## Branched extensions of curves in orientable surfaces

HTML articles powered by AMS MathViewer

- by Cloyd L. Ezell and Morris L. Marx PDF
- Trans. Amer. Math. Soc.
**259**(1980), 515-532 Request permission

## Abstract:

Given a set of regular curves ${f_1} , \ldots , {f_\rho }$ in an orientable surface*N*, we are concerned with the existence and structure of all sense-preserving maps $F: M \to N$ where (a)

*M*is a bordered orientable surface with $\rho$ boundary components ${K_1},\ldots , {K_\rho }$, (b) $F|{K_i} = {f_i}, i = 1, \ldots , \rho$, (c) at each interior point of

*M*, there is an integer

*n*such that

*F*is locally topologically equivalent to the complex map $w = {z^n}$.

## References

- Keith D. Bailey,
*Extending closed plane curves to immersions of the disk with $n$ handles*, Trans. Amer. Math. Soc.**206**(1975), 1–24. MR**370621**, DOI 10.1090/S0002-9947-1975-0370621-3
S. J. Blank, - D. R. J. Chillingworth,
*Winding numbers on surfaces. I*, Math. Ann.**196**(1972), 218–249. MR**300304**, DOI 10.1007/BF01428050 - George K. Francis,
*Assembling compact Riemann surfaces with given boundary curves and branch points on the sphere*, Illinois J. Math.**20**(1976), no. 2, 198–217. MR**402776** - George K. Francis,
*Polymersions with nontrivial targets*, Illinois J. Math.**22**(1978), no. 1, 161–170. MR**463428** - Victor Guillemin and Alan Pollack,
*Differential topology*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974. MR**0348781**
S. Lefshetz, - Morris L. Marx,
*Extensions of normal immersions of $S^{1}$ into $R^{2}$*, Trans. Amer. Math. Soc.**187**(1974), 309–326. MR**341505**, DOI 10.1090/S0002-9947-1974-0341505-0 - Morris L. Marx and Roger F. Verhey,
*Interior and polynomial extensions of immersed circles*, Proc. Amer. Math. Soc.**24**(1970), 41–49. MR**252660**, DOI 10.1090/S0002-9939-1970-0252660-7 - J. R. Quine,
*A global theorem for singularities of maps between oriented $2$-manifolds*, Trans. Amer. Math. Soc.**236**(1978), 307–314. MR**474378**, DOI 10.1090/S0002-9947-1978-0474378-X - Charles J. Titus,
*The combinatorial topology of analytic functions on the boundary of a disk*, Acta Math.**106**(1961), 45–64. MR**166375**, DOI 10.1007/BF02545813 - Charles J. Titus,
*Extensions through codimension one to sense preserving mappings*, Ann. Inst. Fourier (Grenoble)**23**(1973), no. 2, 215–227 (English, with French summary). MR**348770**

*Extending immersions of the circle*, Dissertation, Brandeis University, 1967; cf. Poenaru, Exposé 342, Séminaire Bourbaki, 1967-1968, Benjamin, New York, 1969.

*Topology*, Amer. Math. Soc. Colloq. Publ., no. 12, Amer. Math. Soc., Providence, R. I., 1930.

## Additional Information

- © Copyright 1980 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**259**(1980), 515-532 - MSC: Primary 57M12; Secondary 30C15
- DOI: https://doi.org/10.1090/S0002-9947-1980-0567094-6
- MathSciNet review: 567094