Before I ask my question let me clarify some notation: $f^{i,j}_r$, where $i < j$, refers to the inclusion map $f: H_r(X_i) \hookrightarrow H_r(X_j)$. $X_i$ and $X_j$ are subcomplexes of a filtered complex $X$, where $X_i \subseteq X_j$.
I need clarification/explanation of the following definition. I was able to understand birth events easily, but I'm having some trouble with death events. Why do the two conditions below imply the end of a certain feature?
Suppose an $r$-class $\alpha$ is born at the $i$th level. Then $\alpha$ is said to die at level $j$ iff:
- $f^{i,j}_r(\alpha) \in im(f^{i-1, j}_r)$
- $f^{i,j-1}_r(\alpha) \notin im(f^{i-1, j-1}_r)$
A death event then refers to the death of all homology classes in the coset $\alpha + im(f^{i-1,i}_r)$.