Recognizing certain factors of 𝐸⁴

Rubin, Leonard R.

doi:10.1090/S0002-9939-1970-0266180-7

Recognizing certain factors of $E^{4}$
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by Leonard R. Rubin

Proc. Amer. Math. Soc. 26 (1970), 199-200

DOI: https://doi.org/10.1090/S0002-9939-1970-0266180-7

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Abstract:

It has been proved that for certain peculiar decomposition spaces $Y$ of euclidean $3$-space ${E^3}$ is homeomorphic to euclidean $4$-space, ${E^4}$. In this paper we prove that if a decomposition space $Y$ of ${E^3}$ is generated by a trivial defining sequence whose elements are cubes with handles, and this sequence can be replaced by a toroidal defining sequence, then $Y \times {E^1}$ is homeomorphic to ${E^4}$.

References

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, 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
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