Timeline for Connected sum of surfaces
Current License: CC BY-SA 3.0
3 events
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| Feb 7, 2012 at 14:57 | comment | added | Johannes Ebert | I would say that there is a lot of work swept under the carpet. Unless the involution exists, things can go completely wrong. For example, $CP^2 \sharp CP^2$ is not even homotopy equivalent to the manifold $CP^2 \sharp \bar{CP^2}$ that you obtain if one of the embedded discs is negatively oriented. So the surface case is special. If you make a picture of my construction, you will find it ''intuitive''. P.S: I really think that the smooth structure makes everything easier. | |
| Feb 7, 2012 at 10:48 | comment | added | Baptiste Calmès | Thanks for this answer. However, although I did not mention it in the question, I would like to avoid any argument using a differentiable structure. Somehow, the spirit of my question is: "How come at the beginning of various introductory textbooks on algebraic topology (ex: Massey), this topological fact is stated as true and intuitive, although there seems to be no proof avoiding some kind of classification result, itself quite non-trivial in the non compact case? Am I missing some elementary argument, or is there really a lot of work swept under the carpet at that point?". | |
| Feb 7, 2012 at 0:19 | history | answered | Johannes Ebert | CC BY-SA 3.0 |