I have figured out how to use Mayer Vietoris to find cohomology of $T^2$ here. For a punctured torus, following the hint, let $U$ = $T^2\setminus S^1_1$ and $V$ = $T^2\setminus S^1_2$ such the $S^1_1\cap S^1_2$ is the puncture point. Am I correct in saying $H^0(U)=H^0(V)=H^1(U)=H^1(V) = \mathbb{R}$, $H^2(U)=H^2(V)=0$? Also may I know what $U\cap V$ is diffeomorphic to so as to find $H^0(U\cap V)$ and $H^1(U\cap V)$? $U\cap V = T^2\setminus S_1^1\cup S_2^1$.
Asked
Active
Viewed 833 times
0
-
How about $U$ and $V$ each being the torus minus a circle, with the two circles meeting at your favourite point? – Angina Seng May 29 '19 at 04:34
-
That would be perfect. Let me try to attempt cohomology and update the question. – trickymaverick May 29 '19 at 04:37
-
This question and answer might help you. – jgon May 29 '19 at 04:38
-
The approach outlined in that question/answer is to apply Mayer-Vietoris to $T^2$ with the open sets being $U=T^2\setminus {p}$ and $V$ a small open neighborhood of $p$ homeomorphic to a disk. – jgon May 29 '19 at 04:40
-
@jgon Thanks! Let me try to find the cohomology with this. – trickymaverick May 29 '19 at 04:42
-
@manifolded Either way will work, Lord Shark the Unknown's strategy is also good. Indeed, for this particular case, it might be the easiest strategy. The method I suggested is more general, but a little more work for this particular case. – jgon May 29 '19 at 04:43
-
@jgon true, but the sets looks simpler to find cohomologies for with your link. Would $U\cap V = S^1$ in that case? – trickymaverick May 29 '19 at 04:46
-
@manifolded Yes, well, whichever way you'd like to go about it I guess. – jgon May 29 '19 at 04:48