What are some fields that intersect topology and number theory? I see that number theory is studied from the algebraic and analytics aspects, but I have not seen any approach from topology or axiomatic set theory (using them to investigate the properties or numbers and open problems in number theory).  What are some topics intersecting them?
 A: An arithmetic group is a group determined as the integer points of an algebraic group. One special type of hyperbolic manifolds that are at least somewhat better understood are the manifolds arising as $\Bbb H^3/G$ where $G$ is an arithmetic subgroup of $PSL(2,\Bbb C)$. 
Two such manifolds are commensurable (have common finite sheeted covers) if and only if two invariants agree. A number field and a quaternion algebra associated to the manifold.  (these invariants were generalized to all hyperbolic 3-manifolds in this very famous paper of Neumann and Reid http://www.math.columbia.edu/~neumann/preprints/nrarith.pdf)
For something slightly closer to my own interest, the study of symmetric bilinear forms over $\Bbb Z$ shows up consistently in 3 and 4-manifolds, but is really a part of number theory. For a 4-manifold $X$ the bilinear form (the "intersection form") $H^2(X) \times H^2(X) \to \Bbb Z$ given by the cup product and evaluating on the fundamental class is a very useful invariant.
Freedman proved that this invariant completely determines smooth simply-connected 4-manifolds up to homeomorphism. Donaldson famously showed that if such a form were positive definite (for a closed simply-connected 4-manifold), than the form was diagonizable over $\Bbb Z$.  The reproofs of Donaldson's result via Seiberg-Witten equations and Heegard Floer homology by Kronheimer and Mrowka, and Oszvath and Szabo respectively (https://arxiv.org/pdf/math/0110170v2.pdf for the HF one) in fact requires a nontrivial number theoretic result involving no topology by N. Elkies (https://arxiv.org/abs/math/9906019). It is still a very open question which of the forms can be realized as the intersection form of a closed orientable simply-connected 4-manifold.
A: The mapping class group of a surface connects to some rather deep number theory in more than one way. For example, via Grothendieck-Teichmüller theory. See Chapter 12 of this problem collection edited by Benson Farb.
A: Arithmetic geometry or arithmetic topology:
https://en.wikipedia.org/wiki/Arithmetic_topology
http://ocw.mit.edu/courses/mathematics/18-782-introduction-to-arithmetic-geometry-fall-2013/
In fact, analytic number theory can be roughly considered an intersection between topology (the analytic part) and number theory.
