I am trying to calculate the de Rham cohomology of the connected sum of $S^1 \times S^3$ with $\mathbb CP^2$. But I have trouble in figuring out what is $S^1 \times S^3$ with a disk removed. If I have that figured out, I can proceed using Mayer Vietoris, since $\mathbb CP^2$ with a disk removed is simply $\mathbb CP^1$.
Asked
Active
Viewed 137 times
2
-
1Do you know what $S^1\times S^1$ with a disc removed is? – Michael Albanese Jun 06 '19 at 21:57
-
Yes, that deformation retracts to its boundary which is a torus. – penny Jun 07 '19 at 01:16
-
2No, $S^1\times S^1$ is a torus, it deformation retracts onto $S^1\vee S^1$. – Michael Albanese Jun 07 '19 at 01:20
-
Sorry, I was thing of the wedge sum of two circles but put in the wrong words... – penny Jun 07 '19 at 01:33
-
1OK, we're in agreement. So now, what do you think happens for $S^1\times S^3$? – Michael Albanese Jun 07 '19 at 01:42
-
They way I think about $S^1 \times S^1$ is to use its polygon representation. I am not sure how I can apply that to this case. – penny Jun 07 '19 at 02:30
-
This question and my answer may be useful to you. The trick discussed there is to apply Mayer-Vietoris to the interior of the punctured manifold and to an open ball slightly larger than the one you are removing. – jgon Jun 08 '19 at 01:45