A cellular structure of the space of branched coverings for the two-dimensional sphere. (English. Russian original) Zbl 1481.57024
St. Petersbg. Math. J. 32, No. 5, 885-904 (2021); translation from Algebra Anal. 32, No. 5, 86-113 (2020).
For a closed oriented surface \(\Sigma\), let \(X_{\Sigma,n}\) denote the set of isomorphism classes of \(n\)-sheeted coverings \(f:\Sigma\to S^2\) preserving orientation; here \(S^2\) is the two-dimensional sphere. The main goal of the paper is to define a structure of \(CW\)-complex on the compactification \(\bar{X}_{\Sigma,n}\) of \(X_{\Sigma,n}\) constructed in [V. I. Zvonilov and S. Yu. Orevkov, Proc. Steklov Inst. Math. 298, 118–128 (2017; Zbl 1387.30062); translation from Tr. Mat. Inst. Steklova 298, 127–138 (2017)]. In the cited paper, the authors constructed a cellular decomposition of \(\bar{X}_{\Sigma,n}\). Here they modify its construction and prove that it is a \(CW\)-complex.
Then they give an application of the obtained results to investigation of the spaces of \(j\)-invariants of trigonal curves on a ruled surface. In particular, they determine the fundamental group of the space of nonsingular trigonal curves; this fact plays an important role for understanding the topology of real algebraic varieties, in particular, of surfaces of degree \(5\) in the real projective space.
Then they give an application of the obtained results to investigation of the spaces of \(j\)-invariants of trigonal curves on a ruled surface. In particular, they determine the fundamental group of the space of nonsingular trigonal curves; this fact plays an important role for understanding the topology of real algebraic varieties, in particular, of surfaces of degree \(5\) in the real projective space.
Reviewer: Samyon R. Nasyrov (Kazan)
MSC:
| 57K20 | 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.) |
| 57M10 | Covering spaces and low-dimensional topology |
| 14H30 | Coverings of curves, fundamental group |
| 30F99 | Riemann surfaces |
Keywords:
branched covering; \(CW\)-complex; isomorphism class; cellular decomposition; fundamental group; trigonal curve; real algebraic variety; Hurwitz space; compactificationCitations:
Zbl 1387.30062References:
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