What invariants are of great concern in the field of 3-manifolds and why? How much do we know about them? [closed]

I am curious about 3-manifolds though I know little.

Here I am trying to know what invariants people in this field are interested in.

The following are what I have known and what I particularly want to know. It is also appreciated if anybody provides other information related to the question.

1. It seems that cohomology rings, say, with coefficients in $\mathbb{Z}$ are not quite concerned with (Am I right?). Why? I know that there is little information in the group structure. Does the ring structure give much information? I notice that the homology sphere has a trivial $H^2$ and therefore the multiplication is trivial in this case. But is this phenomenon general?

2. Why are people so interested in the volume (or Gromov norm) of hyperbolic 3-manifolds? Is this because the volume contains much information? I know for any given positive real number $x$, there are finitely many hyperbolic 3-manifolds with volume equal to $x$. And is this invariant well understood? By "well understood", I mean, for example, how does the volume influence the topology, how is it related to other invariants (e.g. the volume conjecture)? Also, has anything new and notable been developed in the recent work on the smallest volume of closed oriented hyperbolic 3-manifolds?

3. Is the fundamental group or Kleinian group still too complicated for us, given the recent work of Agol, Kahn--Markovic and Wise? Is the invariant field helpful to decipher the information encoded in the fundamental group?

• This is far too broad and unfocused. Nov 25, 2013 at 18:57
• You are right @Andy. But I really want to know these. Answers to any part of my question are helpful. Nov 25, 2013 at 19:34
• The trouble is that you are basically asking for a summary of all of 3-manifold topology. That's way too much for a MO question. Nov 25, 2013 at 19:35
• @Andy:Then is it okay if I confine my question to those three listed above and divide this question into three? Maybe the second one is still too broad? Nov 25, 2013 at 19:52
• The first question is probably too elementary (and no, there can definitely be a nontrivial ring structure to the cohomology ring; see my answer mathoverflow.net/a/4479/317; for easier examples, look at the cohomology ring of the 3-torus). The third question is too vague (what do you mean by "too complicated"?). The second question is the closest to a good question, but you should do some more reading to sharpen it before you ask it. Nov 25, 2013 at 19:58