I am trying to do an exercise 2.1.8 from Allen Hatcher : "Algebraic Topology". We want to compute $\Delta$-homology groups of the following $\Delta$-complex:
We have $n$ 3-simplexes $T_{i}$ on the picture above. After that we glue the lower face of $T_i$ with the upper face of $T_{i+1}$ for each $i$. First of all i am trying to understand how much simplexes we will have after gluing. Before gluing, we have $n$ 3-simplexes, $3n$ 2-simplexes, $3n+1$ 1-simplexes and $n+2$ 0-simplexes. After gluing, we have $n$ 3-simplexes, $2n$ 2-simplexes, $n+2$ 1-simplexes and $2$ 0-simplexes. However, seems there is a mistake, because when i try to compute homology groups they are no even close to those that must be. What is wrong there? Am i correct in computing the number of simplexes?
Everything is OK, i correctly computed number of simplexes, the problem was that i misunderstood the notation $Z_n$: i thought that $Z_n$ means $Z\times Z\times Z...\times Z$, but it means $Z/nZ$.