Are the ∑Wi,∑Ri\sum W_i,\sum R_i for H1H_1 here homologous?

The Δ\Delta-chain-complex XX has:
nn three-simplices T0,…,Tn−1T_0,…,T_{n-1}
2n2n two-simplices R0,…,Rn−1R_0,…,R_{n-1} (roof) and W1,…,Wn−1W_1,…,W_{n-1} (walls)
n+2n+2 one-simplices s0,…,sn−1s_0 , . . . , s_{n-1} (slanted), hh (horizontal), and vv (vertical)
22 zero-simplices xx and yy

The boundary maps are:
∂Ti=Wi−Wi−1+Ri−Ri+p\partial T_i=W_i-W_{i-1}+R_i-R_{i+p}
∂Ri=si−si−1+h,∂Wi=v−si+si+p\partial R_i =s_i -s_{i-1} +h, \partial W_i =v-s_i +s_{i+p}
∂si=y−x,∂h=0,∂v=0\partial s_i =y-x,\partial h=0,\partial v=0.

I think the the H2(X;Zn)H_2(X;\mathbb{Z}_n) here =Zn[∑Wi]⨁Zn[∑Ri]=\mathbb{Z}_n[\sum W_i]\bigoplus \mathbb{Z}_n[\sum R_i], but the H2(X;Zn)H_2(X;\mathbb{Z}_n) should be =Zn=\mathbb{Z}_n , i.e∑Wi,∑Ri\sum W_i,\sum R_i should be homologous, I feel confused, can you tell me where I went wrong? Thanks!

And the XX is in the below.




Is it possible to visualize this XX? If you just need to check the calculations with your explicit generators and differentials, you can use some computer algebra system…
– Alejo
Oct 21 at 2:41



@Alejo, thanks I have added the information of XX, I want to know where I went wrong.
– 6666
Oct 21 at 2:48



@Alejo, actually, I find the two cycles ∑Wi,∑Ri\sum W_i,\sum R_i for H1H_1, but the result is actually they are homologous, I can’t see why they are homologous.
– 6666
Oct 21 at 2:51



OK, I suspected it was something like that, but your initial description was confusing. Well, this exercise is basically about annoying calculations… A nice way to calculate the homology groups of lens spaces is something like this:
– Alejo
Oct 21 at 3:21



@Alejo Thank you!
– 6666
Oct 21 at 4:05