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

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?

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

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.