Branched covering representation of nonorientable 4-manifolds

Valentina Bais (SISSA (Trieste))

Monday 1st December, 2025 15:00-16:00
Maths 311B

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According to Berstein and Edmonds, R. H. Fox showed in unpublished work that every closed connected non-orientable 2n-dimensional PL manifold admits a branched covering over the real projective 2n-space, for every n ∈ N.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=4. 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. 

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