The analogy doesn't quite give a number theoretic version of the Poincare conjecture. See Sikora, "Analogies between group actions on 3-manifolds and number fields"
(arXiv:0107210): the author states the Poincare conjecture as "S3 is the only closed 3-manifold with no unbranched covers." The analogous statement in number theory is that Q is the only number field with no unramified extensions, and indeed he points out that there are a few known counterexamples, such as the imaginary quadratic fields with class number 1.
The paper also has a nice but short summary of the so-called "MKR dictionary" relating 3-manifolds to number fields in section 2. Morishita's expository article on the subject, arXiv:0904.3399, has more to say about what knot complements, meridians and longitudes, knot groups, etc. are, but I don't think there's an explanation of what knot surgery would be and so I'm not sure how Kirby calculus fits into the picture.
Edit: An article by B. Morin on Sikora's dictionary (and how it relates to Lichtenbaum's cohomology, p. 28): "he has given proofs of his results which are very different in the arithmetic and in the topological case. In this paper, we show how to provide a unified approach to the results in the two cases. For this we introduce an equivariant cohomology which satisfies a localization theorem. In particular, we obtain a satisfactory explanation for the coincidences between Sikora's formulas which leads us to clarify and to extend the dictionary of arithmetic topology."
I will take the question at face value, but not in the sense of justifying the definition.
A topological space is a convenient way of encoding, or perhaps better, organising, certain types of information. (Vague but true! I will give some instances. the data is sometimes `spatial' but more often than not, is not.)
Perhaps we should not think of spaces as 'god given' merely 'convenient', and there are variants that are more appropriate in various contexts.
A related question, coming from an old Shape Theorist (myself) is : when someone starts a theorem with 'Given a space $X$...', how is the space 'given'? As an algebraic topologist I sometimes need to use CW-complexes, but face the inconvenience that if I could give the CW structure precisely I could probably write down an algebraic model for its homotopy type precisely, and vice versa, so a good model is exactly the same as the one I started with. I hoped for more insight into what the space 'was' from my modelling. Giving the space is the end of the process, not the beginning. Strange. A space is a pseudo-visual way of thinking about 'data', which encodes important features, or at leastsome features that we can analyse, partially.
If someone gives me a compact subspace of $\mathbb{R}^n$, perhaps using some equations and inequalities, can I work out algebraic invariants of its homotopy type, rather than just its weak homotopy type? The answer will usually be no. Yet important properties of $C^*$ algebras on such a space, can sometimes be related to algebraic topological invariants of the homotopy type.
Spaces can arise as ways of encoding actual data as in topological data analysis, where there is a 'cloud' of data points and the practitioner is supposed to say something about the underlying space from which the data comes. There are finitely many data points, but no open sets given, they are for the data analyst to 'divine'.
Not all spatial data is conveniently modelled by spaces as such and directed spaces of various types have been proposed as models for changing data. Models for space-time are like this, but also models for concurrent systems.
Looking at finite topological spaces is again useful for encoding finite data (and I have rarely seen infinite amounts of data). For instance, relations between finite sets of data can be and are modelled in this way. Finite spaces give all homotopy types realisable by finite simplicial complexes. Finite spaces can be given precisely (provided they are not too big!) How do invariants of finite spaces appear in their structure? (Note the problem of infinite intersections does not arise here!!!)
At the other extreme, do we need points? Are locales not cleaner beasties and they can arise in lots of algebraic situations, again encoding algebraic information. Is a locale a space?
I repeat topological spaces are convenient, and in the examples you cite from algebraic geometry they happen to fit for good algebraic reasons. In other contexts they don't. Any Grothendieck topos looks like sheaves on a space, but the space involved will not usually be at all `nice' in the algebraic topological sense, so we use the topos and pretend it is a space, more or less.
Best Answer
Given a convex polytope whose facets are simplices, define the f-vector by f_i = the number of i-dim faces. Which vectors of integers are f-vectors? A list of conditions was conjectured, proven sufficients by direct construction of enough polytopes, and proven necessary by applying hard Lefschetz to the (rationally smooth) toric variety associated to the dual polytope. (A combinatorial proof came later.) See Fulton's book on toric varieties.