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)

 
 

 

Cloverleaf representations of simply connected $ 3$-manifolds


Author: Edwin E. Moise
Journal: Trans. Amer. Math. Soc. 201 (1975), 1-30
MSC: Primary 57C05; Secondary 57A10, 57C40
DOI: https://doi.org/10.1090/S0002-9947-1975-0350745-7
MathSciNet review: 0350745
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ M$ be a triangulated $ 3$-manifold satisfying the hypothesis of the Poincaré Conjecture. In the present paper it is shown that there is a finite linear graph $ {K_1}$ in the $ 3$-sphere, with exactly two components, and a finite linear graph $ {K_2}$ in $ M$, such that when the components of the graphs $ {K_i}$ are regarded as points, the resulting hyperspaces are homeomorphic. $ {K_2}$ satisfies certain conditions which imply that each component of $ {K_2}$ is contractible in $ M$. Thus the conclusion of the theorem proved here is equivalent to the hypothesis of the Poincaré Conjecture.


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

  • [B] R. H. Bing, Conditions under which monotone decompositions of $ {E^3}$ are simply connected, Bull. Amer. Math. Soc. 63 (1957), 143. Abstract #325.
  • [F] Ross Lee Finney III, Some cellular decompositions and pseudo-isotopic mappings of $ n$-manifolds, Dissertation, University of Michigan, Ann Arbor, Mich., 1961.
  • [H] John Hempel, Construction of orientable $ 3$-manifolds, Topology of $ 3$-manifolds and Related Topics (Proc. The Univ. of Georgia Inst., 1961), Prentice-Hall, Englewood Cliffs, N. J., 1962, pp. 207-212. MR 25 #3538. MR 0140115 (25:3538)
  • [L] W. B. R. Lickorish, A representation of orientable combinatorial $ 3$-manifolds, Ann. of Math. (2) 76 (1962), 531-540. MR 27 #1929. MR 0151948 (27:1929)
  • [M] Edwin E. Moise, A monotonic mapping theorem for simply connected $ 3$-manifolds, Illinois J. Math. 12 (1968), 451-474. MR 37 #2200. MR 0226611 (37:2200)
  • [M$ _{5}$] -, Affine structures in $ 3$-manifolds. V. The triangulation theorem and Hauptvermutung, Ann. of Math. (2) 56 (1952), 96-114. MR 14, 72. MR 0048805 (14:72d)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57C05, 57A10, 57C40

Retrieve articles in all journals with MSC: 57C05, 57A10, 57C40


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0350745-7
Keywords: Hyperspaces of the $ 3$-sphere, Poincaré Conjecture
Article copyright: © Copyright 1975 American Mathematical Society

American Mathematical Society