I have already seen an answer to this question, which both places I have found it, it was done pictorially, something I consider useless due to a lack of scalability.(How to learn to deal with problems in the scaling where such pictures fail if you use all the easy examples up by pictures?) I would like to check a few notions of mine:
The question: Consider $[v_0,v_1,v_2]$ a two dimensional simplicial complex and identify $[v_0,v_1]\sim [v_1,v_2]$ preserving orientation.
It seems then we are also identifying $v_0\sim v_1,v_1\sim v_2$ and so have that $\Delta_0(X)=\langle [v_0]\rangle$ and $\Delta_1(X)=\langle [v_0,v_1],[v_0,v_2]\rangle$, so two loops on $v_0$ (since $v_0\sim v_1\sim v_2$), and $\Delta_2(X)=\langle L\rangle$ where $L$ is the $2$-cell. How do I find the boundary of $L$? $L$ seems to be broken up into two components (disconnected) and has boundary $\langle a,b\rangle$ since I can take a cycle around either independently. Is that correct? Although then with the $2$-cell I can retract this to a point?
Secondly if $\Delta_0(X)\cong \Bbb Z$ as above, this means $H_0(X)=\Bbb Z$, but by having two path components it should be $\Bbb Z\times \Bbb Z$? Or because of the $2$-cell I have 1 path component, rather than the two from Van Kampen?