## 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.

### MSC:

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

### Keywords:

4-manifold; branched covering

Zbl 0869.57002
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