A fellow student and I have been studying $\Delta$-complex homology using Hatcher's Algebraic Topology for the past few weeks, but we have hit a barrier in computing $H^\Delta_1(S^2)$, and we are wondering if we have substantial misunderstandings regarding $\Delta$-complex homology, since the results are not as we expect.
We used the $\Delta$-complex above, based on the gluing of the square along $a$ and $b$ as described above, for our computation. We obtain the chain complex
$0 \overset{d_3}\to \Delta_2 \overset{d_2}\to \Delta_1\overset{d_1}\to \Delta_0 \overset{d_0}\to 0$ i.e. $0 \overset{d_3}\to \langle U,L \rangle \overset{d_2}\to \langle a,b,c \rangle \overset{d_1}\to \langle v,w \rangle \overset{d_0}\to 0$
We observe that
- $d_2(U) = a + b - c$
- $d_2(L) = -a - b + c$
- $d_1(a) = w-v$
- $d_1(b) = v-w$
- $d_1(c) = v-v = 0$
And as we understand it, this implies that
- $\operatorname{im} d_2 = \langle a+b-c \rangle$, $\ker d_2 = \langle U+L \rangle$
- $\operatorname{im} d_1 = \langle v-w \rangle$, $\ker d_1 = \langle a+b,c\rangle$
Which in turn implies
- $H^\Delta_0(S^2) = \ker d_0 / \operatorname{im} d_1 = \langle v , w \rangle / \langle v-w \rangle = \langle \bar{v} \rangle \cong \mathbb{Z}$
- $H^\Delta_1(S^2) = \ker d_1 / \operatorname{im} d_2 = \langle a+b,c \rangle / \langle a+b-c \rangle = \langle a+b-c,c \rangle / \langle a+b-c \rangle= \langle \bar{c} \rangle \cong \mathbb{Z}$ (!?)
- $H^\Delta_2(S^2) = \ker d_2 / \operatorname{im} d_3 = \langle U+L \rangle / 0 \cong \mathbb{Z}$
However, of course, we would not expect $\mathbb{Z}$ to be the homology $H_1(S^2)$, but rather $0$; and after several days of attempts we haven't been able to identify our error or misunderstanding. All the more mystifying, this same technique worked perfectly fine for the examples in the textbook which Hatcher does provide, i.e. $\mathbb{T}$, $\mathbb{R}P^2$, and the Klein bottle, using the same method with their respective representations as gluings of a square. We also know that Mayer-Vietoris provides us an easier technique than this for computing these homologies, but wish to understand the concrete methods and why they don't seem to be yielding for us.
Are we misunderstanding something horribly? Are we applying $\Delta$-complex homology to an inappropriate situation or invalid $\Delta$-complex? Is there some obvious error we continue to overlook? And moreover, why is it that this technique seems to work for these other examples ($\mathbb{T}$, $\mathbb{R}P^2$, and $\mathbb{K}$) and yet not for the (presumably) much simpler case of $S^2$?