# 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.

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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: rene.ma.utexas.edu/users/sadun/F10/prelim/hwsol11.pdf
– Alejo
Oct 21 at 3:21

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

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