Algebra/Topology Seminar

Speaker: Valentina Bais

Title: Branched covering representation of nonorientable 4-manifolds

Abstract: According to Berstein and Edmonds, R. H. Fox showed in unpublished work that every closed connected non-orientable 2n-dimensional PL manifold admits a branched covering over the real projective 2n-space, for every n ∈ N. However, no control is given on the degree and on the regularity of the branch set. In a joint work with Riccardo Piergallini and Daniele Zuddas, we improve such result for n=2. More precisely, we show that every closed connected non-orientable PL 4-manifold X is a simple branched covering of RP^4 , where the degree can be chosen to be any number d ⩾ 4 with the same parity of the Stiefel–Whitney number w_1^4 [X]. Moreover, we show that the branch set can be assumed to be non-singular if d ⩾ 5 and to have just nodal singularities if d = 4.