, M. and Miller, H.), vol. 181, pp.1–25 (Providence, RI: AMS, 1995).Google Scholar
4Booth, P. and Heath, P.. On the groups ɛ(X x Y) and ɛBB(X x B Y). In Group of selfequivalences and related topics (ed. Piccinini, R. A.), pp. 17–31. Lecture Notesin Mathematics 1425 (Springer, 1990).CrossRefGoogle Scholar
5Cartan, H.. Classes d'applications d'un espace dans un groupe topologique. Seminaire H. Cartan 1962/1963, exp. 6.Google Scholar
6Didierjean, G.. Homotopie de l'espace des équivalences d'homotopie fibrees. Ann. Inst. Fourier 35 (1985), 33–47.CrossRefGoogle Scholar
7Dold, A.. Partitions of unity in the theory of fibrations. Ann. Math. 78 (1963), 223–255.CrossRefGoogle Scholar
8Dror, E. and Zabrodsky, A.. Unipotency and nilpotency in homotopy equivalences. Topology 18 (1979), 187–197.CrossRefGoogle Scholar
9Dror, E., Dwyer, W. G. and Kan, D. M.. Self-homotopy equivalences of virtually nilpotent spaces. Comm. Math. Helv. 56 (1981), 599–614.CrossRefGoogle Scholar
10Heath, P.. On the group ɛ (X × Y) of self-homotopy equivalences of a product. Quaest. Math. 19 (1996), 433–451.CrossRefGoogle Scholar
11Hilton, P.. Homotopy theory and duality (New York: Gordon and Breach, 1965).Google Scholar
12Kahn, P. J.. Self-equivalences of (n – l)-connected 2n-manifolds. Math. Ann. 180 (1969), 26–47.CrossRefGoogle Scholar
13Koh, S. S.. Note on the homotopy properties of the components of the mapping space X sp Proc. Am. Math. Soc. 11 (1960), 896–904.Google Scholar
14May, J. P.. Simplicial objects in algebraic topology (Princeton, NJ: Van Nostrand, 1967).Google Scholar
15Oka, S., Sawashita, N. and Sugawara, M.. On the group of self-equivalences of a mapping cone. Hiroshima Math. J. 4 (1974), 9–28.CrossRefGoogle Scholar
16Roitberg, J.. Residually finite, Hopfian and coHopfian spaces. In Contemporary Mathematics 37, pp. 131–144 (Providence, RI: AMS, 1985).Google Scholar
17Sawashita, N.. On the group of self-equivalences of the product of spheres. Hiroshima Math. J. 5 (1975), 69–86.CrossRefGoogle Scholar
18Sieradski, A. J.. Twisted self-homotopy equivalences. Pacific J. Math. 3 (1970), 789–802.CrossRefGoogle Scholar
19Toda, H.. Composition methods in homotopy groups of spheres. Annals of Mathematical Studies, vol. 49 (Princeton University Press, 1962).Google Scholar
20Whitehead, G. W.. Elements of homotopy theory. GTM 61 (Springer, 1978).CrossRefGoogle Scholar
21Yamanoshita, T.. On the spaces of self-homotopy equivalences of certain CW-complexes. J. Math. Soc. Jpn 37 (1985), 455–470.CrossRefGoogle Scholar