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.
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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. 

