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The following is in the Bloch's Book: A First Course in Geometric Topology and Differential Geometry: Source: Google Book

and I couldn't find this in errata list. Is that correct? I found two posts here and here that indicate this is an obvious mistake.

Q2: What is the exact definition/realization of the infinite connected sum? I think one candidate could be like this: $$M_1=M, M_2=M\# M,\quad M_3=M\# M\# M,\quad\dots\quad M_n=M\# M\#\dots\# M$$ and tend $n\to \infty$ that seems that this is not the case. why?

C.F.G
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  • Sorry, it is late already, where exactly do you see a mistake? – Dietrich Burde Dec 23 '21 at 20:10
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    This infinite connected sum isn't even compact, is it? – Ayman Hourieh Dec 23 '21 at 20:12
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    The answer here may be of use to you. Q2: For the limit of partial connected sums to work, any connected sum that is eventually constant should converge to that constant -- which is what Bloch is claiming. The Question/Answers you link show this is not the case, so you will not arrive at this limit by partial connected sums. In short: What metric on the set of compact 2-manifolds makes these sequences of eventually constant partial connected sums into Cauchy sequences that do not converge to their constant? – Eric Towers Dec 23 '21 at 20:38
  • See also: Mazur Swindle and associated brief discussion by Terry Tao. – Eric Towers Dec 23 '21 at 20:44

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I agree with the first post you link that connect summing with an infinite amount of spheres would be the same as removing a disk, so as written the argument is not correct.

You can fix the argument though. We would have that $\mathbb{R}^2 \cong A - D^2$, so you just need to show that if $A$ is not a sphere $A -D^2$ can't be homeomorphic to $\mathbb{R}^2$. This follows from an application of what is called the Alexander trick (and it applies to all dimensions). The Alexander trick says you can extend a self homeomorphism of the boundary of a disk to the entire disk.

However, it does not work smoothly. But this is because the result you want to prove actually is not true smoothly. There exist exotic spheres which have inverses under connect sum.

Connor Malin
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