An equality for 2-sided surfaces with a finite number of wild points

Taylor, Michael D.; Rosen, Harvey

doi:10.1090/S0002-9947-1972-0295315-1

An equality for $2$-sided surfaces with a finite number of wild points
HTML articles powered by AMS MathViewer

by Michael D. Taylor and Harvey Rosen

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

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

PDF | Request permission

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 $U \cup S$) 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.

References

R. H. Bing, Locally tame sets are tame, Ann. of Math. (2) 59 (1954), 145–158. MR 61377, DOI 10.2307/1969836
R. H. Bing, Approximating surfaces with polyhedral ones, Ann. of Math. (2) 65 (1957), 465–483. MR 87090
Morton Brown, Locally flat imbeddings of topological manifolds, Ann. of Math. (2) 75 (1962), 331–341. MR 133812, DOI 10.2307/1970177
C. E. Burgess, Characterizations of tame surfaces in $E^{3}$, Trans. Amer. Math. Soc. 114 (1965), 80–97. MR 176456, DOI 10.1090/S0002-9947-1965-0176456-2
C. E. Burgess, Criteria for a $2$-sphere in $S^{3}$ to be tame modulo two points, Michigan Math. J. 14 (1967), 321–330. MR 216481, DOI 10.1307/mmj/1028999781
J. C. Cantrell, Almost locally polyhedral $2$-spheres in $S^{3}$, Duke Math. J. 30 (1963), 249–252. MR 148042, DOI 10.1215/S0012-7094-63-03027-8
R. J. Daverman, Non-homeomorphic approximations of manifolds with surfaces of bounded genus, Duke Math. J. 37 (1970), 619–625. MR 267546, DOI 10.1215/S0012-7094-70-03776-2
Robert J. Daverman, On the number of nonpiercing points in certain crumpled cubes, Pacific J. Math. 34 (1970), 33–43. MR 271920, DOI 10.2140/pjm.1970.34.33
C. H. Edwards Jr., Concentricity in $3$-manifolds, Trans. Amer. Math. Soc. 113 (1964), 406–423. MR 178459, DOI 10.1090/S0002-9947-1964-0178459-X
Sze-tsen Hu, Theory of retracts, Wayne State University Press, Detroit, 1965. MR 0181977
Lloyd L. Lininger, Some results on crumpled cubes, Trans. Amer. Math. Soc. 118 (1965), 534–549. MR 178460, DOI 10.1090/S0002-9947-1965-0178460-7
D. R. McMillan Jr., Neighborhoods of surfaces in $3$-manifolds, Michigan Math. J. 14 (1967), 161–170. MR 212778

An upper bound on the number of wild points on a

sphere

Raymond Louis Wilder, Topology of manifolds, American Mathematical Society Colloquium Publications, Vol. XXXII, American Mathematical Society, Providence, R.I., 1963. MR 0182958