I'm having trouble understanding the definitions I've been reading, of what has been called an 'induced coboundary operator' or a 'connecting homomorphism' depending on what source you're reading.
Firstly, the what I've been working with is the Cech Homology groups are induced by the coboundary operator.
$$\check{H}^p(\mathcal{U},\mathscr{E}) = \frac{\ker(\delta:C^p(\mathcal{U},\mathscr{E}) \xrightarrow{} C^{p+1}(\mathcal{U},\mathscr{E}))}{\delta C^{p-1}(\mathcal{U},\mathscr{E})}$$
Where $\mathcal{U} = \{U_\alpha\}$ is a locally finite open cover of our manifold $M$.
Initially we start with a short exact sequence of Sheaves
$$0 \xrightarrow{} \mathscr{E} \xrightarrow{\alpha} \mathscr{F} \xrightarrow{\beta} \mathscr{G}\xrightarrow{} 0$$
This induces a map on the Cech cochain complexes
$$C^p(\mathcal{U},\mathscr{E}) \xrightarrow{\alpha} C^p(\mathcal{U},\mathscr{F}) \xrightarrow{\beta} C^p(\mathcal{U},\mathscr{G})$$
Which consequently induces maps on the cohomology groups
$$\check{H}^p(\mathcal{U},\mathscr{E}) \xrightarrow{\alpha_*} \check{H}^p(\mathcal{U},\mathscr{F}) \xrightarrow{\beta_*} \check{H}^p(\mathcal{U},\mathscr{G})$$
All these definitions are fine so far, the problem I encounter is the source I'm reading: "Griffiths & Harris – Principles of Algebraic Geometry", gives a very loose definition of the induced coboundary map
$$\delta_*:\check{H}^p(\mathcal{U},\mathscr{G}) \xrightarrow{} \check{H}^{p+1}(\mathcal{U},\mathscr{E})$$
The definition is contained within a proof of another theorem, it involves diagram chasing, yet is rather difficult to interpret.
Best Answer
I managed to adapt a proof I found in here involving relative homology groups.
I will append a subscript $p$ to denote the maps from the respective cochain complexes and cohomology groups.
We can define the map $\delta_{*p}: \check{H}^{p}(\mathcal{U},\mathscr{G})\xrightarrow{} \check{H}^{p+1}(\mathcal{U},\mathscr{E})$ by taking $\sigma \in C^{p}(\mathcal{U},\mathscr{G})$ with $\delta_p \sigma = 0$, thus representing a cohomology class $[\sigma] \in \check{H}^{p}(\mathcal{U},\mathscr{G})$.
Since the map $\beta_p$ is surjective, there is a $\tau \in C^{p}(\mathcal{U},\mathscr{F})$ such that $\beta_p(\tau) = \sigma$.
Now $\delta_p(\tau) \in \ker(\beta_{p+1})$, since the diagram is commutative and $$\beta_{p+1} \delta_p \tau = \delta_p \beta_p(\tau) = \delta_p \sigma = 0$$ Using the exactness of the rows and injectivity of $\alpha$, we can say that there exists a unique $\lambda \in C^{p+1}(\mathcal{U},\mathscr{E})$ such that $\alpha_{p+1} (\lambda) = \delta_p(\tau)$ and again using commutivity, we can see that $$\alpha_{p+2} \delta_{p+1} \lambda = \delta_{p+1} \alpha_{p+1} \lambda = \delta_{p+1} \delta_p \tau = 0$$ Again, given $\alpha$ is injective, this implies $\delta_{p+1} \lambda = 0$ and $[\lambda] \in \check{H}^{p+1}(\mathcal{U},\mathscr{E})$.
We define $\delta_{*p} [\sigma] = [\lambda] \in \check{H}^{p+1}(\mathcal{U},\mathscr{E})$ $\square$