Example of induced homomorphism in algebraic topology I would like to understand what induced homomorphism are, as they appear in the definition of the Mayer-Vietoris sequence. Since an homology group $\tilde{H}_n$ is a quotient group defined as $\frac{\text{Ker } \partial_n}{\text{Im } \partial_{n+1}}$, how is the induced homomorphism related to the objects in $C_n$ ? The notations used are those of the Wikipedia article Singular Homology.
Could someone gives an explicit expression of an induced homomorphism in a very simple example ?

Given a continuous map $f : X \to Y$, there is an induced chain map $f_\# : C_n(X) \to C_n(Y)$ of singular chains. Since $C_n(X)$ is a free abelian group whose basis is the set of singular simplices $\sigma : \Delta^n \to X$, in order to define $f_\#$ all you need to do is to define it's value $f_\#(\sigma)$ on cach basis element, and then $f_\#$ is uniquely extended by linearity. Namely:
$$f_\#(\sigma) = f \circ\sigma : \Delta^n \to Y
$$
The key property of this induced map is the identity
$$\partial \circ f_\# = f_\# \circ \partial : C_n(X) \to C_{n-1}(Y)
$$
Once this identity is proved, the induced map on homology may be defined:
$$f_*[c] = [f_\#(c)]
$$
for each $n$-cycle $c \in C_n(X)$ with homology class $[c] \in H_n(X)$.
Here is an example: take the doubling map
$$f : S^1 \to S^1, \quad f(z)=z^2
$$
A cycle whose homology class generates $H_1(S^1)$ is the singular 1-simplex $\sigma : [0,1] \to S^1$ defined by $$\sigma(t) = e^{2 \pi i t}
$$ 
which wraps the interval once around the circle. The image of this cycle under $f_*$ is
$$f_*[\sigma] = [f_\#(\sigma)] = [f \circ \sigma]
$$
where
$$f \circ \sigma(t) = e^{4 \pi i t}
$$
which wraps the interval twice around the circle. From this you can conclude that $f_* : H_1(S^1) \to H_1(S^1)$ is multiplication by $2$ acting on $H_1(S^1) \approx \mathbb{Z}$.
Any algebraic topology textbook, such as Hatcher, will explain induced homomorphisms in this language.
