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Globally invertible differentiable or holomorphic maps. (English) Zbl 1165.32305

Fix an integer \(n\geq 5\). Let \(M_{\text{top}}\) be a compact connected topological \(n\)-dimensional manifold which admits a differentiable structure and with \(\pi_i(M_{\text{top}},P)=0\) for every \(i\geq 2\), where \(P\in M_{\text{top}}\) is any point. The author proves that there exists a differentiable structure \(M\) on \(M_{\text{top}}\) and a locally invertible differentiable map \(f:M\rightarrow M\) with \(\deg (f)\geq 2\) if and only if \(\pi_1(M_{\text{top}},P)\) contains a proper subgroup \(H\) of finite index with \(H\cong \pi_1(M_{\text{top}},P)\) as abstract groups.
Then the author considers the same problem for holomorphic maps. Let \(X\) be a compact complex surface such that there exists a holomorphic locally invertible map \(\pi \colon X\rightarrow X\) with \(\deg (\pi )>1\). The author proves that \(X\) belongs to one of the following classes: (i) \(X\cong E\times B\) with \(E\) an elliptic curve and \(B\) a smooth curve of genus \(\geq 2\); (ii) \(X\) is a torus; (iii) \(X\) is a hyperelliptic surface; (iv) \(X\) is a minimal ruled surface over an elliptic curve; (v) \(X\) is one of the non-Kähler surfaces without curves and with \(b_1(X)=1\), constructed by Inoue. Moreover, every product \(E\times B\) as above, every torus, every hyperelliptic surface and every Inoue surface as in (v) has such a nontrivial covering. Some, but not all, minimal ruled surfaces over an elliptic curve have such a nontrivial covering and they are completely described.

MSC:

32J15 Compact complex surfaces
58C10 Holomorphic maps on manifolds
58C25 Differentiable maps on manifolds
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References:

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