Remarks on Congruence of 3–manifolds

We give two proofs that the 3–torus is not weakly d–congruent to #S × S, if d > 2. We study how cohomology ring structure relates to weak congruence. We give an example of three 3–manifolds which are weakly 5–congruent but are not 5–congruent. Let d be an integer greater than one. In [G], we considered two equivalence relations generated by restricted surgeries on oriented closed 3–manifolds. Weak type–d surgery is q/ds Dehn surgery along a simple closed curve. Here s and q (which must be relatively prime to d and s) may vary but d is held fixed. The label q/ds indicating which surgery is given with respect to some meridional and a longitudal pair on the boundary of a solid torus neighborhood of the surgery curve. A meridian bounds a disk in the solid torus which meets the surgery curve transversely in one point, and a longitude meets the meridian transversely in one point in the boundary torus. The set of surgeries described as weak type–d surgeries does not depend on the choice of meridional and a longitudinal pair. If q = ±1 (mod d), we say the surgery is type–d surgery. This concept is also independent of the choice of meridian and longitude. The equivalence relation on the set of closed oriented 3–manifolds generated by weak type–d surgery is called weak d–congruence. The equivalence relation generated by type–d surgery is called d–congruence. The equivalence relation d–congruence is coaser [G] than an equivalence relation which was first considered by Lackenby [L] : congruence modulo d. It is not known that d–congruence is strickly coaser than congruence modulo d, but this seems likely. The notion of d-congruence of 3-manifolds is closely related to the notion of td-move (now called d-move) equivalence of links [P1, remark before proof of Theorem p.639]: A d-move between links implies that there is 1/d Dehn surgery relating the double branched covers of S along the links. Similarly, weak d-equivalence of 3-manifolds is closely related to rational move equivalence of links as analyzed in [P2, footnote 5], [P3, footnote 22] and [DP, DIP]. Completing this circle of ideas, we note that d-move equivalence of links is a special case of congruence modulo (d, q) of links due to Fox [F1]. Lackenby’s study of congruence modulo (d, q) of links lead him to define congruence modulo d of 3-manifolds. We will give two proofs of the following theorem. The first proof will use Burnside groups and second will use cohomology ring structure. We let T 3 denotes the 3– torus. Theorem 1. T 3 is not weakly d–congruent to #S × S for any d > 2. Date: November 15, 2007. partially supported by NSF-DMS-0604580.