Composites of open maps

Decomposability problem on branched coverings

“…In a decomposition of a branched covering which is not a covering at least one of its components is a branched covering which is not a covering. Moreover, since the degree of a decomposable covering is the product of the degrees of its components (see [2], theorem 2.3), we are interested in branched coverings with a non-prime degree.…”

“…Example 2.8. A primitive branched covering like (M, φ, T 1 , {x}, 4) is indecomposable: notice that possible admissible data are [1,3] and [2,2]. But [1,3] is not realized by a decomposable primitive branched covering on T 1 by Corollary 2.7, and every factorization of [2,2] has either a trivial or a non-admissible first factor on T 1 .…”

“…In 1973, Baildon [1] showed that if an open surjective map of finite order between closed surfaces is a composition of k simple open surjective maps, then its order is equal to 2 k . In 2002, Krzempek [15] constructed covering maps on locally arcwise connected continua that are not factorizable into covering maps of order ≤ n − 1, for all n. In 2002, Bogataya, Bogatyȋ and Zieschang [2] extended Baildon's theorem to compositions of arbitrary open maps and showed that the order of a product (composition) is the product of the orders. Moreover, they gave an example of a 4fold covering of a surface of genus 2 by a surface of genus 5 that cannot be represented as a composition of two non-trivial open maps.…”

Indecomposable branched coverings over the projective plane by surfaces $M$ with $χ(M) \leq 0$

“…Such collections are called either admissible data if χ(N) ≤ 0, or nonorientable-admissible if N = RP 2 and the covering surface is nonorientable. Due to [8], these conditions are either ν(D) ≡ 0 (mod 2) if χ(N) ≤ 0, or d − 1 ≤ ν(D) ≡ 0 (mod 2) if N = RP 2 and the covering surface is nonorientable, see (4) for N = RP 2 .…”

Проблема Разложимости Разветвленных Накрытий

“…В 2002 г. С. Богатая, С. Богатый и Х. Цишанг обобщили теорему Бэйлдона для композиций произвольных открытых отображений и показали, что порядок произведения (композиции) равен произведению порядков (см. [5]). Они также построили пример четырехлистного накрытия поверхности рода два поверхностью рода пять, которое не может быть разложено в композицию двух нетривиальных открытых отображений.…”