Open and discrete maps with piecewise linear branch set images are piecewise linear maps

Abstract

The image of the branch set of a piecewise linear (PL)-branched cover between PL n-manifolds is a simplicial (n − 2)-complex. We demonstrate that the reverse implication also holds: an open and discrete map f : S n → S n with the image of the branch set contained in a simplicial (n − 2)-complex is equivalent up to homeomorphism to a PL-branched cover.

Quasiregular curves and cohomology

“…For an example of a quasiregular curve against a power of a Kähler form, recall that there exists a nonconstant quasiregular map 𝑓 ∶ ℝ 2𝑚 → ℂ𝑃 𝑚 for 𝑚 ⩾ 1 by [10,Proposition 1.3]. For 𝓁 ⩾ 𝑚 and 𝜄 ∶ ℂ𝑃 𝑚 → ℂ𝑃 𝓁 the natural embedding (see Lemma 7.4), the composition 𝜄•𝑓 ∶ ℝ 2𝑚 → ℂ𝑃 𝓁 is a nonconstant quasiregular 𝜔 ∧𝑚 sym -curve.…”

“…For an example of a quasiregular curve against a power of a Kähler form, recall that there exists a nonconstant quasiregular map $$f\colon \mathbb {R}^{2m} \rightarrow \mathbb {C}P^m$$ for $$m\geqslant 1$$ by [10, Proposition 1.3]. For $$\ell \geqslant m$$ and $$\iota \colon \mathbb {C}P^m \rightarrow \mathbb {C}P^\ell$$ the natural embedding (see Lemma 7.4), the composition $$\iota \circ f\colon \mathbb {R}^{2m}\rightarrow \mathbb {C}P^\ell$$ is a nonconstant quasiregular $$\omega _\mathrm{sym}^{\wedge m}$$‐curve.…”