Semicellularity, decompositions and mappings in manifolds

Coram, Donald

doi:10.1090/s0002-9947-1974-0356068-3

Trans. Amer. Math. Soc.

1974

DOI: 10.1090/s0002-9947-1974-0356068-3

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Semicellularity, decompositions and mappings in manifolds

Donald Coram¹

Abstract: If X is an arbitrary compact set in a manifold, we give algebraic criteria on X and on its embedding to determine that X has an arbitrarily small, closed neighborhood each component of which is a p-connected, piecewise linear manifold which collapses to a ^-dimensional subpolyhedron from some p and q. This property generalizes cellularity.The criteria are in terms of UV properties and Alexander-Spanier cohomology.These criteria are then applied to decide when the components of a given compact set in a manifold… Show more

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Cited by 7 publications

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“…By [C,Corollary 13] there is a map/: S2k+X -> S2*+1 such that one point inverse is a A:-sphere, the other nondegenerate point inverses are (k -l)-connected polyhedra, and the image of the nondegenerate point inverses is a tame arc A. Let g: S2k+X -> Sk+X be a Ar-sphere mapping [L4], [C-D4], [F-W].…”

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confidence: 99%

Finiteness Theorems for Approximate Fibrations

Coram¹,

Duvall²

1982

Transactions of the American Mathematical Society

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Abstract. This paper concerns conditions on the point inverses of a mapping between manifolds which insure that it is an approximate fibration almost everywhere. The primary condition is it,-movability, which says roughly that nearby point inverses include isomorphically on the ith shape group into a mutual neighborhood. Suppose/: Mm-> N" is a UVX mapping which is w.-movable for i < k -1, and n > k + 1. An earlier paper proved that / is an approximate fibration when m < 2k -1. If instead m = 2k, this paper proves that there is a locally finite set S c N such th!df\f~l(N -5) is an approximate fibration. Also if m = 2k + 1 and all of the point inverses are FANR's with the same shape, then there is a locally finite set E c N such that f\f~\N -E) is an approximate fibration.

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Finiteness Theorems for Approximate Fibrations

Coram¹,

Duvall²

1982

Transactions of the American Mathematical Society

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“…Introduction. Bing [Bi;m = 3], Bean [Be;777 = 3], and Coram [Co;m > 4] showed that if A" is a proper, closed subset of a PL manifold Mm such that for each open, connected subset U C M either U -(X n U) is connected or X n bd(U) ¥=0, then there is a monotone mapping/from M onto the m-sphere with each component of X being a point-inverse of /. These results are generalized in two ways.…”

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confidence: 99%

Extending Monotone Decompositions of Manifolds

Walsh¹

1979

Proceedings of the American Mathematical Society

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Abstract. Let Mm (m > 3) be a compact, connected PL manifold and let X Q M be a proper, closed subset of the interior of M such that for each open, connected subset U Ç M either U -(X n U) is connected or X n bd(t7) ^0. Let P be a connected and simply connected polyhedron with dim P > 3. There exists a monotone mapping / from M onto P with each component of X being a point-inverse of/. In the case with M oriented and P the 771-sphere, there exists such a monotone mapping of each degree. that for each open, connected subset U C M either U -(X n U) is connected or X n bd(U) ¥=0, then there is a monotone mapping/from M onto the m-sphere with each component of X being a point-inverse of /. These results are generalized in two ways. First, if P is a simply connected polyhedron with dim P > 3, then there is a monotone mapping / from M onto P with each component of X being a point-inverse of /. Second, if M is oriented, then there is such a monotone mapping onto the m-sphere of each degree. The mappings constructed by Bing, Bean and Coram have degree ± 1 ; PL monotone mappings between manifolds of the same dimension have degree ± 1 and their mappings are "almost" PL.In general, the assumption that P be simply connected cannot be omitted since monotone mappings between ANR's induce surjections between fundamental groups. Adopting a point of view used in [Wa-1], , a generalization to the nonsimply connected case is possible; such a generalization will appear in a subsequent paper.Since there are limitations placed on those compacta which are monotone images of 2-dimensional manifolds [Ho], [Mo], [R-S], [Yo], the assumption that Mm have dimension at least three is essential. However, the first result holds for certain simply connected 2-dimensional polyhedra and, if X is connected, for simply connected 1-dimensional polyhedra.The main tool used in the proofs of the above results is an extension theorem in ; roughly, the theorem says that if a mapping from the boundary of an w-manifold (777 > 3) to a simply connected ANR has an

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“…Beginning with McMillan's cellularity criterion [17], loop shrinking conditions have proven useful in the study of compacta wildly embedded in PL manifolds. The cellularity criterion was applied to embeddings of non cell-like compacta in [5], [6], [10], [ll], [20], [21], and Daverman [8] used a similar condition to study embeddings of S . In this paper…”

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A loop condition for embedded compacts

Coram¹,

Daverman²,

Duvall³

1975

Proc. Amer. Math. Soc.

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In this paper we consider a version ( SLC ) ({\text {SLC}}) of the cellularity criterion which applies to embeddings of arbitrary compacta. It is shown that if the continuum X ⊂ E n , dim ⁡ X ⩽ n − 3 X \subset {E^n},\;\dim X \leqslant n - 3 , has the shape of a finite k k -complex K , 2 k + 1 ⩽ n K,\;2k + 1 \leqslant n , and satisfies the SLC {\text {SLC}} , then X X has arbitrarily small neighborhoods which are regular neighborhoods of a copy of K K .

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Semicellularity, decompositions and mappings in manifolds

Cited by 7 publications

References 23 publications

Finiteness Theorems for Approximate Fibrations

Finiteness Theorems for Approximate Fibrations

Extending Monotone Decompositions of Manifolds

A loop condition for embedded compacts

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