Homology of the Klein Bottle using M-V sequences

[quasar987]

About the proof of the homology of the Klein Bottle here: http://en.wikipedia.org/wiki/Mayer–Vietoris_sequence#Klein_bottle.

I do not see how the conclusion follows from the fact that we can "choose" {(1,0), (1,-1)} as a basis for Z²...

 

[matt grime]

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

 

[owlpride]

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?

 

[quasar987]

Thanks. I had forgotten that given a short exact sequence 0-->A-f->B-->C-->0, we have C=B/Im(f).

Actually, there is no need to talk about basis here since Im(alpha) is clearly just 2Z, so H_1(K)=(Z+Z)(2Z)=Z+Z/2Z.