0
$\begingroup$

enter image description here

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?

$\endgroup$
  • $\begingroup$ Do you know exact sequences? $\endgroup$ – Randall Dec 12 '18 at 3:10
  • $\begingroup$ yes, how do I use them here? $\endgroup$ – user352102 Dec 12 '18 at 3:12
  • $\begingroup$ It's the easiest way to get at the relative homology. Look up the long exact sequence of a pair. $\endgroup$ – Randall Dec 12 '18 at 3:18
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.