The \(D(2)\)-problem for dihedral groups of order \(4n\). (English) Zbl 1262.57002

The following question was first posed by Wall and it is known as the \(D(2)\)-problem: Let \(X\) be a finite connected \(3\)-dimensional \(CW\)-complex, with universal cover \(\widetilde{X},\) such that \(H_{3}(\widetilde {X};\mathbb{Z})=H^{3}(X;\mathcal{B})=0\) for all coefficient systems \(\mathcal{B}\) on \(X.\) Is it true that \(X\) is homotopy equivalent to a finite \(2\)-dimensional \(CW\)-complex?
It is said that the \(D(2)\) property holds for a finitely presented group \(G\) if the above question is answered in the affirmative for every \(X\) with \(\pi_{1}(X)\simeq G.\) It is also said that torsion-free cancellation holds for a group ring \(\mathbb{Z}(G)\) if \(X\oplus M\simeq X\oplus N\Longrightarrow M\simeq N\) for any \(\mathbb{Z}(G)\)-lattices \(X,\) \(M\) and \(N.\) In the present work it is proven that the \(D(2)\)-propery holds for \(D_{4n},\) the dihedral group of order \(4n,\) provided that \(\mathbb{Z}(D_{4n})\) satisfies torsion-free cancellation.


57M05 Fundamental group, presentations, free differential calculus
55P15 Classification of homotopy type
Full Text: DOI


[1] W J Browning, Homotopy types of certain finite CW-complexes with finite fundamental group, PhD thesis, Cornell University (1978)
[2] P J Davis, Circulant matrices, Wiley-Interscience (1979) · Zbl 0418.15017
[3] S Endô, T Miyata, On the class groups of dihedral groups, J. Algebra 63 (1980) 548 · Zbl 0436.16006
[4] K W Gruenberg, Homotopy classes of truncated projective resolutions, Comment. Math. Helv. 68 (1993) 579 · Zbl 0806.55006
[5] M Gutierrez, M P Latiolais, Partial homotopy type of finite two-complexes, Math. Z. 207 (1991) 359 · Zbl 0712.55007
[6] I Hambleton, M Kreck, Cancellation of lattices and finite two-complexes, J. Reine Angew. Math. 442 (1993) 91 · Zbl 0779.57002
[7] F E A Johnson, Stable modules and the \(D(2)\)-problem, London Math. Soc. Lect. Note Series 301, Cambridge Univ. Press (2003) · Zbl 1055.57002
[8] M P Latiolais, When homology equivalence implies homotopy equivalence for \(2\)-complexes, J. Pure Appl. Algebra 76 (1991) 155 · Zbl 0761.55005
[9] W H Mannan, The \(D(2)\) property for \(D_8\), Algebr. Geom. Topol. 7 (2007) 517 · Zbl 1151.57001
[10] W H Mannan, Realizing algebraic 2-complexes by cell complexes, Math. Proc. Cambridge Philos. Soc. 146 (2009) 671 · Zbl 1165.57301
[11] R G Swan, Torsion free cancellation over orders, Illinois J. Math. 32 (1988) 329 · Zbl 0666.16002
[12] C T C Wall, Finiteness conditions for \(\mathrm{CW}\)-complexes, Ann. of Math. 81 (1965) 56 · Zbl 0152.21902
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.