Let's say I have a Drupal website and I want there a function that would work like this:
Paste an URL to a .doc file, so on submit Drupal downloads that file, uploads to its local server and generates a link to download from the website.

I was wondering if this needs custom coding or can this be achieve with a module?


Question: What does a singular simplex with real coefficient mean 
Answer: As Lee suggests, you should look at Allen Hatcher's book.  In particular, chapter 2 and section 3.3.

In practice, you can often think of $\sum a_{i}\sigma_{i}$ as coming from the following procedure.  Let $\phi:X\rightarrow M$ be a degree $d$ map of $n$-manifolds.  Then any singular cocycle (with integer coefficients) $\alpha$ representing the fundamental class of $X$ will push forward to a singular cocycle $\phi_{\ast}\alpha$ representing $d$ times the fundamental class of $M$.  Thus, $\frac{1}{d}\phi_{\ast}\alpha$ will be a cocycle (with rational coefficients) representing the fundamental class of $M$.

The prototypical examples are as follows:

If $M$ is a torus, then there are covering maps $\phi: M\rightarrow M$ of arbitrarily high degree, hence $M$ has no simplicial volume.  (We can iterate the above procedure, starting with any cocycle $\alpha$ representing the fundamental class of $M$, getting representatives $\alpha,\frac{1}{d}\phi_{\ast}(\alpha),\frac{1}{d^{2}}\phi_{\ast}^{2}(\alpha),\ldots$ with volume going to $0$.)

On the other hand, if $M$ is a higher genus surface, then this strategy is obstructed by Euler characteristic.  Namely, if $\phi:Y\rightarrow M$ is a degree $d$ covering map, then $\chi(Y)=d\chi(X)$, so the genus of $Y$ grows linearly with $d$.  In fact, you can show that $X$ has nontrivial simplicial volume using hyperbolic geometry.