Abstract:We show that every closed connected non-orientable PL $4$-manifold $X$ is a simple branched covering of $\RP^4$. We also show that $X$ is a simple branched covering of the twisted $S^3$-bundle $S^1 \simtimes S^3$ if and only if the first Stiefel--Whitney class $w_1(X)$ admits an integral lift. In both cases, the degree of the covering can be any number $d \geq 4$, provided that $d$ has the same parity of the Stiefel--Whitney number $w_1^4[X]$ in the case of $\RP^4$. Moreover, the branch set can be assumed to be non-singular if $d \geq 5$ and to have just nodal singularities if $d=4$.
| Subjects: | Geometric Topology (math.GT) |
| Cite as: | arXiv:2509.09319 [math.GT] |
| (or arXiv:2509.09319v2 [math.GT] for this version) | |
| https://doi.org/10.48550/arXiv.2509.09319 arXiv-issued DOI via DataCite |
From: Valentina Bais [view email]
[v1]
Thu, 11 Sep 2025 10:08:16 UTC (627 KB)
[v2]
Mon, 25 May 2026 21:06:23 UTC (1,287 KB)
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