Algebraic Topology: Connected Sum & Reference Help

[julypraise]

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 $abcd$ whereas I think it should be $aba$ 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 $D$ as a disk and putting two of them as $D = a, D=b$, I get $ab$ 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?

 

[mathwonk]

a connected sum is an identification space or quotient topological space. to prove a certain construction gives another explicit surface one can compute the homology and use classification of surfaces. have you read andrew wallace on classification of surfaces? he is usually pretty clear. or seifert and threllfall?

 

[julypraise]

a connected sum is an identification space or quotient topological space. to prove a certain construction gives another explicit surface one can compute the homology and use classification of surfaces. have you read andrew wallace on classification of surfaces? he is usually pretty clear. or seifert and threllfall?

Thanks for the reply mathwonk! I actually thought I wouldn't get any answer, anyway thanks. Okay as for the textbooks that you'v recommneded, I would look them up. But I'm very new to this topic. So obviously I haven't read them at all. Actually I'm not even really studying 'real' algebraic topology. I'm doing a topology course in undergraduate course now, which is closer to combinatorial topology. And my lecturer is doing everything in an intuitive manner, very flufffy personally... And he uses this method of 'cutting and pasting', exactly the same thing described in the Munkre's second-half. But I really feel uneasy of this method; I alsmot feel like this is some kind of chemisty stuff, doing things very fluffy, so many exceptions, pictures, etc.. I think just for the sake of explanation, this would be good. But I believe no way this can be used in actual proofs. But it seems Munkres is doing that.

Have you read this second-half of Munkres's carefully? If you have, what do you make of it? Do you think the treatment is rigorous and good? And it seems Massey's book is also using this method. And if you have read this book too, what do you make of it? Do you recommend it?

 

[mathwonk]

i have not read those books and agree they are a little imprecise. but gluing is a very precise process.

to glue two spaces you begin with their disjoint union, then define an equivalence relation on that, and then the new space is the set of equivalence classes.

those equivalence classes are the "points" and then you have to define neighborhoods of those points. in guess a neighborhood of such an equivalence class is just a union of neighborhoods of the points in the class, mod equivalence.

i.e. if you identify two edges of two rectangles, to make one rectangle, then a nbhd of a point A=B on the identified edge is made from two nbhds, one of A and one of B.

so each individual nbhd looks like a half disc, but after identifying them it looks like a full disc. that's why gluing bordered surfaces gives a surface, usually with fewer borders.

to make the whole process more combinatorial and less messy, they make up a shorthand for these rectangles with identifications, but i never read it.

the hard theorem is that any surface can be cut apart into triangles, i.e. becomes a disjoint union of cells after removal of a finite number of arcs and loops. then you conclude that every surface can be reconstructed from such triangles or cells by reassembling them. so numbering all possible ways of reassembling them gives you a dictionary of all possible surfaces.

i have never studied this topic but it is considered a great result by topologists and very useful to learning topology. it is just a gap in my background, one of many. but still i know something about it of course after all these years.

 

[Bacle2]

Yes, there is a result that every closed surface is triangulable; see, e.g:

http://en.wikipedia.org/wiki/Triangulation_(topology ), below "Explicit methods of

triangulation" . As the article says, triangulation allows you to use simplicial homology theory, which is simple

compared with other homology theories.

To extend on Mathwonk's comment a bit, the topology given in the process is the

quotient topology : a subset of the surface you get by identification is open iff.(def.)

its inverse image is open in the unglued space. I think the topology in the unglued

space is just the subspace topology, but I'm not 100%. Look up Munkres' section on

the quotient topology for more details.

 

[julypraise]

i have not read those books and agree they are a little imprecise. but gluing is a very precise process.

to glue two spaces you begin with their disjoint union, then define an equivalence relation on that, and then the new space is the set of equivalence classes.

those equivalence classes are the "points" and then you have to define neighborhoods of those points. in guess a neighborhood of such an equivalence class is just a union of neighborhoods of the points in the class, mod equivalence.

i.e. if you identify two edges of two rectangles, to make one rectangle, then a nbhd of a point A=B on the identified edge is made from two nbhds, one of A and one of B.

so each individual nbhd looks like a half disc, but after identifying them it looks like a full disc. that's why gluing bordered surfaces gives a surface, usually with fewer borders.

to make the whole process more combinatorial and less messy, they make up a shorthand for these rectangles with identifications, but i never read it.

the hard theorem is that any surface can be cut apart into triangles, i.e. becomes a disjoint union of cells after removal of a finite number of arcs and loops. then you conclude that every surface can be reconstructed from such triangles or cells by reassembling them. so numbering all possible ways of reassembling them gives you a dictionary of all possible surfaces.

i have never studied this topic but it is considered a great result by topologists and very useful to learning topology. it is just a gap in my background, one of many. but still i know something about it of course after all these years.

Thanks mathwonk. I will definitely keep in mind your insight and comment when I study this in a more rigorous setting.

Yes, there is a result that every closed surface is triangulable; see, e.g:

http://en.wikipedia.org/wiki/Triangulation_(topology ), below "Explicit methods of

triangulation" . As the article says, triangulation allows you to use simplicial homology theory, which is simple

compared with other homology theories.

To extend on Mathwonk's comment a bit, the topology given in the process is the

quotient topology : a subset of the surface you get by identification is open iff.(def.)

its inverse image is open in the unglued space. I think the topology in the unglued

space is just the subspace topology, but I'm not 100%. Look up Munkres' section on

the quotient topology for more details.

And thanks Bacle2. Indeed the quotient topology section in Munkres seems quite good; it does not use that fluffy method in the proof.