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Minimal 2-complexes and the D(2)-problem

Author: F. E. A. Johnson
Journal: Proc. Amer. Math. Soc. 132 (2004), 579-586
MSC (2000): Primary 55M05, 57M20; Secondary 16D70
Published electronically: September 5, 2003
MathSciNet review: 2022384
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Abstract: We show that when $n\geq 5$ there is a minimal algebraic $2$-complex over the quaternion group $Q(2^n)$ which is not homotopy equivalent to the Cayley complex of the standard minimal presentation. This raises the possibility that Wall's D(2)-property might fail for $Q(2^n)$.

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  • 1. W. Browning, Homotopy types of certain finite C.W. complexes with finite fundamental group. Ph.D. Thesis, Cornell University, 1978.
  • 2. M. N. Dyer and A. J. Sieradski, Trees of homotopy types of two-dimensional CW-complexes. Comment. Math. Helv. 48 (1973), 31-44. MR 51:14074
  • 3. D.B.A. Epstein, Finite presentations of groups and 3-manifolds. Quart. J. Math. Oxford Ser. (2) 12 (1961), 205-212. MR 26:1867
  • 4. M. Gutierrez and M. P. Latiolais, Partial homotopy type of finite two-complexes. Math. Zeit. 207 (1991) 359-378. MR 92h:55007
  • 5. F.E.A. Johnson, Stable modules and the structure of Poincaré $3$-complexes. Geometry and Topology, Aarhus (Proceedings of the $6^{th}$ Aarhus Topology Conference). Contemp. Math. 258 (2000), 227-248. MR 2001e:57001
  • 6. F.E.A. Johnson, Stable modules and Wall's D(2) problem. Comment. Math. Helv. 78 (2003), 18-44.
  • 7. R. G. Swan, Periodic resolutions for finite groups. Ann. of Math. 72 (1960) 267-291. MR 23:A2205
  • 8. R. G. Swan, Projective modules over binary polyhedral groups. J. reine angew. Math. 342 (1983) 66-172. MR 84j:16003
  • 9. M. F. Vigneras, Simplification pour les ordres des corps de quaternions totalement définis. J. reine angew. Math. 286/287 (1976) 257-277. MR 55:2851
  • 10. C.T.C. Wall, Finiteness conditions for CW-complexes. Ann. of Math. 81 (1965) 56-69. MR 30:1515

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Additional Information

F. E. A. Johnson
Affiliation: Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom

Keywords: Algebraic $2$-complexes, non-cancellation, minimal presentations
Received by editor(s): December 28, 2000
Received by editor(s) in revised form: August 22, 2002
Published electronically: September 5, 2003
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2003 American Mathematical Society

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