Document Zbl 0962.20026 - zbMATH Open
Automorphism groups of nilpotent groups and spaces. (English) Zbl 0962.20026
In this article the following main results are obtained.
Theorem. There exist finitely generated, torsion-free nilpotent groups \(G_1\), \(G_2\) in the same localization genus whose automorphism groups \(\operatorname{Aut}(G_1)\), \(\operatorname{Aut}(G_2)\) are not isomorphic.
Corollary. There exist finite, aspherical CW-complexes (indeed, closed smooth manifolds) \(Y_1\), \(Y_2\) in the same localization genus whose automorphism groups \(\operatorname{Aut}(Y_1)\), \(\operatorname{Aut}(Y_2)\) are not isomorphic. More precisely, \(Y_1\), \(Y_2\) are closed, smooth manifolds which are Eilenberg-MacLane spaces \(K(G_1,1)\), \(K(G_2,1)\), where \(G_1\), \(G_2\) are groups of the kind occuring in the theorem.
MSC:
| 20F28 | Automorphism groups of groups |
| 20F18 | Nilpotent groups |
| 55P10 | Homotopy equivalences in algebraic topology |
| 55P60 | Localization and completion in homotopy theory |
References:
| [1] | Arkowitz, M.; Lupton, G., Equivalence classes of homotopy-associative comultiplications of finite complexes, J. Pure Appl. Algebra, 102, 109-136 (1995) · Zbl 0862.57026 |
| [2] | Baumslag, G., Automorphism groups of nilpotent groups, Amer. J. Math., 91, 1003-1011 (1969) · Zbl 0208.03203 |
| [3] | G. Baumslag, Lecture Notes on Nilpotent Groups, AMS Regional Conference Series in Mathematics (2), 1971.; G. Baumslag, Lecture Notes on Nilpotent Groups, AMS Regional Conference Series in Mathematics (2), 1971. · Zbl 0241.20001 |
| [4] | Donaldson, S. K., An application of gauge theory to the topology of 4-manifolds, J. Diff. Geom., 18, 279-315 (1983) · Zbl 0507.57010 |
| [5] | P. Hall, Nilpotent groups, Canadian Mathematical Congress, Summer Seminar, University of Alberta, 1957.; P. Hall, Nilpotent groups, Canadian Mathematical Congress, Summer Seminar, University of Alberta, 1957. · Zbl 0211.34201 |
| [6] | P. Hilton, G. Mislin, J. Roitberg, Localization of nilpotent groups and spaces, Notas de Matemática, North Holland Math. Studies, vol. 15, North-Holland, Amsterdam, 1975.; P. Hilton, G. Mislin, J. Roitberg, Localization of nilpotent groups and spaces, Notas de Matemática, North Holland Math. Studies, vol. 15, North-Holland, Amsterdam, 1975. · Zbl 0323.55016 |
| [7] | J. Milnor, On simply connected 4-manifolds, Symp. Internacional de Topologia Algebraica, Mexico, 1958, pp. 122-128.; J. Milnor, On simply connected 4-manifolds, Symp. Internacional de Topologia Algebraica, Mexico, 1958, pp. 122-128. · Zbl 0105.17204 |
| [8] | P.F. Pickel, On the isomorphism problem for finitely generated, torsion-free nilpotent groups, Rice University doctoral dissertation, 1970.; P.F. Pickel, On the isomorphism problem for finitely generated, torsion-free nilpotent groups, Rice University doctoral dissertation, 1970. |
| [9] | Roitberg, J., Genus and symmetry in homotopy theory, Math. Ann., 305, 381-386 (1996) · Zbl 0864.55004 |
| [10] | Wilkerson, C. W., Applications of minimal simplicial groups, Topology, 15, 111-130 (1976) · Zbl 0345.55011 |
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