Extending cell-like maps on manifolds

Authors:
B. J. Ball and R. B. Sher

Journal:
Trans. Amer. Math. Soc. **186** (1973), 229-246

MSC:
Primary 57A60

MathSciNet review:
0328950

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Abstract | References | Similar Articles | Additional Information

Abstract: Let *X* be a closed subset of a manifold *M* and be a cell-like upper semicontinuous decomposition of *X*. We consider the problem of extending to a cell-like upper semicontinuous decomposition *G* of *M* such that . Under fairly weak restrictions (which vanish if or and we show that such a *G* exists if and only if the trivial extension of , obtained by adjoining to the singletons of , has the desired property. In particular, the nondegenerate elements of Bing's dogbone decomposition of are not elements of any cell-like upper semicontinuous decomposition *G* of such that . Call a cell-like upper semicontinuous decomposition *G* of a metric space *X simple* if and say that the closed set *Y* is *simply embedded* in *X* if each simple decomposition of *Y* extends trivially to a simple decomposition of *X*. We show that tame manifolds in are simply embedded and, with some additional restrictions, obtain a similar result for a locally flat *k*-manifold in an *m*-manifold . Examples are given of an everywhere wild simply embedded simple closed curve in and of a compact absolute retract which embeds in yet has no simple embedding in .

**[1]**Steve Armentrout,*Upper semi-continuous decompositions of 𝐸³ with at most countably many non-degenerate elements*, Ann. of Math. (2)**78**(1963), 605–618. MR**0156331****[2]**Steve Armentrout,*Concerning cellular decompositions of 3-manifolds that yield 3-manifolds*, Trans. Amer. Math. Soc.**133**(1968), 307–332. MR**0230296**, 10.1090/S0002-9947-1968-0230296-7**[3]**Steve Armentrout,*Concerning cellular decompositions of 3-manifolds with boundary*, Trans. Amer. Math. Soc.**137**(1969), 231–236. MR**0236931**, 10.1090/S0002-9947-1969-0236931-2**[4]**Steve Armentrout,*Cellular decompositions of 3-manifolds that yield 3-manifolds*, Bull. Amer. Math. Soc.**75**(1969), 453–456. MR**0239578**, 10.1090/S0002-9904-1969-12218-4**[5]**-,*Cellular decompositions of*3-*manifolds that yield*3-*manifolds*, Mem. Amer. Math. Soc. No.**107**(1970).**[6]**Ralph J. Bean,*Repairing embeddings and decompositions in 𝑆³*, Duke Math. J.**36**(1969), 379–385. MR**0243499****[7]**R. H. Bing,*Upper semicontinuous decompositions of 𝐸³*, Ann. of Math. (2)**65**(1957), 363–374. MR**0092960****[8]**R. H. Bing,*A decomposition of 𝐸³ into points and tame arcs such that the decomposition space is topologically different from 𝐸³*, Ann. of Math. (2)**65**(1957), 484–500. MR**0092961****[9]**R. H. Bing,*Extending monotone decompositions of 3-manifolds*, Trans. Amer. Math. Soc.**149**(1970), 351–369. MR**0263051**, 10.1090/S0002-9947-1970-0263051-1**[10]**Karol Borsuk,*Theory of retracts*, Monografie Matematyczne, Tom 44, Państwowe Wydawnictwo Naukowe, Warsaw, 1967. MR**0216473****[11]**H. G. Bothe,*Ein homogen wilder Knoten*, Fund. Math.**60**(1967), 271–283 (German). MR**0216494****[12]**William S. Boyd Jr.,*Repairing embeddings of 3-cells with monotone maps of 𝐸³*, Trans. Amer. Math. Soc.**161**(1971), 123–144. MR**0282352**, 10.1090/S0002-9947-1971-0282352-5**[13]**H. Cook,*Continua which admit only the identity mapping onto non-degenerate subcontinua*, Fund. Math.**60**(1967), 241–249. MR**0220249****[14]**James Dugundji,*Topology*, Allyn and Bacon, Inc., Boston, Mass., 1966. MR**0193606****[15]**E. Dyer and M.-E. Hamstrom,*Completely regular mappings*, Fund. Math.**45**(1958), 103–118. MR**0092959****[16]**William Haver,*A characterization theorem for cellular maps*, Bull. Amer. Math. Soc.**76**(1970), 1277–1280. MR**0267538**, 10.1090/S0002-9904-1970-12638-6**[17]**R. C. Lacher,*Cell-like mappings. I*, Pacific J. Math.**30**(1969), 717–731. MR**0251714****[18]**H. W. Lambert,*Some comments on the structure of compact decompositions of 𝑆³*, Proc. Amer. Math. Soc.**19**(1968), 180–184. MR**0225307**, 10.1090/S0002-9939-1968-0225307-4**[19]**Jack W. Lamoreaux,*Decomposition of metric spaces with a 0-dimensional set of non-degenerate elements*, Canad. J. Math.**21**(1969), 202–216. MR**0240777****[20]**Louis F. McAuley,*Some fundamental theorems and problems related to monotone mappings.*, Proc. First Conf. on Monotone Mappings and Open Mappings (SUNY at Binghamton, Binghamton, N.Y., 1970) State Univ. of New York at Binghamton, N.Y., 1971, pp. 1–36. MR**0287518****[21]**D. R. McMillan Jr.,*Some topological properties of piercing points*, Pacific J. Math.**22**(1967), 313–322. MR**0216486****[22]**R. L. Moore,*Concerning upper semi-continuous collections of continua*, Trans. Amer. Math. Soc.**27**(1925), no. 4, 416–428. MR**1501320**, 10.1090/S0002-9947-1925-1501320-8**[23]**Victor Nicholson,*Mapping cylinder neighborhoods*, Trans. Amer. Math. Soc.**143**(1969), 259–268. MR**0248788**, 10.1090/S0002-9947-1969-0248788-4**[24]**R. B. Sher,*Determining the cellularity of a 𝑖-complex by properties of its arcs.*, Proc. Amer. Math. Soc.**26**(1970), 491–498. MR**0270353**, 10.1090/S0002-9939-1970-0270353-7**[25]**Richard B. Sher,*Realizing cell-like maps in Euclidean space*, General Topology and Appl.**2**(1972), 75–89. MR**0303546****[26]**L. C. Siebenmann,*Approximating cellular maps by homeomorphisms*, Topology**11**(1972), 271–294. MR**0295365****[27]**Carl D. Sikkema,*A duality between certain spheres and arcs in 𝑆³*, Trans. Amer. Math. Soc.**122**(1966), 399–415. MR**0199851**, 10.1090/S0002-9947-1966-0199851-5**[28]**Jussi Väisälä,*The invariance of domain under acyclic mappings*, Duke Math. J.**33**(1966), 679–681. MR**0200929****[29]**G. T. Whyburn,*A Continuum Every Subcontinuum of Which Separates the Plane*, Amer. J. Math.**52**(1930), no. 2, 319–330. MR**1506757**, 10.2307/2370686**[30]**-,*Analytic topology*, Amer. Math. Soc. Colloq. Publ., vol. 28, Amer. Math. Soc., Providence, R.I., 1942. MR**4**, 86.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1973-0328950-3

Keywords:
Extension,
proper map,
cell-like space,
cell-like map,
monotone map,
trivial extension,
upper semicontinuous decomposition,
dogbone space,
tame,
locally flat

Article copyright:
© Copyright 1973
American Mathematical Society