Decompositions of continua over the hyperbolic plane

Rogers, James T.

doi:10.1090/S0002-9947-1988-0965753-1

Decompositions of continua over the hyperbolic plane
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Trans. Amer. Math. Soc. 310 (1988), 277-291 Request permission

Abstract:

The following theorem is proved. Theorem. Let $X$ be a homogeneous continuum such that ${H^1}(X) \ne 0$. If $\mathcal {G}$ is the collection of maximal terminal proper subcontinua of $X$, then (1) The collection $\mathcal {G}$ is a monotone, continuous, terminal decomposition of $X$, (2) The nondegenerate elements of $\mathcal {G}$ are mutually homeomorphic, indecomposable, cell-like, terminal, homogeneous continua of the same dimension as $X$, (3) The quotient space is a homogeneous continuum, and (4) The quotient space does not contain any proper, nondegenerate, terminal subcontinuum. This theorem is related to the Jones’ Aposyndetic Decomposition Theorem. The proof involves the hyperbolic plane and a subset of the circle at $\infty$, called the set of ends of a component of the universal cover of $X$.

References

David P. Bellamy and Lewis Lum, The cyclic connectivity of homogeneous arcwise connected continua, Trans. Amer. Math. Soc. 266 (1981), no. 2, 389–396. MR 617540, DOI 10.1090/S0002-9947-1981-0617540-5
R. H. Bing, A homogeneous indecomposable plane continuum, Duke Math. J. 15 (1948), 729–742. MR 27144
R. H. Bing, A simple closed curve is the only homogeneous bounded plane continuum that contains an arc, Canadian J. Math. 12 (1960), 209–230. MR 111001, DOI 10.4153/CJM-1960-018-x
J. H. Case and R. E. Chamberlin, Characterizations of tree-like continua, Pacific J. Math. 10 (1960), 73–84. MR 111000
Edward G. Effros, Transformation groups and $C^{\ast }$-algebras, Ann. of Math. (2) 81 (1965), 38–55. MR 174987, DOI 10.2307/1970381
J. Grispolakis and E. D. Tymchatyn, On confluent mappings and essential mappings—a survey, Rocky Mountain J. Math. 11 (1981), no. 1, 131–153. MR 612135, DOI 10.1216/RMJ-1981-11-1-131
Charles L. Hagopian, Homogeneous plane continua, Houston J. Math. 1 (1975), 35–41. MR 383369
Charles L. Hagopian, Indecomposable homogeneous plane continua are hereditarily indecomposable, Trans. Amer. Math. Soc. 224 (1976), no. 2, 339–350 (1977). MR 420572, DOI 10.1090/S0002-9947-1976-0420572-1
Charles L. Hagopian, Atriodic homogeneous continua, Pacific J. Math. 113 (1984), no. 2, 333–347. MR 749539
Charles L. Hagopian, A characterization of solenoids, Pacific J. Math. 68 (1977), no. 2, 425–435. MR 458381
F. Burton Jones, The aposyndetic decomposition of homogeneous continua, Proceedings of the 1983 topology conference (Houston, Tex., 1983), 1983, pp. 51–54. MR 738469
F. Burton Jones, On a certain type of homogeneous plane continuum, Proc. Amer. Math. Soc. 6 (1955), 735–740. MR 71761, DOI 10.1090/S0002-9939-1955-0071761-1
F. Burton Jones, Homogeneous plane continua, Proceedings of the Auburn Topology Conference (Auburn Univ., Auburn, Ala., 1969; dedicated to F. Burton Jones on the occasion of his 60th birthday), Auburn Univ., Auburn, Ala., 1969, pp. 46–56. MR 0391040
F. Burton Jones, Concerning aposyndetic and non-aposyndetic continua, Bull. Amer. Math. Soc. 58 (1952), 137–151. MR 48797, DOI 10.1090/S0002-9904-1952-09582-3
J. Krasinkiewicz, Mapping properties of hereditarily indecomposable continua, Houston J. Math. 8 (1982), no. 4, 507–516. MR 688251
J. Krasinkiewicz, On one-point union of two circles, Houston J. Math. 2 (1976), no. 1, 91–95. MR 397679
Wayne Lewis, The pseudo-arc of pseudo-arcs is unique, Houston J. Math. 10 (1984), no. 2, 227–234. MR 744907
Wayne Lewis, Continuous curves of pseudo-arcs, Houston J. Math. 11 (1985), no. 1, 91–99. MR 780823
T. Maćkowiak and E. D. Tymchatyn, Continuous mappings on continua. II, Dissertationes Math. (Rozprawy Mat.) 225 (1984), 57. MR 739739
D. R. McMillan Jr., Acyclicity in three-manifolds, Bull. Amer. Math. Soc. 76 (1970), 942–964. MR 270380, DOI 10.1090/S0002-9904-1970-12510-1
Piotr Minc and James T. Rogers Jr., Some new examples of homogeneous curves, Proceedings of the 1985 topology conference (Tallahassee, Fla., 1985), 1985, pp. 347–356. MR 876903
James T. Rogers Jr., Solenoids of pseudo-arcs, Houston J. Math. 3 (1977), no. 4, 531–537. MR 464193
James T. Rogers Jr., Decompositions of homogeneous continua, Pacific J. Math. 99 (1982), no. 1, 137–144. MR 651491
James T. Rogers Jr., Homogeneous, separating plane continua are decomposable, Michigan Math. J. 28 (1981), no. 3, 317–322. MR 629364
James T. Rogers Jr., Homogeneous hereditarily indecomposable continua are tree-like, Houston J. Math. 8 (1982), no. 3, 421–428. MR 684167
James T. Rogers Jr., Cell-like decompositions of homogeneous continua, Proc. Amer. Math. Soc. 87 (1983), no. 2, 375–377. MR 681852, DOI 10.1090/S0002-9939-1983-0681852-7
James T. Rogers Jr., Homogeneous curves that contain arcs, Topology Appl. 21 (1985), no. 1, 95–101. MR 808728, DOI 10.1016/0166-8641(85)90062-8
James T. Rogers Jr., Hyperbolic ends and continua, Michigan Math. J. 34 (1987), no. 3, 337–347. MR 911809, DOI 10.1307/mmj/1029003616
James T. Rogers Jr., Homogeneous continua, Proceedings of the 1983 topology conference (Houston, Tex., 1983), 1983, pp. 213–233. MR 738476
James T. Rogers Jr., Orbits of higher-dimensional hereditarily indecomposable continua, Proc. Amer. Math. Soc. 95 (1985), no. 3, 483–486. MR 806092, DOI 10.1090/S0002-9939-1985-0806092-1
Gordon Thomas Whyburn, Analytic Topology, American Mathematical Society Colloquium Publications, Vol. 28, American Mathematical Society, New York, 1942. MR 0007095
G. T. Whyburn, Semi-locally connected sets, Amer. J. Math. 61 (1939), 733–749. MR 182, DOI 10.2307/2371330