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Nilpotent subgroups of the group of self-homotopy equivalences. (English) Zbl 0735.55003

For a space \(X\) let \({\mathcal E}(X)\) be the group of homotopy classes of self homotopy equivalences. If \(E_ *\) is a homology theory, then \({\mathcal E}(X)\) acts naturally on \(E_ *(X)\). The main result of the paper is as follows:
Let \(E\) be a connective, reduced, multiplicative homology theory such that \(E_ *(S^ 0)\approx {\mathbb{Z}}_ P\) (integers localized at some set of primes \(P\)). Let \(X\) be a connected, nilpotent, finite dimensional \(CW\)-complex and let \(X_ P\) be its \(P\)-localization. Then any subgroup \(G\subset{\mathcal E}(X)\) acting nilpotently on \(E_ *(X_ P)\) is nilpotent.
Let \({\mathcal E}_ E(X)\subset{\mathcal E}(X)\) be the subgroup of all elements acting trivially on \(E_ *(X)\). Then for \(X\) as above it follows that \({\mathcal E}_{MU}(X)\), \({\mathcal E}_{MSp}(X)\) and \({\mathcal E}_{BP}(X_{(p)})\) are nilpotent groups. Here \(p\) is a prime and \(BP\) is the corresponding Brown-Peterson theory.
It is interesting to note that the connectivity of \(E\) is crucial. The authors give an example of a space \(X\) where \(\tilde K_ *(X)=0\) whence \({\mathcal E}_ K(X)={\mathcal E}(X)\), but where this group is not nilpotent.
The method of proof is to use the Adams spectral sequence and the Atiyah- Hirzebruch spectral sequence in order to reduce the problem from general \(E\) to ordinary homology. Then the results of E. Dror and A. Zabrodsky [Topology 18, 187-197 (1979; Zbl 0417.55008)] are applied.

MSC:

55P10 Homotopy equivalences in algebraic topology

Citations:

Zbl 0417.55008
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References:

[1] Adams, J. F., On the group J(X)-IV, Topology, 5, 21-71 (1966) · Zbl 0145.19902 · doi:10.1016/0040-9383(66)90004-8
[2] J. F. Adams,Stable Homotopy and Generalised Homology, Chicago Univ. Press, 1974. · Zbl 0309.55016
[3] Bousfield, A. K., The localization of spectra with respect to homology, Topology, 18, 257-281 (1979) · Zbl 0417.55007 · doi:10.1016/0040-9383(79)90018-1
[4] Dror, E.; Zabrodsky, A., Unipotency and nilpotency in homotopy equivalences, Topology, 18, 187-197 (1979) · Zbl 0417.55008 · doi:10.1016/0040-9383(79)90002-8
[5] Hilton, P. J.; Mislin, G.; Roitberg, J., Localization of nilpotent groups and spaces (1975), Amsterdam: North-Holland, Amsterdam · Zbl 0323.55016
[6] Johnson, D. C.; Wilson, W. S., BP-operation and Morava’s extraordinary K-theories, Math. Z., 144, 55-75 (1975) · Zbl 0309.55003 · doi:10.1007/BF01214408
[7] Shimada, N.; Yagita, N., Multiplications in the complex bordism theory with singularities, Publ. RIMS, Kyoto Univ., 12, 259-293 (1976) · Zbl 0341.57010 · doi:10.2977/prims/1195190968
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