On embedding polyhedra and manifolds
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- by Krešo Horvatić PDF
- Trans. Amer. Math. Soc. 157 (1971), 417-436 Request permission
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
- James W. Alexander, The combinatorial theory of complexes, Ann. of Math. (2) 31 (1930), no. 2, 292–320. MR 1502943, DOI 10.2307/1968099 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.
- M. L. Curtis, On $2$-complexes in 4-space, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 204–207. MR 0140114
- C. H. Edwards Jr., Unknotting polyhedral homology manifolds, Michigan Math. J. 15 (1968), 81–95. MR 226629 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. —, Über $n$-dimensionaler Komplexe, die im ${R^{2n + 1}}$ absolut selbstverschlungen sind, Erg. Math. Kolloq. 6 (1934), 4-7.
- V. K. A. M. Gugenheim, Piecewise linear isotopy and embedding of elements and spheres. I, II, Proc. London Math. Soc. (3) 3 (1953), 29–53, 129–152. MR 58204, DOI 10.1112/plms/s3-3.1.29
- J. F. P. Hudson and E. C. Zeeman, On regular neighbourhoods, Proc. London Math. Soc. (3) 14 (1964), 719–745. MR 166790, DOI 10.1112/plms/s3-14.4.719
- J. F. P. Hudson, Piecewise linear embeddings, Ann. of Math. (2) 85 (1967), 1–31. MR 215308, DOI 10.2307/1970522
- M. C. Irwin, Embeddings of polyhedral manifolds, Ann. of Math. (2) 82 (1965), 1–14. MR 182978, DOI 10.2307/1970560
- R. C. Kirby and L. C. Siebenmann, On the triangulation of manifolds and the Hauptvermutung, Bull. Amer. Math. Soc. 75 (1969), 742–749. MR 242166, DOI 10.1090/S0002-9904-1969-12271-8
- W. B. R. Lickorish, The piecewise linear unknotting of cones, Topology 4 (1965), 67–91. MR 203736, DOI 10.1016/0040-9383(65)90049-2
- William S. Massey, Algebraic topology: An introduction, Harcourt, Brace & World, Inc., New York, 1967. MR 0211390 K. Menger, Dimensions theorie, Teubner, Leipzig, 1928.
- R. Penrose, J. H. C. Whitehead, and E. C. Zeeman, Imbedding of manifolds in euclidean space, Ann. of Math. (2) 73 (1961), 613–623. MR 124909, DOI 10.2307/1970320 H. Seifert and W. Threlfall, Lehrbuch der Topologie, Teubner, Leipzig, 1934; reprint, Chelsea, New York, 1947.
- Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
- Ralph Tindell, A counterexample on relative regular neighborhoods, Bull. Amer. Math. Soc. 72 (1966), 892–893. MR 198477, DOI 10.1090/S0002-9904-1966-11603-8 E. R. Van Kampen, Komplexe in euklidische Raumen, Abh. Math. Sem. Univ. Hamburg 9 (1932), 72-78. —, Berichtung dazu, Abh. Math. Sem. Univ. Hamburg 9 (1932), 152-153.
- C. T. C. Wall, All $3$-manifolds imbed in $5$-space, Bull. Amer. Math. Soc. 71 (1965), 564–567. MR 175139, DOI 10.1090/S0002-9904-1965-11332-5
- C. Weber, Plongements de polyhèdres dans le domaine métastable, Comment. Math. Helv. 42 (1967), 1–27 (French). MR 238330, DOI 10.1007/BF02564408
- Wen Tsün Wu, On the realization of complexes in euclidean spaces. I, Sci. Sinica 7 (1958), 251–297. MR 99026 E. C. Zeeman, Seminar on combinatorial topology, Mimeographed Notes, Inst. Hautes Études Sci., Paris, 1963.
Additional Information
- © Copyright 1971 American Mathematical Society
- 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