A Nonshrinkable Decomposition of S n Involving a Null Sequence of Cellular Arcs

Daverman, R. J.; Walsh, J. J.

doi:10.2307/1998727

Transactions of the American Mathematical Society

1982

DOI: 10.2307/1998727

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A Nonshrinkable Decomposition of S n Involving a Null Sequence of Cellular Arcs

R. J. Daverman¹,

J. J. Walsh²

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1985

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“…In general, it is not possible to replace the 2-cell with an arc. Pointed out in [DW,§5] is that for n > 5 the decomposition of S" having nondegenerate elements contained in a finite graph satisfies the Disjoint Disks Property. The latter is a property which R. D. Edwards showed implies the shrinkability of cell-like decompositions [Ed].…”

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A nonshrinkable decomposition of 𝑆³ whose nondegenerate elements are contained in a cellular arc

Row¹,

Walsh²

1985

Trans. Amer. Math. Soc.

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A decomposition G of S1 is constructed with the following properties: (1) The set NG of all nondegenerate elements consists of a null sequence of arcs and J-CL(U{ g e NG}) is a simple closed curve. (2) Each arc contained in J is cellular. (3) J is the boundary of a disk Q that is locally flat except at points of J. (4) The decomposition G is not shrinkable; that is, the decomposition space is not homeomorphic to S3.

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A nonshrinkable decomposition of 𝑆³ whose nondegenerate elements are contained in a cellular arc

Row¹,

Walsh²

1985

Trans. Amer. Math. Soc.

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A characterization of manifold decompositions satisfying the disjoint triples property

Garity¹

1981

Proc. Amer. Math. Soc.

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Abstract. A metric space X satisfies the Disjoint Triples Property (DD3) if maps 7j, f2 and f3 from B2 into X are approximate by maps /,, f2 and /3 with H ?_ i MB2) = 0-Those CE decompositions of manifolds satisfying DD3 and yielding finite-dimensional nonmanifold decomposition spaces are shown to be precisely those intrinsically O-dimensional decompositions which yield nonshrinkable null cellular decompositions under amalgamation. This characterization results in another proof of the fact that E"/G X £ ' is secretly O-dimensional where G is a CE use decomposition of E", n > 4, with E"/G finite dimensional. Introduction. Recent work of Cannon [C], Edwards[Ed] and Quinn [Q] characterizes topological «-manifolds, n > 5, in terms of a simple general position property, the Disjoint Disks Property (DDP). Daverman introduced a property closely related to the DDP, the Disjoint Triples Property (hereafter referred to as DD3), and showed that certain decompositions satisfying DD3 can be amalgamated so as to yield nonshrinkable null cellular decompositions. See [Dl].We show that finite-dimensional nonshrinkable decompositions satisfying DD3 can be characterized as those intrinsically O-dimensional decompositions having amalgamations as above. Thus, the class of finite-dimensional decompositions satisfying DD3 but not DDP has the minimal amount of complexity required to yield nonmanifold decompositions in the sense that nonshrinkable null cellular decompositions form a prototype for this class.

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A Nonshrinkable Decomposition of S n Involving a Null Sequence of Cellular Arcs

Cited by 2 publications

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A nonshrinkable decomposition of 𝑆³ whose nondegenerate elements are contained in a cellular arc

A nonshrinkable decomposition of 𝑆³ whose nondegenerate elements are contained in a cellular arc

A characterization of manifold decompositions satisfying the disjoint triples property

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