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I have the following question concerning excision: consider the torus $T^{2}$ and a disk $D^{2}$ in the torus. Is it possible to say, by excision, that $H_{*}(T^{2}, D^{2}) = H_{*}(T^{2} - D^{2})$? If yes, why? Because I don't see it since $\bar{D^{2}}$ is not contained in $int(D^{2})$. How can I apply excision to calculate $H_{*}(T^{2} - D^{2})$ ?

beno

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    how can I apply excision here to calculate for example $H_{*}(T^{2} - D^{2})$ ?2012-01-15
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    Remove a slightly smaller subset $S\subset D^2$, so that the hypothesis of the excision theorem hold *and* such that the inclusion $T^2-D^2\to T^2-S$ is an deformation retract.2012-01-15
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    what is $R^{2}$?2012-01-15
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    i just got $ H_{n}(T^{2}, D^{2}) = H_{n}(T^{2} - S, D^{2} - S) $, for all $n$. How to go on further?2012-01-15
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    can i do like this: since $(T^{2}-D^{2}, \emptyset) \rightarrow (T^{2}-S, D^{2}-S)$ is a deformation retract, then $H_{n}(T^{2}-S, D^{2}-S) = H_{n}(T^{2}-D^{2})$ ??2012-01-15
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    @beno: Are you allowed to use the long exact sequence of a pair?2012-01-15
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    sure. of which pair?2012-01-15

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