Branched coverings of \(\mathbb{C} \mathrm{P}^2\) and other basic 4-manifolds. (English) Zbl 1487.57026

In [R. Piergallini, Topology 34, No. 3, 497–508 (1995; Zbl 0869.57002)], the first author proved that any closed orientable PL \(4\)-manifold is a branched covering of \(S^4\). In this paper, it is shown that there is a closed PL \(4\)-manifold which is not a branched covering of \(\mathbb{C}P^2\), namely, a closed connected oriented PL \(4\)-manifold \(M\) is a branched covering of \(\mathbb{C}P^2\) if and only if \(b_2^+(M) \geq 1\), where \(b_2^+(M)\) means the maximal dimension of a vector subspace of \(H_2(M;\mathbb{R})\), in which the intersection form of \(M\) is positive definite. Furthermore, in general, necessary and sufficient conditions for a closed connected oriented PL \(4\)-manifold to be a branched covering of \(\#_m \mathbb{C}P^2 \#_n \overline{\mathbb{C}P^2}\), \(\#_n (S^2 \times S^2)\) and \(\#_n(S^3 \times S^1)\) are given.


57M12 Low-dimensional topology of special (e.g., branched) coverings
57K40 General topology of 4-manifolds
57Q05 General topology of complexes


Zbl 0869.57002
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