I've been referring to this set of notes on persistent homology, and am confused with the definition and intuition for the death of a homology class for the persistent homology of a filtration.
Given a filtration $X^0 \subseteq X^1 \subseteq X^2\subseteq \cdots$ of simplicial complexes, the inclusions induce homomorphisms on homology grousp $f^{i, j}_n : H^i_n(X^i) \rightarrow H^j_n(X^j)$, and we define $H^{i, j}_n = im(f^{i,j}_n)$. Standard stuff.
I am confused about this definition about death (pg 3, defn 39):
A $p$-th homology class $[c]$ born at stage $i$ dies entering $X_j$ if it merges with a class born earlier. Formally, if $f^{i, j-1}_p([c]) \not \in H^{i-1, j-1}_p$, but $f^{i, j}([c]) \in H^{i, j-1}_p$
My doubt isn't necessarily about the definition itself, but rather the intuition/reasoning behind the definition. As I understand it, we want to mark the death of a homology class born at stage $i$ when it becomes trivial with respect to some stage $j$. The best example I can think of is conducting a Rips complex sequence on a point cloud of data. Say at stage $\epsilon_i$, we have a $n$-hole in complex, corresponding to some $[c]$ in the $n$-th homology of $X^i$. At some future stage $j$, when $\epsilon_j$ becomes big enough, this $n$-hole gets filled with an $n$-simplex (when all the vertices are <$\epsilon_j$ away from each other). Thus the $n$-hole dies because, well, there no longer is an $n$-hole. So $[c]$ is trivial with respect to $X^j$'s $n$th homology. Is this the wrong interpretation?
I don't see how this matches up with the definition above, which is measuring when the homology class merges with some homology class before. What is our $n$-hole merging with? Isn't it simply disappearing?
A similar example: suppose at some stage $i$ we have a cluster of points $B$, which shows up a single connected component in $0$th homology group. As our distance parameter $\epsilon$ grows bigger, we will eventually at some future stage disappear by overlapping with some other component, and now we're left with one less component; i.e, the homology element associated with $B$ disappears, and thus "dies". I don't see how this squares with the idea of merging with something that was already present previously.
To put it succinctly, in what sense is $[c]$ dying if it is still a (possibly) non trivial element of $H^{i, j-1}_p$?
So my question is: Am I completely wrong in my interpretation? Are the two interpretations equivalent and if so, how?
Best Answer
Your first example is of a class that "merged" with the zero class. That is, it's a smallest $j$ such that $f^{i,j}([c])=0$. Of course it is in $H^{i-1,j}$ then, since it is in the image of the "ancient" class $0$. If it is not in the image of $0$ at any previous stages it may have not died until then - thus making its death time $j$. But it could have died earlier, by merging with another, non-zero class.
This actually happens in your second example. Suppose, for radical simplicity, that we just start with two points, $p_1$ and $p_2$, and at stage $j$ they get connected by a segment. We could say that the class $[p_1-p_2]$ has merged the class $[0]$, thus giving us the death of $[p_1-p_2]$ at time $j$. But what about the classes $[p_1]$ and $[p_2]$? Neither of them got killed, they just merged. As noted in the notes you linked, one can choose to say that the younger of $p_1$ and $p_2$ is killed, and the older survives. Now, this only works if there is a younger and and older. The notes you linked do say "for simplicity we assume that we add one simplex at a time into a filtration" - but unfortunately they only say this after the definition of death. However, without this the definition will indeed be deficient -- the two points having been born simultaneously will never die, which is not what we want! But if all the ages are distinct then this is perfectly fine. Of course one could artificially impose the "one at a time" condition, at least for finite complexes -- just insert artificially some extra timestamps and "split" adding of all the new simplexes at some time $t$ into multiple additions, one by one.