Is there an "arithmetic cobordism category"?

Question

This question is a clumsy attempt to apply a certain analogy. I hope that if the answer is negative it comes with a clarification of the scope and limitations of the analogy.

Arithmetic topology is based on an analogy between number fields and 3-manifolds where primes are something like knots, the Legendre symbol is something like a linking number, etc. In quantum topology, on the other hand, one way to study 3-manifolds is to study 3d TQFTs, e.g. functors $Z : 3\text{Cob} \to \text{Vect}$. These functors assign to every 3-manifold, interpreted as a cobordism from the empty 2-manifold to itself, a morphism $k \to k$ where $k$ is the base field, and therefore give $k$-valued invariants of 3-manifolds.

If the analogy between number fields and 3-manifolds is strong enough, there might conceivably exist an "arithmetic cobordism category" whose morphisms are number fields and whose objects are... whatever boundaries of number fields are in arithmetic topology. (One might need to adapt this construction depending on whether number fields are considered to have "boundaries" at all.) It might conceivably be possible to adapt constructions of 3d TQFTs to the arithmetic case and therefore to find "quantum invariants" of number fields.

So is any construction like this possible, or am I just talking nonsense?

This post imported from StackExchange MathOverflow at 2014-09-25 20:29 (UTC), posted by SE-user Qiaochu Yuan

Comments

Something TQFT-like in number theory cropped up here - londonnumbertheory.wordpress.com/2010/05/10/…

This post imported from StackExchange MathOverflow at 2014-09-25 20:29 (UTC), posted by SE-user David Corfield

---

This goes into that direction: math.uiuc.edu/K-theory/0547

This post imported from StackExchange MathOverflow at 2014-09-25 20:29 (UTC), posted by SE-user Thomas Riepe

---

In the analogy 3manifolds = numberfields, 1-manifolds = finitefields, you're asking 2manifolds = ?. You could also ask 0manifolds = ?.

This post imported from StackExchange MathOverflow at 2014-09-25 20:29 (UTC), posted by SE-user André Henriques

---

Number fields are not (compact) 3-manifolds. Rings of integers and S-integers are. Local fields are 2-manifolds, the boundary around a knot. Rings of integers in them are the tubular neighborhood of the knot. That doesn't give a lot of 2-manifolds to work with. Not enough for Heegaard splittings. But you could try to approach the Casson invariant some other way, without mentioning Heegaard splittings or the TQFT more generally.

This post imported from StackExchange MathOverflow at 2014-09-25 20:29 (UTC), posted by SE-user Ben Wieland