Prime -manifolds and the doubling operation

Author:
Jonathan L. Gross

Journal:
Proc. Amer. Math. Soc. **27** (1971), 375-380

MSC:
Primary 57.01

MathSciNet review:
0271948

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

**[1]**Jonathan L. Gross,*A unique decomposition theorem for 3-manifolds with connected boundary*, Trans. Amer. Math. Soc.**142**(1969), 191–199. MR**0246303**, 10.1090/S0002-9947-1969-0246303-2**[2]**Jonathan L. Gross,*The decomposition of 3-manifolds with several boundary components.*, Trans. Amer. Math. Soc.**147**(1970), 561–572. MR**0258047**, 10.1090/S0002-9947-1970-0258047-X**[3]**Jonathan L. Gross,*An infinite class of irreducible homology 3-spheres*, Proc. Amer. Math. Soc.**25**(1970), 173–176. MR**0268895**, 10.1090/S0002-9939-1970-0268895-3**[4]**J. Milnor,*A unique decomposition theorem for 3-manifolds*, Amer. J. Math.**84**(1962), 1–7. MR**0142125****[5]**C. D. Papakyriakopoulos,*On solid tori*, Proc. London Math. Soc. (3)**7**(1957), 281–299. MR**0087944****[6]**C. D. Papakyriakopoulos,*On Dehn’s lemma and the asphericity of knots*, Ann. of Math. (2)**66**(1957), 1–26. MR**0090053****[7]**Friedhelm Waldhausen,*On irreducible 3-manifolds which are sufficiently large*, Ann. of Math. (2)**87**(1968), 56–88. MR**0224099**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1971-0271948-8

Keywords:
Connected sum,
pasting,
irreducible

Article copyright:
© Copyright 1971
American Mathematical Society