Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Approximating topological surfaces in $ 4$-manifolds


Author: Gerard A. Venema
Journal: Trans. Amer. Math. Soc. 265 (1981), 35-45
MSC: Primary 57Q35; Secondary 57N45
DOI: https://doi.org/10.1090/S0002-9947-1981-0607105-3
MathSciNet review: 607105
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {M^2}$ be a compact, connected $ 2$-manifold with $ \partial {M^2} \ne \emptyset $ and let $ h:{M^2} \to {W^4}$ be a topological embedding of $ {M^2}$ into a $ 4$-manifold. The main theorem of this paper asserts that if $ {W^4}$ is a piecewise linear $ 4$-manifold, then $ h$ can be arbitrarily closely approximated by locally flat PL embeddings. It is also shown that if the $ 4$-dimensional annulus conjecture is correct and if $ W$ is a topological $ 4$-manifold, then $ h$ can be arbitrarily closely approximated by locally flat embeddings. These results generalize the author's previous theorems about approximating disks in $ 4$-space.


References [Enhancements On Off] (What's this?)

  • [1] R. H. Bing, Vertical general position, Geometric Topology, Lecture Notes in Math., vol. 438, Springer-Verlag, Berlin and New York 1975, pp. 16-41. MR 0394685 (52:15484)
  • [2] M. Brown and H. Gluck, Stable structures on manifolds. I, II, and III, Ann. of Math. (2) 79 (1964), 1-58. MR 0158383 (28:1608a)
  • [3] J. C. Cantrell and C. H. Edwards, Jr., Almost locally polyhedral curves in Euclidean $ n$-space, Trans. Amer. Math. Soc. 107 (1963), 451-457. MR 0149453 (26:6941)
  • [4] R. J. Daverman and W. T. Eaton, An equivalence for the embeddings of cells in a manifold, Trans. Amer. Math. Soc. 145 (1969), 369-382. MR 0250280 (40:3519)
  • [5] S. Kinoshita, On diffeomorphic approximations of polyhedral surfaces in $ 4$-space, Osaka Math. J. 12 (1960), 191-194. MR 0130687 (24:A547)
  • [6] Y. Matsumoto, Wild embeddings of piecewise linear manifolds in codimension two, Geometric Topology, Academic Press, New York, 1979, pp. 393-428. MR 537743 (80i:57012)
  • [7] Y. Matsumoto and G. A. Venema, Failure of the Dehn lemma on contractible $ 4$-manifolds, Invent. Math. 51 (1979), 205-218. MR 530628 (80k:57013)
  • [8] R. T. Miller, Approximating codimension $ 3$ embeddings, Ann. of Math. (2) 95 (1972), 406-416. MR 0307246 (46:6366)
  • [9] R. B. Sher, Tame polyhedra in wild cells and spheres, Proc. Amer. Math. Soc. 30 (1971), 169-174. MR 0281178 (43:6897)
  • [10] G. A. Venema, A topological disk in a $ 4$-manifold can be approximated by piecewise linear disks, Bull. Amer. Math. Soc. 83 (1971), 386-387. MR 0431190 (55:4192)
  • [11] -, Approximating disks in $ 4$-space, Michigan Math. J. 25 (1978), 19-27. MR 497879 (80d:57011)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57Q35, 57N45

Retrieve articles in all journals with MSC: 57Q35, 57N45


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0607105-3
Keywords: Surface, $ 4$-manifold, topological embedding, piecewise linear approximation, locally flat approximation
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society