Question: I have the following setup:

There is a collection of items I and a collection of partial rankings V. That is, an element of V is a total ordering on a subset of I. There is no expectation of consistency among the elements of V: it may be that x < y for one element and y < x for another. 

I would like to assign a score $s : I \to \mathbb{R}$ which in some sense captures these rankings. That is, I would like s(x) < s(y) to mean "x tends to be less than y for elements of V which have both in their domain". I'm not sure of what a good way to do this is.

[Arrow's impossibility theorem](http://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem) puts some constraints on what can be achieved here, because given a set of votes and a scoring function like this we could use the scoring function to define a total order on the items, which is then constrained by the theorem.

I suppose I'm really looking for references rather than an answer to this question (although both would be appreciated): I'm sure there's a body of theory around this, but I have no idea what it is like or what it's called, so I'm at a bit of a loss as to where to start looking for a solution.

Answer: 
Any two compact surfaces (without boundary) of the same genus are diffeomorphic. However, if S is a surface of genus g > 0, there are uncountably many non-isomorphic complex (or, equivalently, algebraic) structures on S.
For a closed analytic subset Z ⊂ S of a (say compact) complex manifold 
with complement U=S-Z one has additivity of the (topological) Euler characteristic:

Χ(S)=Χ(Z)+Χ(U).

This is wrong for if S and Z are topological spaces or smooth manifolds.
Indeed, take for Z a point on a circle S.
This (surprising) difference was recently pointed out to me by Manfred Lehn.

Of course there is also no additivity of Poincare-Polynomials or other 
"motivic" invariants of complex varieties.