Timeline for Interesting examples of vacuous / void entities
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| Dec 21, 2020 at 3:26 | comment | added | Tom Goodwillie | Yes. $BSO(n)$ is a $2$-sheeted cover of $BO(n)$ even when $n=0$. (But to make sense of this you might have to think of $SO(0)$ as a groupoid, not a group.) | |
| Feb 21, 2019 at 21:51 | comment | added | Dustin Clausen | (And instead of GL(n)/SL(n) one takes the homotopy fiber of "BSL(n)" --> BGL(n).) | |
| Feb 21, 2019 at 21:38 | comment | added | Dustin Clausen | I had similar thoughts to Allen, but I resolved them like this. More natural than SL(n) is "BSL(n)", the homotopy fiber of the determinant map BGL(n) --> BGL(1). For n>0 this is connected and indeed is homotopy equivalent with BSL(n), but for n=0 it's homotopically two points. | |
| Feb 10, 2014 at 9:21 | history | edited | Pietro Majer | CC BY-SA 3.0 |
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| Nov 10, 2011 at 4:44 | comment | added | Allen Knutson | I wasn't convinced about orienting points -- as $GL(0)/SL(0)$ has one component, not two -- until I hit the statement "If $A = B\oplus C$, orienting any two uniquely determines an orientation on the third." | |
| Nov 15, 2010 at 14:25 | history | edited | Pietro Majer | CC BY-SA 2.5 |
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| Nov 15, 2010 at 14:10 | comment | added | Pietro Majer | I used to work with a notion of orientation (in a context of Fredholm pairs and Fredholm operators) and trivial cases where actually the key point of the whole story. But this question seems of general interest, so I add some remark on this point. | |
| Nov 15, 2010 at 14:03 | history | edited | Pietro Majer | CC BY-SA 2.5 |
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| Nov 15, 2010 at 4:48 | comment | added | Michael Hardy | I don't know what a 1-dimensional framed bordism is, but I'm glad someone mentioned that there's a (nontrivial?) reason to think about the concept. | |
| Nov 15, 2010 at 4:35 | comment | added | David Roberts♦ | And this is essential when considering 1-dimensional framed bordism. | |
| Nov 14, 2010 at 7:07 | comment | added | Pietro Majer | Talking about a point $p$ as a connected real manifold $M=${p} of dimension $0$. It has two canonical orientations, $+1$ and $-1$, the generators of the top-degree exterior power $\Lambda^{top}(T_p M)=\mathbb{R}^1$. It seems to me another instance of a trivial case of a mathematical notion, as interesting and useful as the other ones (and slightly paradoxical as well). | |
| Nov 13, 2010 at 23:35 | history | answered | Pietro Majer | CC BY-SA 2.5 |