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

I need to show that where is two distinct points is homotopically equivalent to . 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!

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

where

is two distinct points is homotopically equivalent to

. 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 along those lines by a factor of , etc. From there you'll have to make the shapes right.

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

along those lines by a factor of

, 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.