where is the integer coefficient of basis element in a given formal sum, the first sum is in , and the second sum is in . This surjection is the unique group homomorphism which extends the function , and so its construction can be seen as an instance of the universal property. When and are as above, the kernel of the surjection from to is also free abelian, as it is a subgroup of (the subgroup of elements mapped to the identity). Therefore, these groups form a short exact sequence in which and are both free abelian and is isomorphic to the factor group . This is a free resolution of . Furthermore, assuming the axiom of choice, the free abelian groups are precisely the projective objects in the category of abelian groups. Applications Algebraic topology In algebraic topology, a formal sum of -dimensional simplices is called a -chain, and the free abelian group having a collection of -simplices as its basis is called a chain group. The simplices are generally taken from some topological space, for instance as the set of -simplices in a simplicial complex, or the set of singular -simplices in a manifold. Any -dimensional simplex has a boundary that can be represented as a formal sum of -dimensional simplices, and the universal property of free abelian groups allows this boundary operator to be extended to a group homomorphism from -chains to -chains. The system of chain groups linked by boundary operators in this way forms a chain complex, and the study of chain complexes forms the basis of homology theory. Algebraic geometry and complex analysis Every rational function over the complex numbers can be associated with a signed multiset of complex numbers , the zeros and poles of the function (points where its value is zero or infinite). The multiplicity of a point in this multiset is its order as a zero of the function, or the negation of its order as a pole. Then the function itself can be recovered from this data, up to a scalar factor, as If these multisets are interpreted as members of a free abelian group over the complex numbers, then the product or quotient of two rational functions corresponds to the sum or difference of two group members. Thus, the multiplicative group of rational functions can be factored into the multiplicative group of complex numbers (the associated scalar factors