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## Main Question or Discussion Point

I was working on some algebraic topology matters, thinkgs like the connected sum of some surfaces is some other surface. And for this study, I was using the Munkres's famous textbook 'Topology' the algebraic topology part. My qeustions are as follows:

Q1) Munkres introduces 'labelling scheme' and 'cutting and pasting' techniques. And he uses them in the proofs of the theorems. But I don't think this is a rigorous way for several reasons:

Reason 1) Not every surface can be clearly labelled by 'labelling shceme'. Among the examples are Mobius band and annulus. Actually, Munkres states that Mobius band can be labelled as [itex]abcd[/itex] whereas I think it should be [itex] aba [/itex] because it has only one actual boundary. Especially when I think of this together with 'cutting and pasting' method, I feel more confident the latter is the case as the mobius band is the connected sum of the projective plane and disk. And especially for annulus, I don't know how to label this though I know one way to do it, but it brings kind of a contradction. As for this contradiction, I will explain below.

Reason 2) The method of 'cutting and pasting' seems not rigorous; it gives only a scratch of the description of this method, and only intuitive, not formalistic at all. And this method, it seems, makes some contradiction related with the point mentioned above about annulus, which I explain right now: Considering a 'kind of' proof given in the page 462, which is of the fact that Klein bottle is the connected sum of two projective planes, I followed the method described here exactly to construct annulus as the connected sum of two disks. Thus setting [itex]D[/itex] as a disk and putting two of them as [itex] D = a, D=b [/itex], I get [itex] ab [/itex] as the conncted sum of two disks. (I know this application is not rigorous at all because the textbook itself does not define connected sum for general cases except for intuitive uses.) But from this I cannot obtain $$ ac^{-1}bc $$ which is also an annulus.

Sorry for long reasons. Summarizing questions:

Q1-1) Do you think annulus can be labelled by 'labelling scheme'? If can, what is it? And what about Mobius band?

Q1-2) Do you think my usage of 'cutting and pasting' for the construction of annulus is valid? If not, where am I wrong?

Q2) Are there any textbooks or papers that uses that method of 'cutting and pasting' in a rigorous way, (as defining the theory very clearly)?

Q3) For actual algebraic topologists, what is the general formalistic way of defining the meaning of connected sum and proving some sufrace is the connected sum of some surfaces? (You don't have to give me full description; just give me some hints or reference texts.)

Q4) Do you think the last second-half of Munkres's text is rigorous? I mean, does it contain no error at all (except for typos)? Or maybe I should ask you rather, are there many schools that use this book for introductory algebraic courses? Do you think it is good for this type of course?

Q5) Could you give me good reference texts in this topic, one for intutive udnerstanding and one for complete formalistic construction of the theory?

Q1) Munkres introduces 'labelling scheme' and 'cutting and pasting' techniques. And he uses them in the proofs of the theorems. But I don't think this is a rigorous way for several reasons:

Reason 1) Not every surface can be clearly labelled by 'labelling shceme'. Among the examples are Mobius band and annulus. Actually, Munkres states that Mobius band can be labelled as [itex]abcd[/itex] whereas I think it should be [itex] aba [/itex] because it has only one actual boundary. Especially when I think of this together with 'cutting and pasting' method, I feel more confident the latter is the case as the mobius band is the connected sum of the projective plane and disk. And especially for annulus, I don't know how to label this though I know one way to do it, but it brings kind of a contradction. As for this contradiction, I will explain below.

Reason 2) The method of 'cutting and pasting' seems not rigorous; it gives only a scratch of the description of this method, and only intuitive, not formalistic at all. And this method, it seems, makes some contradiction related with the point mentioned above about annulus, which I explain right now: Considering a 'kind of' proof given in the page 462, which is of the fact that Klein bottle is the connected sum of two projective planes, I followed the method described here exactly to construct annulus as the connected sum of two disks. Thus setting [itex]D[/itex] as a disk and putting two of them as [itex] D = a, D=b [/itex], I get [itex] ab [/itex] as the conncted sum of two disks. (I know this application is not rigorous at all because the textbook itself does not define connected sum for general cases except for intuitive uses.) But from this I cannot obtain $$ ac^{-1}bc $$ which is also an annulus.

Sorry for long reasons. Summarizing questions:

Q1-1) Do you think annulus can be labelled by 'labelling scheme'? If can, what is it? And what about Mobius band?

Q1-2) Do you think my usage of 'cutting and pasting' for the construction of annulus is valid? If not, where am I wrong?

Q2) Are there any textbooks or papers that uses that method of 'cutting and pasting' in a rigorous way, (as defining the theory very clearly)?

Q3) For actual algebraic topologists, what is the general formalistic way of defining the meaning of connected sum and proving some sufrace is the connected sum of some surfaces? (You don't have to give me full description; just give me some hints or reference texts.)

Q4) Do you think the last second-half of Munkres's text is rigorous? I mean, does it contain no error at all (except for typos)? Or maybe I should ask you rather, are there many schools that use this book for introductory algebraic courses? Do you think it is good for this type of course?

Q5) Could you give me good reference texts in this topic, one for intutive udnerstanding and one for complete formalistic construction of the theory?