I think that my objection might be disjoint: My main problem is that 3-manifolds aren’t popping up all over mathematics and in the sciences, in diverse contexts. That is what I would expect of a “mathematical primitive”- I would expect it to make veiled appearances all over the place. Perhaps Monsterous Moonshine and its cousins “justify” sporatic simple groups. They are clearly more than just artifacts of suboptimal definitions. But I don’t see a parallel for 3-manifolds- the only place they seem to appear is inside geometric topology.

To sharpen the philosophical objection further, there are several related objects which DO appear in other contexts, but they’re no longer “just” 3-manifolds. For example, there is a whole industry devoted to the analogy between 3-manifolds and number fields, which would remove my objection, except that the 3-manifolds are equipped with a flow. I found the answers here, and linked references, quite illuminating in this respect.

So my personal speculation would be that 3-manifolds could perhaps be thought of as a partial manifestation of some more “fundamental” structure, such as some sort of spacial algebra of the sort that Habiro and Asaeda have looked at… or maybe something completely different. I really don’t know… Dror Bar-Natan may have vaguely similar doubts when during talks he says phrases like, “I don’t know what a 3-manifold is”.

]]>Thanks!

Fyodor Gainullin ]]>

For me, what would “convince” me that 3-manifolds “exist” in some philosophical sense would be if one could find a strong tie-in with the main body of mathematics whose “existence” is somehow well-established. For links we sort-of have this via quantum invariants, which provide insights into number theory (multiple zetas etc), field with one element, Lie theory etc. and also somehow reflect quite concrete physical objects which one could almost hold in one’s hand. I don’t know of a parallel sense in which 3-manifolds tie in with the rest of mathematics except as examples; for example, 3-manifold quantum invariants tend to come from link invariants, and the tie-ins with the rest of math happen already on the level of links (e.g. the MOO invariant, as discussed in the post).

]]>But I think this is exactly the kind of thing that makes mathematics entertaining. One studies PDEs not because one hopes to have a single unifying perspective on the subject that answers every question efficiently. You tend to study particular families that are useful. 3-manifolds are of that variety of question. They’re a useful family, rather than a generality. They’re also not so small that one can quickly bludgeon the subject to death.

]]>http://www.worldscientific.com/worldscibooks/10.1142/9348

New Ideas in Low Dimensional Topology

Edited by: Louis Kauffman & Vassily Olegovich Manturov

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http://www.worldscientific.com/worldscibooks/10.1142/9048

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