For an $n$dimensional orientable closed manifold $M$, the simplicial volume is the infimum of the $l^1$norm of the elements $\sum a_i \sigma_i$ ($a_i \in \mathbb{R}$) which represent the fundamental class. My question is: since $\sigma_i$ is a continuous map from the $n$dimensional simplex to $M$, what does $a_i \sigma_i$ mean? Is it the same kind of map? I don't know how to make sense of this expression.

closed as too localized by Fernando Muro, Anton Petrunin, Misha, Mark Grant, Dmitri Pavlov May 23 '13 at 3:33
This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.
It's only a formal (i.e. symbolic) sum, understood as an element of the $\mathbb R$module generated by the simplexes $\sigma_i$ (similar to vectors from a basis in linear algebra), where addition is by definition componentwise, commutative and distributive. It does not possess a clear geometrical meaning. In particular it does not represent a map into $M$. Try to look for references on singular homology. 


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. 

