Imbedding compact 3-manifolds in 𝐸³

Knoblauch, Tom

doi:10.1090/S0002-9939-1975-0368010-6

Imbedding compact $3$-manifolds in $E^{3}$
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by Tom Knoblauch

Proc. Amer. Math. Soc. 48 (1975), 447-453

DOI: https://doi.org/10.1090/S0002-9939-1975-0368010-6

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Abstract:

We show that in a large finite disjoint collection of compacta in a closed orientable $3$-manifold there is a compactum that imbeds in ${E^3}$. However, given a closed $3$-manifold ${M^3}$, there is a pair of compact $3$-manifolds $(L,N)$ such that $L$ contains infinitely many disjoint copies of $N$ but $N$ does not imbed in ${M^3}$.

References

Wolfgang Haken, Some results on surfaces in $3$-manifolds, Studies in Modern Topology, Math. Assoc. America, Buffalo, N.Y.; distributed by Prentice-Hall, Englewood Cliffs, N.J., 1968, pp. 39–98. MR 0224071
Wolfgang Heil, On $P^{2}$-irreducible $3$-manifolds, Bull. Amer. Math. Soc. 75 (1969), 772–775. MR 251731, DOI 10.1090/S0002-9904-1969-12283-4

Lehrbuch der Topologie