relative persistent homology of a filtration of a 2-sphere cell complex

Question

I am reading this paper and am trying to understand the relative persistent homology of the 2-sphere cell complex filtration shown above.

I am not familar with how to compute relative homology. Please help me understand the relative homology groups.

It is stated that $Pers(H_*(S_6,\mathbb{S}))= \{[-\infty,1)_0, [2,3)_1, [4,5)_2, [-\infty,6)_2\}$

meaning that:

at step 1, the relative homology group $H_0(\mathbb{S},point)$ is zero. In fact, the zeroth relative homology stays 0? see: here.

at step 2, we create a new 1-dimensional homology class. Is $H_1(\mathbb{S}, two points)$ 1 dimensional?

at step 3, that 1-class in step 2 is destroyed. Is $H_1(\mathbb{S}, edge)$= 0?

at step 4, we create two new 2-dimensional void

at step 5 we destroy one of the 2-dimensional void.

finally at step 6, we kill the original void of $\mathbb{S}$

How do you compute relative homology in general? Are my above interpretations correct?

---

edit: I believe $H_2(\mathbb{S}, S_1)$= $H_2(\mathbb{S}, S_2)$=$H_2(\mathbb{S}, S_3)$ all have dimension 1 ? so only one new void is created at step 4?

Answer 1

One of the cool things about topology is that you can often think algebraically, as described by Randall above, or you can often think geometrically, as I'll describe below. Let $X$ be a space and let $A\subseteq X$ be a subspace. If the pair of spaces $(X,A)$ is "a good pair", then the relative homology $H_*(X,A)$ is isomorphic to the reduced homology $\tilde{H}_*(X/A)$ of the quotient space $X/A$. Indeed, see Proposition 2.22 of Hatcher's book "Algebraic Topology". We now consider the above example.

• Note that $\mathbf{S}/S_1$ is homeomorphic to the 2-sphere $S^2$, hence contributing to the interval $[-\infty,6)_2$.

• Note that $\mathbf{S}/S_2$ is homeomorphic to the wedge sum $S^2\vee S^1$ of a 2-sphere with a 1-sphere, hence contributing to the interval $[-\infty,6)_2$ and also to the interval $[2,3)_1$.

• Note that $\mathbf{S}/S_3$ is homeomorphic to the 2-sphere $S^2$, hence contributing to the interval $[-\infty,6)_2$.

• Note that $\mathbf{S}/S_4$ is homeomorphic to the wedge sum $S^2\vee S^2$ of two 2-spheres, hence contributing to the intervals $[-\infty,6)_2$ and $[4,5)_2$.

• Note that $\mathbf{S}/S_5$ is homeomorphic to the 2-sphere $S^2$, hence contributing to the interval $[-\infty,6)_2$.

• Note that $\mathbf{S}/S_6$ is homeomorphic to the single point $*$, and hence all persistent homology intervals have ended by filtration parameter 6.