Timeline for The Wedge Sum of path connected topological spaces
Current License: CC BY-SA 2.5
20 events
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| Jul 11, 2010 at 12:48 | history | edited | Dmitri Pavlov |
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| Jul 4, 2010 at 2:57 | comment | added | Tom Goodwillie | Jeff, I was responding to a slightly more general question, but it was relevant. Your impulse to look at a 'dumbbell' was a good one. When you attach a line segment to $X$ at $x_0$ and use the other end of the segment as basepoint, then the new space is homotopy equivalent to $X$ but not necessarily homotopy equivalent in the based sense. | |
| Jul 4, 2010 at 1:39 | vote | accept | Jeff | ||
| Jul 3, 2010 at 22:35 | comment | added | Autumn Kent | Oops, in light of Hatcher's comment, I guess I must have been assuming some nice things about X and Y without realizing. Sorry. | |
| Jul 3, 2010 at 22:30 | answer | added | Allen Hatcher | timeline score: 20 | |
| Jul 3, 2010 at 21:40 | comment | added | Jeff | I think I understand my lack of clarity. "Base Point" has many uses in topology, and so I apologize. The edit is above, but basically base point means the points we are identifying to form a wedge sum. Not the "Base point" of a loop, or anything like that. Sorry. | |
| Jul 3, 2010 at 21:38 | history | edited | Jeff | CC BY-SA 2.5 |
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| Jul 3, 2010 at 21:35 | comment | added | Autumn Kent | Tom, I think he's not varying the homeomorphism type of the two factors, only the placement of the basepoint. | |
| Jul 3, 2010 at 21:34 | comment | added | Jeff | You claimed that my professor's assertion was false, so I was referring to this claim you made. My claim was that the wedge sum does not depend on where you connect the two spaces, up to homotopy equivalence. I do not understand why your response has 3 spaces in it. The question is if you wedge X and Y (path connected) together at x0 and y0, how is this homotopy equivalent to the same wedge, but done bringing together x1 and y1? You said this is not true in general. So, I am wondering if you have a counterexample? Thanks. | |
| Jul 3, 2010 at 21:30 | comment | added | Tom Goodwillie | I mean, what's the question? True: If $X$, $Y$, and $Z$ are based spaces and $X$ and $Y$ are homotopy equivalent in the based sense then $X\vee Z$ and $Y\vee Z$ are homotopy equivalent in the same sense. False: If $X$, $Y$, and $Z$ are path connected spaces and $X$ and $Y$ are homotopy equivalent then $X\vee Z$ and $Y\vee Z$ are homotopy equivalent no matter what base points you use to stick things together. | |
| Jul 3, 2010 at 21:21 | comment | added | Jeff | @Tom Goodwillie, meaning not even if X and Y are path connected? If so, do you have a counterexample? I was worried myself after so much effort that it would be false. | |
| Jul 3, 2010 at 21:20 | comment | added | Jeff | Yes, I know this... I have tried to use that in many ways. I have tried to show mutual homotopy equivalence to the disjoint union of X and Y (if these are the two spaces you are wedging.) Of course, that didn't work. It would be a cruel world in topology if it did, since it is blatantly false. I have tried to show homotopy to the dumb-bell like shape. (The wedge product except the point at which the spaces meet is extended to a line.) I believe it is harder than it first appears to think about this beyond the intuitive level, and to actually rigorously construct f and g. | |
| Jul 3, 2010 at 21:19 | comment | added | Tom Goodwillie | It's not true in general. | |
| Jul 3, 2010 at 21:17 | comment | added | Autumn Kent | Homotopy equivalence is a transitive relation. | |
| Jul 3, 2010 at 21:08 | comment | added | Jeff | One week's worth of effort, as well as the failure to find anything useful on wikipedia, google-book's preview, and in Caltech's own library says otherwise. Also, none of my peers who have taken Math 109a can handle it either. I have asked, trust me. Do you have a hint? I first tried elementary things like inclusion maps and projections (remember the wedge product is a quotient space.) I then thought about examples (I can do well-behaved cases in R^n) and I thought about showing a homotopy with a dumb-bell like object where the wedged point becomes a line. | |
| Jul 3, 2010 at 21:04 | comment | added | Autumn Kent | Ok, but mathoverflow isn't for homework level problems either, and this is usually a homework problem. You can do it. | |
| Jul 3, 2010 at 20:59 | comment | added | Jeff | This is not homework. It is an exercise left by my professor, which was not assigned for homework. If you don't believe me, you can look at exercise 2.35 in his set of lecture notes: math.caltech.edu/~ma109a/109anotes.pdf You may notice the class is over. I want this fact proven for research I'm doing this summer. | |
| Jul 3, 2010 at 20:57 | comment | added | Autumn Kent | 1. Mathoverflow isn't for homework. 2. This is a fun homework assignment and you should think about it more. | |
| Jul 3, 2010 at 20:52 | history | edited | Jeff |
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| Jul 3, 2010 at 20:45 | history | asked | Jeff | CC BY-SA 2.5 |