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**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.

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.

Reviewer: Charalampos Charitos (Athens)

### MSC:

57M05 | Fundamental group, presentations, free differential calculus |

55P15 | Classification of homotopy type |

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\textit{S. O'Shea}, Algebr. Geom. Topol. 12, No. 4, 2287--2297 (2012; Zbl 1262.57002)

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