TL;DR
This paper investigates the properties of 3-manifolds under weak congruence, providing proofs and examples that distinguish between different levels of congruence and their relation to cohomology structures.
Contribution
It offers new proofs regarding the non-congruence of specific 3-manifolds and presents examples illustrating differences between weak and strong congruence.
Findings
The 3-torus is not weakly d-congruent to the connected sum of three S^1xS^2's for d>2.
Cohomology ring structure relates to weak congruence properties.
Existence of 3-manifolds weakly 5-congruent but not 5-congruent.
Abstract
We give two proofs that the 3-torus is not weakly d-congruent to the connected sum of three S^1xS^2'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.
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