# Formal finite sum for integration on k-chains

This question is related to the integration of differential forms on chains, as exposed in the document at: http://www.math.upenn.edu/~ryblair/Math%20600/papers/Lec17.pdf . The author gives the following definitions:

Let U be an open subset of $\mathbb{R}^n$. A singular k-cube in U is a continuous map $c:[0,1]^k\rightarrow U$

A (singular) k-chain in U is a formal finite sum of singular k-cubes in U with integer coefficients, such as $2c^1+ 3c^2−4c^3$.

Such a decomposition is used later for the integration of a differential form $\omega$, in:

$\int_c \omega = \sum a_i\int_{c_i} \omega$

What is, in simple terms, the definition (or at least an intuition) of a formal finite sum of functions in this context ? Is a Free Abelian Group involved, with which group operation ?

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Singular $k$-chains form a free abelian group generated by singular $k$-cubes. You can define integral of differential forms over singular $k$-cubes, and the integral over chains is just the $\mathbb{Z}$-linear extension of the integral maps over cubes.
OK, but what is the meaning of $2c^1$ for example ? How is it related to $c^1$ ? – vkubicki Mar 31 '13 at 10:07
As $c_i$ are functions, what is the group operation ? – vkubicki Mar 31 '13 at 10:42
@johndeas, "formal" means that we consider expressions of the form $\sum n_ic_i$, where $n_i$ are integers and $c_i$ are your cubes. The group operation does not change $c_i$, we merely add up the coefficients of $c_i$. An analogy would the free abelian group generated by fruits. An element in this group could then be "1 apple + 1 orange", and the sum of "1 apple + 1 orange" + "3 orange + 1 kiwi + 1 banana" is just "1 apple + 4 orange + 1 kiwi + 1 banana". – user27126 Mar 31 '13 at 15:58
I am sorry I still do not get it. What kind of object is $2c^1$ for example, if it is not a function ? – vkubicki Mar 31 '13 at 23:19
In the case of $2c^1$, does this mean that you integrate two times over the same place ? – vkubicki Mar 31 '13 at 23:36