On the nonexistence of certain branched covers. (English) Zbl 1270.57011
Geom. Topol. 16, No. 3, 1321-1344 (2012); retracted.
The main result in the paper under review is the following: every branched covering \(T^n \to N\) from the \(n\)-torus \(T^n\) to a closed oriented \(n\)-manifold \(N\) is an ordinary covering, provided that \(H^r(N; \mathbb R) \cong H^r(T^n; \mathbb R)\) for some \(1 \leq r < n\).
In particular, for \(n = 4\) and \(N \cong \#_3(S^2 \times S^2)\), the connected sum of three copies of \(S^2 \times S^2\), the authors can conclude that there is no branched cover \(T^4 \to \#_3(S^2 \times S^2)\), while it is known that there are PL maps \(T^4 \to \#_3(S^2 \times S^2)\) of arbitrarily large degree [H. Duan and S. Wang, Acta Math. Sin., Engl. Ser. 20, No. 1, 1–14 (2004; Zbl 1060.57018)].
This shows that the picture changes completely when passing from dimension 3 to 4. In fact, every \(\pi_1\)-surjective map \(M \to N\) between closed, orientable 3-manifolds of degree \(> 3\) is homotopic to a branched cover [A. L. Edmonds, Math. Ann. 245, 273–279 (1979; Zbl 0406.57003)].
[Editorial remark: This article was retracted by the authors due to a gap in the proof of Proposition 3.1. For the retraction note (received 14 February 2019), see https://msp.org/gt/2012/16-3/gt-v16-n3-x02-Branched_covers_and_cohomology_retraction.pdf.]
In particular, for \(n = 4\) and \(N \cong \#_3(S^2 \times S^2)\), the connected sum of three copies of \(S^2 \times S^2\), the authors can conclude that there is no branched cover \(T^4 \to \#_3(S^2 \times S^2)\), while it is known that there are PL maps \(T^4 \to \#_3(S^2 \times S^2)\) of arbitrarily large degree [H. Duan and S. Wang, Acta Math. Sin., Engl. Ser. 20, No. 1, 1–14 (2004; Zbl 1060.57018)].
This shows that the picture changes completely when passing from dimension 3 to 4. In fact, every \(\pi_1\)-surjective map \(M \to N\) between closed, orientable 3-manifolds of degree \(> 3\) is homotopic to a branched cover [A. L. Edmonds, Math. Ann. 245, 273–279 (1979; Zbl 0406.57003)].
[Editorial remark: This article was retracted by the authors due to a gap in the proof of Proposition 3.1. For the retraction note (received 14 February 2019), see https://msp.org/gt/2012/16-3/gt-v16-n3-x02-Branched_covers_and_cohomology_retraction.pdf.]
Reviewer: Riccardo Piergallini (Camerino)
MSC:
| 57M12 | Low-dimensional topology of special (e.g., branched) coverings |
Keywords:
branched coveringReferences:
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