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)

 
 

 

On embedding polyhedra and manifolds


Author: Krešo Horvatić
Journal: Trans. Amer. Math. Soc. 157 (1971), 417-436
MSC: Primary 57.01
DOI: https://doi.org/10.1090/S0002-9947-1971-0278314-4
MathSciNet review: 0278314
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is well known that every $ n$-polyhedron PL embeds in a Euclidean $ (2n + 1)$-space, and that for PL manifolds the result can be improved upon by one dimension. In the paper are given some sufficient conditions under which the dimension of the ambient space can be decreased. The main theorem asserts that, for there to exist an embedding of the $ n$-polyhedron $ X$ into $ 2n$-space, it suffices that the integral cohomology group $ {H^n}(X - \operatorname{Int} A) = 0$ for some $ n$-simplex $ A$ of a triangulation of $ X$. A number of interesting corollaries follow from this theorem. Along the line of manifolds the known embedding results for PL manifolds are extended over a larger class containing various kinds of generalized manifolds, such as triangulated manifolds, polyhedral homology manifolds, pseudomanifolds and manifolds with singular boundary. Finally, a notion of strong embeddability is introduced which allows us to prove that some class of $ n$-manifolds can be embedded into a $ (2n - 1)$-dimensional ambient space.


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

  • [1] J. W. Alexander, The combinatorial theory of complexes, Ann. of Math. 31 (1930), 292-320. MR 1502943
  • [2] P. S. Alexandroff and H. Hopf, Topologie. Vol. 1, Springer, Berlin, 1935; reprint, Chelsea, New York, 1965; newer ed., Die Grundlehren der math. Wissenschaften, Band 45, Springer-Verlag, Berlin, 1967. MR 32 #3023.
  • [3] M. L. Curtis, On $ 2$-complexes in $ 4$-space, Topology of $ 3$-manifolds and related topics (Proc. Univ. of Georgia Inst., 1961), Prentice-Hall, Englewood Cliffs, N. J., 1962, pp. 204-207. MR 25 #3537. MR 0140114 (25:3537)
  • [4] C. H. Edwards, Jr., Unknotting polyhedral homology manifolds, Michigan Math. J. 15 (1968), 81-95. MR 37 #2218. MR 0226629 (37:2218)
  • [5] A. I. Flores, Über die Existenz $ n$-dimensionaler Komplexe, die nicht in den $ {R^{2n}}$ topologisch einbettbar sind, Erg. Math. Kolloq. 5 (1933), 17-24.
  • [6] -, Über $ n$-dimensionaler Komplexe, die im $ {R^{2n + 1}}$ absolut selbstverschlungen sind, Erg. Math. Kolloq. 6 (1934), 4-7.
  • [7] V. K. A. M. Gugenheim, Piecewise linear isotopy and embedding of elements and spheres. I, Proc. London Math. Soc. (3) 3 (1953), 29-53. MR 15, 336. MR 0058204 (15:336d)
  • [8] J. F. P. Hudson and E. C. Zeeman, On regular neighborhoods, Proc. London Math. Soc. (3) 14 (1964), 719-745. MR 29 #4063. MR 0166790 (29:4063)
  • [9] L. S. Husch, Jr., Piecewise linear embeddings in codimensions 0 and 1, Doctoral Thesis, Florida State University, Tallahassee, Florida, 1967. MR 0215308 (35:6149)
  • [10] M. C. Irwin, Embeddings of polyhedral manifolds, Ann. of Math. (2) 82 (1965), 1-14. MR 32 #460. MR 0182978 (32:460)
  • [11] R. C. Kirby and L. C. Siebenmann, On the triangulation of manifolds and the Hauptvermutung, Bull. Amer. Math. Soc. 75 (1969), 742-749. MR 0242166 (39:3500)
  • [12] W. B. R. Lickorish, The piecewise linear unknotting of cones, Topology 4 (1965), 67-91. MR 34 #3585. MR 0203736 (34:3585)
  • [13] W. S. Massey, Algebraic topology: An introduction, Harcourt, Brace & World, New York, 1967. MR 35 #2271. MR 0211390 (35:2271)
  • [14] K. Menger, Dimensions theorie, Teubner, Leipzig, 1928.
  • [15] R. Penrose, J. H. C. Whitehead and E. C. Zeeman, Embedding of manifolds in euclidean space, Ann. of Math. (2) 73 (1961), 613-623. MR 23 #A2218. MR 0124909 (23:A2218)
  • [16] H. Seifert and W. Threlfall, Lehrbuch der Topologie, Teubner, Leipzig, 1934; reprint, Chelsea, New York, 1947.
  • [17] E. H. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. MR 35 #1007. MR 0210112 (35:1007)
  • [18] R. Tindell, A counterexample on relative regular neighborhoods, Bull. Amer. Math. Soc. 72 (1966), 892-893. MR 33 #6635. MR 0198477 (33:6635)
  • [19] E. R. Van Kampen, Komplexe in euklidische Raumen, Abh. Math. Sem. Univ. Hamburg 9 (1932), 72-78.
  • [20] -, Berichtung dazu, Abh. Math. Sem. Univ. Hamburg 9 (1932), 152-153.
  • [21] C. T. C. Wall, All $ 3$-manifolds imbed in $ 5$-space, Bull. Amer. Math. Soc. 71 (1965), 566-567. MR 30 #5324. MR 0175139 (30:5324)
  • [22] C. Weber, Plongements de polyèdres dans le domaine métastable, Comment. Math. Helv. 42 (1967), 1-27. MR 38 #6606. MR 0238330 (38:6606)
  • [23] Wu Wen-tsün, a) On the realization of complexes in euclidean spaces. I, Sci. Sinica 7 (1958), 251-297. MR 20 #5471. b) II, Sci. Sinica 7 (1958), 365-387. MR 22 #3000. c) III, Sci. Sinica 8 (1959), 133-150. MR 20 #4825b. MR 0099026 (20:5471)
  • [24] E. C. Zeeman, Seminar on combinatorial topology, Mimeographed Notes, Inst. Hautes Études Sci., Paris, 1963.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57.01

Retrieve articles in all journals with MSC: 57.01


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0278314-4
Keywords: PL category, PL embeddings, polyhedra, PL manifolds, generalizations of manifolds, strong embeddability
Article copyright: © Copyright 1971 American Mathematical Society

American Mathematical Society