Timeline for 3-manifold with boundary containing a projective plane
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Sep 6, 2021 at 8:43 | comment | added | HJRW | @JoshHowie: So it is! That will teach me to trust the MR review. Thanks for the explanation. | |
| Sep 4, 2021 at 18:08 | history | edited | Josh Howie | CC BY-SA 4.0 |
added note about Livesay's theorem
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| Sep 4, 2021 at 18:05 | comment | added | Josh Howie | Epstein's result is that such a manifold is homotopy equivalent to $P^2\times I$ minus some balls. See the comments on p477 of his paper, where he states that to upgrade to homeomorphism, one needs to prove two conjectures: one is the Poincaré Conjecture, the other is the result of Livesay. | |
| Sep 4, 2021 at 13:50 | comment | added | HJRW | Surely the bottom line is that the desired result is in fact Epstein's theorem (plus the Poincare conjecture)? From the MR review of Epstein's paper: "In the second chapter the author shows, surprisingly, that any compact non-orientable 3-manifold with finite fundamental group has to be (modulo the Poincaré conjecture) homeomorphic to $P^2×I$ minus a number of 3-balls". | |
| Sep 1, 2021 at 20:53 | comment | added | Greg Friedman | If this new argument is now correct, the comment stating that there is a gap/mistake should probably be deleted or modified to avoid future confusion. | |
| Sep 1, 2021 at 0:46 | vote | accept | Zhiqiang | ||
| Aug 29, 2021 at 17:15 | history | edited | Josh Howie | CC BY-SA 4.0 |
Stated the theorem and references from which the result follows immediately, and corrected the proof of the special case asked by OP.
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| Aug 29, 2021 at 13:07 | comment | added | Allen Hatcher | It is a theorem of Livesay (1963 Annals of Math., pp. 582-593) that $M=P^2\times I$ if $N=S^2\times I$. A shorter proof was later published by Rubinstein in the 1976 Proceedings of the AMS, pp. 317-320. | |
| Aug 29, 2021 at 6:13 | comment | added | Adterram | The fact that $k=2$ follows from the Lefschetz fixed point theorem. Indeed, since the deck transformation of the covering map has no fixed point, its Lefschetz number must be $0$. By direct calculation, $k=2$. What I do not understand is how to deduce $M=P^2 \times I$ if we know $N=S^2 \times I$? | |
| Aug 28, 2021 at 19:23 | history | edited | YCor | CC BY-SA 4.0 |
added "yes"
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| Aug 28, 2021 at 18:09 | history | answered | Josh Howie | CC BY-SA 4.0 |