Prime 3-manifolds and the doubling operation

Gross, Jonathan L.

doi:10.1090/S0002-9939-1971-0271948-8

Prime $3$-manifolds and the doubling operation
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by Jonathan L. Gross PDF

Proc. Amer. Math. Soc. 27 (1971), 375-380 Request permission

Abstract:

Examples are given to show that, in general, one may not factor a compact $3$-manifold $M$ with nonvacuous boundary into primes relative to the multi-disk sum (a boundary pasting operation) by factoring the double of $M$ into primes relative to the connected sum. Necessary and sufficient conditions for the double of a compact $3$-manifold with nonvacuous boundary to be prime relative to the connected sum are established.

References

Jonathan L. Gross, A unique decomposition theorem for $3$-manifolds with connected boundary, Trans. Amer. Math. Soc. 142 (1969), 191–199. MR 246303, DOI 10.1090/S0002-9947-1969-0246303-2
Jonathan L. Gross, The decomposition of $3$-manifolds with several boundary components, Trans. Amer. Math. Soc. 147 (1970), 561–572. MR 258047, DOI 10.1090/S0002-9947-1970-0258047-X
Jonathan L. Gross, An infinite class of irreducible homology $3$-spheres, Proc. Amer. Math. Soc. 25 (1970), 173–176. MR 268895, DOI 10.1090/S0002-9939-1970-0268895-3
J. Milnor, A unique decomposition theorem for $3$-manifolds, Amer. J. Math. 84 (1962), 1–7. MR 142125, DOI 10.2307/2372800
C. D. Papakyriakopoulos, On solid tori, Proc. London Math. Soc. (3) 7 (1957), 281–299. MR 87944, DOI 10.1112/plms/s3-7.1.281
C. D. Papakyriakopoulos, On Dehn’s lemma and the asphericity of knots, Ann. of Math. (2) 66 (1957), 1–26. MR 90053, DOI 10.2307/1970113
Friedhelm Waldhausen, On irreducible $3$-manifolds which are sufficiently large, Ann. of Math. (2) 87 (1968), 56–88. MR 224099, DOI 10.2307/1970594