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ddo

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## Homework Statement

I'm trying to calculate singular homology groups of the torus and Klein's bottle using the Mayer-Vietoris sequence.

## The Attempt at a Solution

I represent both spaces as a rectangle with identified edges. Then I take the sets:

U=rectangle without the boundary

V=rectangle without the middlepoint

so U is contractible thus H_n(U)=0 for n>0, H_0(U)=Z

V=S1vS1 so H_1(V)=ZxZ, H_n(V)=0

and their intersection = S1, H_n(S1)=0, H_1(S1)=Z, H_0(S1)=Z

Now from the M-V sequence for n>2 we get an exact sequence

0->0x0->H_n(T)->0, so H_n(T)=0.

But I don't know what to do for smaller n...

Please help!