A generalization of the Cheeger-Gromoll splitting theorem

Gregory J. Galloway

doi:10.1007/bf01191365

A generalization of the Cheeger-Gromoll splitting theorem

Archiv der Mathematik ◽

10.1007/bf01191365 ◽

1986 ◽

Vol 47 (4) ◽

pp. 372-375 ◽

Cited By ~ 2

Author(s):

Gregory J. Galloway

Keyword(s):

Splitting Theorem

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The universal splitting property. II

Journal of Symbolic Logic ◽

10.2307/2274097 ◽

1984 ◽

Vol 49 (1) ◽

pp. 137-150 ◽

Cited By ~ 11

Author(s):

M. Lerman ◽

J. B. Remmel

Keyword(s):

Turing Degree ◽

Splitting Theorem ◽

Splitting Property ◽

Exhaustive Study

We say that a pair of r.e. sets B and C split an r.e. set A if B ∩ C = ∅ and B ∪ C = A. Friedberg [F] was the first to study the degrees of splittings of r.e. sets. He showed that every nonrecursive r.e. set A has a splitting into nonrecursive sets. Generalizations and strengthenings of Friedberg's result were obtained by Sacks [Sa2], Owings [O], and Morley and Soare [MS].The question which motivated both [LR] and this paper is the determination of possible degrees of splittings of A. It is easy to see that if B and C split A, then both B and C are Turing reducible to A (written B ≤TA and C ≤TA). The Sacks splitting theorem [Sa2] is a result in this direction, as are results by Lachlan and Ladner on mitotic and nonmitotic sets. Call an r.e. set A mitotic if there is a splitting B and C of A such that both B and C have the same Turing degree as A; A is nonmitotic otherwise. Lachlan [Lac] showed that nonmitotic sets exist, and Ladner [Lad1], [Lad2] carried out an exhaustive study of the degrees of mitotic sets.The Sacks splitting theorem [Sa2] shows that if A is r.e. and nonrecursive, then there are r.e. sets B and C splitting A such that B <TA and C <TA. Since B is r.e. and nonrecursive, we can now split B and continue in this manner to produce infinitely many r.e. degrees below the degree of A which are degrees of sets forming part of a splitting of A. We say that an r.e. set A has the universal splitting property (USP) if for any r.e. set D ≤T A, there is a splitting B and C of A such that B and D are Turing equivalent (written B ≡TD).

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