I am curious about the following statement :
Statement : Infinite connected sum of $S^n$ is homeomorphic to $\mathbb{R}^n.$
Any hint, proof of reference will be appreciated.
Thank you.
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What is your definition of an infinite connected sum? – Eric Wofsey Oct 14 '15 at 15:11
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The infinite connected sum should be homeomorphic to an infinite cylinder, not $\Bbb R^n$. See this from Tao's blog – Ben Grossmann Oct 14 '15 at 15:13
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2@Omnomnomnom: that would be a bi-infinite connected sum. There's different ways of taking an infinite connect sum. – Cheerful Parsnip Oct 14 '15 at 16:13
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1In fact, you could get a punctured sphere with arbitrarily many punctures $\geq 1$. – Cheerful Parsnip Oct 14 '15 at 16:15
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1In further fact, you can get $S^n\setminus C$ where $C$ is a Cantor set. :) – Cheerful Parsnip Oct 14 '15 at 18:23
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Think of a nested countable sequence of $(n-1)$-spheres in $\mathbb R^n$ starting with the unit sphere and going out to $\infty$. The annuli between each sphere are homeomorphic to $S^n$ with two disks removed. So this realizes the infinite connected sum.
Cheerful Parsnip
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Dear Cheerful Parsnip, you are saying that infinite connected sum is this: in each step $n$ we have connected sum of $n$ spheres and a cap removed from last one? Why not this one: $M_1=M, M_2=M# M,\ M_3=M# M# M,\ \dots\ M_n=M# M#\dots# M$ and $n\to\infty$? see this post – C.F.G Dec 23 '21 at 21:06
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@C.F.G no I am not saying that. I am saying line up a countable sequence of spheres in a line and then tube them together. As I mentioned above, infinite connected sum is actually not well defined in general. – Cheerful Parsnip Dec 24 '21 at 01:00