A general class of factors of 𝐸⁴

Rubin, Leonard R.

doi:10.1090/S0002-9947-1972-0295314-X

A general class of factors of $E^4$
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by Leonard R. Rubin

Trans. Amer. Math. Soc. 166 (1972), 215-224

DOI: https://doi.org/10.1090/S0002-9947-1972-0295314-X

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Erratum: Trans. Amer. Math. Soc. 177 (1973), 505.

Abstract:

In this paper we prove that any upper semicontinuous decomposition of $E^n$ which is generated by a trivial defining sequence of cubes with handles determines a factor of $E^{n + 1}$. An important corollary to this result is that every 0-dimensional point-like decomposition of $E^3$ determines a factor of $E^4$. In our approach we have simplified the construction of the sequence of shrinking homeomorphisms by eliminating the necessity of shrinking sets piecewise in a collection of n-cells, the technique employed by R. H. Bing in the original result of this type.

References

W. R. Alford and R. B. Sher, Defining sequences for compact $0$-dimensional decompositions of $E^{n}$, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 17 (1969), 209–212 (English, with Russian summary). MR 254824
J. J. Andrews and M. L. Curtis, $n$-space modulo an arc, Ann. of Math. (2) 75 (1962), 1–7. MR 139153, DOI 10.2307/1970414
J. J. Andrews and Leonard Rubin, Some spaces whose product with $E^{1}$ is $E^{4}$, Bull. Amer. Math. Soc. 71 (1965), 675–677. MR 176454, DOI 10.1090/S0002-9904-1965-11394-5
R. H. Bing, A decomposition of $E^3$ into points and tame arcs such that the decomposition space is topologically different from $E^3$, Ann. of Math. (2) 65 (1957), 484–500. MR 92961, DOI 10.2307/1970058
R. H. Bing, The cartesian product of a certain nonmanifold and a line is $E^{4}$, Ann. of Math. (2) 70 (1959), 399–412. MR 107228, DOI 10.2307/1970322
R. H. Bing, Point-like decompositions of $E^{3}$, Fund. Math. 50 (1961/62), 431–453. MR 137104, DOI 10.4064/fm-50-4-431-453
John L. Bryant, Euclidean space modulo a cell, Fund. Math. 63 (1968), 43–51. MR 230298, DOI 10.4064/fm-63-1-43-51

Euclidean n-space modulo an

cell

James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606
R. C. Lacher, Cell-like mappings. I, Pacific J. Math. 30 (1969), 717–731. MR 251714, DOI 10.2140/pjm.1969.30.717
H. W. Lambert and R. B. Sher, Point-like $0$-dimensional decompositions of $S^{3}$, Pacific J. Math. 24 (1968), 511–518. MR 225308, DOI 10.2140/pjm.1968.24.511
D. R. McMillan Jr., A criterion for cellularity in a manifold, Ann. of Math. (2) 79 (1964), 327–337. MR 161320, DOI 10.2307/1970548
Ronald H. Rosen, $E^{4}$ is the cartesian product of a totally non-euclidean space and $E^{1}$, Ann. of Math. (2) 73 (1961), 349–361. MR 124888, DOI 10.2307/1970337
Leonard Rubin, The product of an unusual decompostion space with aline is $E^{4}$, Duke Math. J. 33 (1966), 323–329. MR 195078
Leonard R. Rubin, The product of any dogbone space with a line is $E^{4}$, Duke Math. J. 37 (1970), 189–192. MR 267548
Leonard R. Rubin, Recognizing certain factors of $E^{4}$, Proc. Amer. Math. Soc. 26 (1970), 199–200. MR 266180, DOI 10.1090/S0002-9939-1970-0266180-7
Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto-London, 1966. MR 0210112