An equality for -sided surfaces with a finite number of wild points

Authors:
Michael D. Taylor and Harvey Rosen

Journal:
Trans. Amer. Math. Soc. **167** (1972), 347-358

MSC:
Primary 57A10

MathSciNet review:
0295315

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let *S* be a 2-sided surface in a 3-manifold that is wild from one side *U* at just *m* points. It is shown that the minimal genus possible for all members of a sequence of surfaces in *U* converging to *S* (where these surfaces each separate the same point from *S* in ) is equal to the sum of the genus of *S* and a certain multiple of the sum of *m* special topological invariants associated with the wild points. In this equality, the sum of these invariants is multiplied by just one of the numbers 0, 1, or 2, dependent upon the genus and orientability class of *S* and the value of *m*. As an application, an upper bound is given for the number of nonpiercing points that a 2-sided surface has with respect to one side.

**[1]**R. H. Bing,*Locally tame sets are tame*, Ann. of Math. (2)**59**(1954), 145–158. MR**0061377****[2]**R. H. Bing,*Approximating surfaces with polyhedral ones*, Ann. of Math. (2)**65**(1957), 465–483. MR**0087090****[3]**Morton Brown,*Locally flat imbeddings of topological manifolds*, Ann. of Math. (2)**75**(1962), 331–341. MR**0133812****[4]**C. E. Burgess,*Characterizations of tame surfaces in 𝐸³*, Trans. Amer. Math. Soc.**114**(1965), 80–97. MR**0176456**, 10.1090/S0002-9947-1965-0176456-2**[5]**C. E. Burgess,*Criteria for a 2-sphere in 𝑆³ to be tame modulo two points*, Michigan Math. J.**14**(1967), 321–330. MR**0216481****[6]**J. C. Cantrell,*Almost locally polyhedral 2-spheres in 𝑆³*, Duke Math. J.**30**(1963), 249–252. MR**0148042****[7]**R. J. Daverman,*Non-homeomorphic approximations of manifolds with surfaces of bounded genus*, Duke Math. J.**37**(1970), 619–625. MR**0267546****[8]**Robert J. Daverman,*On the number of nonpiercing points in certain crumpled cubes*, Pacific J. Math.**34**(1970), 33–43. MR**0271920****[9]**C. H. Edwards Jr.,*Concentricity in 3-manifolds*, Trans. Amer. Math. Soc.**113**(1964), 406–423. MR**0178459**, 10.1090/S0002-9947-1964-0178459-X**[10]**Sze-tsen Hu,*Theory of retracts*, Wayne State University Press, Detroit, 1965. MR**0181977****[11]**Lloyd L. Lininger,*Some results on crumpled cubes*, Trans. Amer. Math. Soc.**118**(1965), 534–549. MR**0178460**, 10.1090/S0002-9947-1965-0178460-7**[12]**D. R. McMillan Jr.,*Neighborhoods of surfaces in 3-manifolds*, Michigan Math. J.**14**(1967), 161–170. MR**0212778****[13]**M. D. Taylor,*An upper bound on the number of wild points on a*2-*sphere*, Ph.D. Thesis, Florida State University, Tallahassee, Fla., 1969.**[14]**Raymond Louis Wilder,*Topology of manifolds*, American Mathematical Society Colloquium Publications, Vol. XXXII, American Mathematical Society, Providence, R.I., 1963. MR**0182958**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
57A10

Retrieve articles in all journals with MSC: 57A10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1972-0295315-1

Keywords:
2-sided surfaces in 3-manifolds,
surfaces with finitely many wild points,
convergent sequence of surfaces,
limiting genus,
local enveloping genus,
number of nonpiercing points of surfaces

Article copyright:
© Copyright 1972
American Mathematical Society