2
$\begingroup$

I'm looking for examples of 3-manifolds with unusual rational cohomology rings. I'm curious about what the cup product structure can actually look like, and I'd like some examples to play with. Does anyone have some suggestions about where to look? What's your favorite cohomology ring, do you know any manifolds realizing it, and have you run into any odd ones that, for whatever reason, caused a problem with something you were trying to prove?

I know there are some results about realizing a given cohomology ring (ex the arxiv post last month from Jim Fowler and Zhixu Su), which is not exactly what I'm looking for, but I'd be glad to hear if anyone has a favorite paper or result in this area.

Improve this question
$\endgroup$
1

2 Answers 2

Reset to default
3
$\begingroup$

Doesn't Sullivan's paper On the intersection ring of compact three manifolds, Topology 1975; 14(3):275-277 characterize the cohomology ring? I think he shows that any skew tri-linear form is realized. Turaev extended this (Cohomology rings, linking forms and invariants of spin structures in three-dimensional manifolds, Mat. Sb. (N.S.) 120(162) (1983), no. 1, 68–83) extends this considerably, giving realization results for the linking form on torsion.

Not every cohomology ring is realized by specific types of manifolds; eg there are restrictions on the cohomology rings of Seifert fibered spaces. See eg Cochran-Tanner, http://arxiv.org/abs/1207.5042. This paper has a number of nice examples.

Improve this answer
$\endgroup$
2
$\begingroup$

I'm not sure if this is quite what you're looking for, because:

(a) it's a $3$-manifold with boundary, and

(b) it's not the ring structure which is interesting;

but the Borromean Rings link complement is a $3$-manifold with interesting rational cohomology. All cup products are trivial, but it supports a non-trivial triple Massey product. This is an algebraic manifestation of the rings being linked but pairwise unlinked.

The original reference is the classic paper

W. S. Massey, Higher order linking numbers, in: Conference on Algebraic Topology (Chicago, 1968), 174–205.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .