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I want to find the homology groups of the Klein bottle by Mayer-Vietoris. For this I want to describe the klein bottle as an adjunction space. I think it can written as a pushout $S^1\cup_f D^2$ but I don't know what $f$ should be.

How can I write the Klein bottle as an adjunction space ?

Please first give me a hint.

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The Klein bottle is the quotient of a square by the equivalence relation where you identify top and bottom, and left and right, but where one of those two identifications has a twist. If you look just at the boundary of the square, the identification gives you a wedge of two circles. So when you glue in the rest of the square you get an adjunction space $(S^1\vee S^1)\cup_f D^2$.

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  • $\begingroup$ I think $f$ should go form S^1 v S^1 to the S^1 but i can't write it .Could you give hint please ? $\endgroup$
    – bytrz
    Aug 5, 2015 at 8:41
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    $\begingroup$ @rmznyzgyr $f$ goes from $S^1$ to $S^1\vee S^1$. If you denote going around the first copy of $S^1$ by the symbol $a$ and going around the second copy by $b$, then the map is $aba^{-1}b$. $\endgroup$
    – Cheerful Parsnip
    Aug 5, 2015 at 18:06

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