# Homology of the Klein Bottle using M-V sequences

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matt grime
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The homology group is Z + Z quotiented by the image of alpha. In the given bases label them e,f, what is this? It is <e,f>/(2f=0) i.e. Z + Z/2Z

How do you compute the quotient

$$\frac{\ker(d_n)}{\text{im}(d_{n+1})}$$

? If you can express $$\ker(d_n)$$ and $$\text{im}(d_{n+1})$$ in terms of the same basis, then modding out is straight forward. That's why Wiki is choosing a non-standard basis for Z². Why don't you write out the sequence and the maps?

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