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Could someone please check my homology computations and see if I have made a mistake in the following? Also, I'd appreciate it if I could sample your thoughts about the last note!

Problem Consider a 2-simplex $\Delta = [v_0, v_1, v_2]$ with edges $e_1 = [v_0,v_1]$, $e_2 = [v_1,v_2]$ and $e_3 = [v_0,v_2]$. Let $\Delta_1$ be obtained by identifying edges $e_1$ and $e_3$, and identifying $v_1$ and $v_2$. Compute the relative homology groups $H_i(\Delta_1,\{v_0\})$, $H_i(\Delta_1,\{v_0,v_1\})$ and $H_i(\Delta_1,\{v_0,v_1\}\cup e_2)$.

Solution

$\mathbf{H_i(\Delta_1,\{v_0\})}$ $$ \partial_2 \Delta_1 = e_1+e_2-e_1 = e_2\\ \partial_1 e_1 = v_1, \partial_1 e_2 = 0\\ \partial_0 v_1 = 0 $$ Thus, $$ H_2(\Delta_1,\{v_0\}) = 0\\ H_1(\Delta_1,\{v_0\}) = 0\\ H_0(\Delta_1,\{v_0\}) = 0 $$

$\mathbf{H_i(\Delta_1,\{v_0,v_1\})}$ $$ \partial_2 \Delta_1 = e_1+e_2-e_1 = e_2\\ \partial_1 e_1 = 0, \partial_1 e_2 = 0 $$ Thus, $$ H_2(\Delta_1,\{v_0,v_1\}) = 0\\ H_1(\Delta_1,\{v_0,v_1\}) = <e_1>\\ H_0(\Delta_1,\{v_0,v_1\}) = 0 $$

$\mathbf{H_i(\Delta_1,\{v_0,v_1\}\cup e_2)}$ $$ \partial_2 \Delta_1 = e_1-e_1 = 0\\ \partial_1 e_1 = 0 $$ Thus, $$ H_2(\Delta_1,\{v_0,v_1\}\cup e_2) = <\Delta_1>\\ H_1(\Delta_1,\{v_0,v_1\}\cup e_2) = <e_1>\\ H_0(\Delta_1,\{v_0,v_1\}\cup e_2) = 0 $$

Note:

  1. I see that $\{v_0\} \subset \{v_0,v_1\} \subset \{v_0,v_1\}\cup e_2$, and the homology groups are getting bigger and bigger. Because the boundary homomorphisms are mapping into smaller and smaller images, and therefore the kernel of the homomorphisms is bigger and bigger?

  2. Are $H_i(\Delta_1,\{v_0\})$ isomorphic to those for a cone? (I am guessing excision of $v_0$ is applicable here.)

  3. Are $H_i(\Delta_1,\{v_0,v_1\}\cup e_2)$ isomorphic to those for a sphere with two points identified (after excising the identified point)? Fundamental group of $S^2$ with north and south pole identified

Thank you.

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First of all, let me upload a picture I just have drawn and which, I think, will help us:

Your identifications

The picture represents the identifications you describe. Now let's begin with the computations from a conceptual approach.

  • $H_i(\Delta_1,\{v_0\})$ is the reduced homology of the cone, $\widetilde{H}_i(Cone)$, which is trivial in all dimensions since the cone is contractible. I think this answers your second question.
  • Regarding your third question, the response is affirmative and I think you can see it visually as a consequence of Theorem 2.13 combined with Proposition 2.22 in Hatcher. The moral of the story is that if you work with goods pairs (nice spaces and nice subspaces) (you can find the formal definition in [Theorem 2.13, Hatcher]) which is the case here, then relative homology "is like" reduced homology of the quotient of the space by the subspace.
  • Regarding the computations of $H_i(\Delta_1,\{v_0,v_1\})$ it is enough to use the idea of the previous point and you can see $H_i(\Delta_1,\{v_0,v_1\})$ as $\widetilde{H}_i(\mathbb{S}^1)$.

I hope this helps!

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