Verify that R^3\{2 points} is homotopy equivalent to S^2 V S^2

• Nov 22nd 2011, 12:57 AM
alsn
Verify that R^3\{2 points} is homotopy equivalent to S^2 V S^2
I need to show that $\mathbb{R}^3\backslash A$ where $A$ is two distinct points is homotopically equivalent to $S^2\vee S^2$. I can see this geometrically but am finding it difficult to come up with an explicit proof. Trying to find a deformation retraction to the two spheres but I can't work out the actual equations. Help please!
• Nov 22nd 2011, 06:53 AM
Drexel28
Re: Verify that R^3\{2 points} is homotopy equivalent to S^2 V S^2
Quote:

Originally Posted by alsn
I need to show that $\mathbb{R}^3\backslash A$ where $A$ is two distinct points is homotopically equivalent to $S^2\vee S^2$. I can see this geometrically but am finding it difficult to come up with an explicit proof. Trying to find a deformation retraction to the two spheres but I can't work out the actual equations. Help please!

You're actually supposed to find the explicit deformation retract--blah, sounds like a Hatcher problem. Try to find an explicit map that does the following. First try extending from each of the balls outwards (in each direction not facing the other ball) a line and imagine that you are shrinking $\mathbb{R}^3$ along those lines by a factor of $t$, etc. From there you'll have to make the shapes right.
• Nov 22nd 2011, 12:02 PM
alsn
Re: Verify that R^3\{2 points} is homotopy equivalent to S^2 V S^2
Quote:

Originally Posted by Drexel28
You're actually supposed to find the explicit deformation retract--blah, sounds like a Hatcher problem. Try to find an explicit map that does the following. First try extending from each of the balls outwards (in each direction not facing the other ball) a line and imagine that you are shrinking $\mathbb{R}^3$ along those lines by a factor of $t$, etc. From there you'll have to make the shapes right.

I just emailed my lecturer and he said it doesn't have to be an explicit equation. pheww. so i'm guessing i just have to describe the deformation retraction... which seems pretty easy.
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