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A. Church, in his classical paper An unsolvable problem in elementary number theory in American Journal of Mathematics Vol. 58 No. 2. (1936), pp. 345-363, (available here), wrote:

There is a class of problems of elementary number theory which can be stated in the form that it is required to find an effectively calculable function $f$ of $n$ positive integers, such that $f(x_1,x_2,\dots,x_n)=2$ is a necessary and sufficient condition for the truth of a certain proposition of elementary number theory involving $x_1,x_2,\dots,x_n$ as free variables.

As an example he gave:

[...] [A] problem of this class is, for instance, the problem of topology, to find a complete set of effectively calculable invariants of closed three-dimensional simplicial manifolds under homeomorphisms. This problem can be interpreted as a problem of elementary number theory in view of the fact that topological complexes are representable by matrices of incidence. In fact, as is well known, the property of a set of incidence matrices that it represent a closed three-dimensional manifold, and the property of two sets of incidence matrices that they represent homeomorphic complexes, can both be described in purely number-theoretic terms.

I'd like to understand the gist of this example, although it's not needed in the rest of the paper. What's a "topological complex"? Is that a simplicial complex? In which sense can it be represented as incidence matrices, assuming it's the same concept from graph theory?

Carl Mummert
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Jacopo Notarstefano
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    I assume it's a simplicial complex, yes. The incidence matrices tell you which $n$-simplices are faces of which $(n+1)$-simplices, generalizing the incidence matrices of graphs (which tell you which vertices are faces of which edges). – Qiaochu Yuan Mar 24 '13 at 01:48

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Just so that this question does not show up as unanswered:

I assume it's a simplicial complex, yes. The incidence matrices tell you which n-simplices are faces of which (n+1)-simplices, generalizing the incidence matrices of graphs (which tell you which vertices are faces of which edges). – Qiaochu Yuan Mar 24, 2013 at 1:48

This is certainly correct, and the existence of the necessary matrices is a consequence of the assumption the complexes are finite.

Carl Mummert
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