In proving $\mathbb{R}^n$ minus finitely many points is simply connected for $n\geq 3$, we use the fact above. This is intuitively very much clear but it would be very helpful if someone can help me to find the explicit retraction map. I couldn't able to find the map anywhere . Any help is very much appreciated.
Asked
Active
Viewed 225 times
0
-
Are you aware of how to do it for a single puncture? The case of multiple punctures is similar, you just have to pick a point in $\mathbb{R}^n$ which is not co-linear with two of the punctures and then 'section off' $\mathbb{R}^n$ so that each partition contains exactly one puncture. Then, do the exact same procedure as for a single puncture (where you make sure the spheres you're retracting onto all have a point at the new point you've picked out), except now you need to make sure that the boundaries between partitions are all retracted onto the new point you've picked out. – Dan Rust Jun 19 '22 at 10:17
-
https://math.stackexchange.com/questions/1186956/prove-that-mathbb-r-n-without-a-finite-number-of-points-is-simply-connected . See a few of the answers here. – Mr.Gandalf Sauron Jun 19 '22 at 12:30