Connected sum of 2 annuli (reading John. M. Lee's Introduction to topological manifolds)

Question

I am a beginner in topology and have been reading John M.Lee's Introduction to Topological Manifolds for the past 2 weeks. My question is concerned with the concept of connected sums, and in turn have other related questions. I have probably used incorrect terms unknowingly; please correct me.

My aim: To obtain the connected sum of 2 annuli in $\mathbb{R}^2$. I need a big picture understanding of the concept, and want to clarify if I'm on the right path, before ironing out the details.

What I've done: I'm able to define the topology of 1 annulus using open sets, by using the exponential quotient map. Chapter 6 of the book, under Connected Sums of Surfaces mentions that one needs to use a connected surface for the connected sum. So, I verify on pg. 90 of the book, under example 4.16(c) that $\mathbb{R}^2\setminus {0}$ is path-connected, and thus is connected. Since the annulus is homeomorphic or topologically equivalent to $\mathbb{R}^2\setminus {0}$, the annulus is also path-connected, and thus is connected. Thus, it is a valid candidate to be used for the connected sum operation.

Where I'm getting stuck: I'm having trouble defining the second annulus. I could define the second annulus in a manner similar to the first, and with a shifted origin. But, if that is so, it means that the annulus is located on the same space. Then that means that I am puncturing the already origin-punctured plane, yet again. Am I thinking correctly so far? If so, this topological space is not connected anymore; right? If not, how would I reason it? The confusion is that the annuli need to be a disjoint union of topological spaces, but I don't know how to treat them separately. They look like 2 annuli in the same space in my mind, so I'm confused about how each of their relationships to the punctured plane looks like. I think they should look like a disjoint union of annuli and equivalently a disjoint union of punctured planes; but because the punctured plane goes to infinity in both directions, I am now terribly confused.

What I think I should do: Define the second annulus the same way as the first using the exponential quotient map and the homeomorphism to $\mathbb{R}^2\setminus {0}$. Then, as part of the connected sum operation, remove an open coordinate ball (in this case a disc) from each annulus. It leaves behind a boundary homeomorphic to $\mathbb{S}^1$ in each. Then, when defining the equivalence class to connect the 2 annuli along that left-behind boundary, use:

1. A translation to map the two left-over boundaries to overlap. Translation is invariant under homeomorphism, so this should be fine.
2. How do I move the rest of the second annulus? The equivalence class applies only to the left-over boundary after the ball removal. So, can I define another map to map the remaining part of the annulus?
3. Once we have the quotient map, we verify that they do locally resemble a homeomorphic space, and if necessary, to preserve orientation, a composition of homeomorphisms can be used to achieve the desired adjunction space. Right?

Topology is not my background, so please excuse the crduity of my understanding/the thought process. I want to understand it systematically. Any help is greatly appreciated!

Thank you.

Answer 1

Here is a pointer in the correct direction. If you take the plane, and remove three distinct points, then the result is homeomorphic to the connect sum of a pair of annuli.

Answer 2

Disclaimer: I'm not sure about your definition of an annulus, so I assume you mean $S^1$. If you reject the temptation of puting everything down in coordinates, things might become clearer. I just read the definition of a connected sum, and it seems to me that the connected sum of two annuli would be another annuli, no? I imagined two rings and I cut out a small segment from each of them. Next I glue the two pairs of exposed points. Then I get another ring, which is an annulus in your words.