Timeline for Reduction of structure group and classifying spaces
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| S Dec 7 at 18:59 | history | suggested | Arnav Das | CC BY-SA 4.0 |
Corrected reference, rephrased "homotopy lift"
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| Dec 7 at 13:05 | review | Suggested edits | |||
| S Dec 7 at 18:59 | |||||
| Dec 6 at 15:15 | comment | added | Arnav Das | A quicker question: what makes the dashed morphism $G$-equivariant (and thereby a morphism of principal $G$-bundles)? | |
| Dec 5 at 20:59 | comment | added | Arnav Das | Thank you for the edit. Upon a second read, I don't believe you are using the choice of homotopy $h: B\phi \circ \tau_H \to \tau_G$ in your argument (or have explicated the existence of one being a condition for any syllogism). Where does it come in? I think it's relevant to ask, e.g. what's to stop us from applying the $(-)\times_H G$ functor on arbitrary $H$-bundles and then running the rest of your argument? My understanding was that a given homotopy $h$ leads to a (unique?) isomorphism $\theta_h : Q \times_H G \to P$. Is this what you show? Could you make this point explicit? | |
| Dec 4 at 13:22 | comment | added | Baylee Schutte | I don't know if you can modify the proof of the reverse implication to remove the admissibility assumption. For now, I've edited the last paragraph of my answer to reflect this. | |
| Dec 4 at 13:18 | history | edited | Baylee Schutte | CC BY-SA 4.0 |
added 32 characters in body
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| S Dec 4 at 12:55 | review | First answers | |||
| Dec 4 at 13:42 | |||||
| S Dec 4 at 12:55 | history | edited | Baylee Schutte | CC BY-SA 4.0 |
deleted 1 character in body
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| Dec 4 at 12:49 | comment | added | Arnav Das | Thank you. In Mitchell's notes, he does not seem to have considered arbitrary homomorphisms $\phi : H \to G$, only subgroup inclusions (upon a cursory look, a remark is also made about admissability of the subgroup on p. 21). Have you a reference for a proof of the reverse implication where $\phi$ is arbitrary (or does the same argument therein hold with slight modifications)? | |
| S Dec 4 at 12:33 | review | First answers | |||
| Dec 4 at 12:40 | |||||
| S Dec 4 at 12:33 | history | answered | Baylee Schutte | CC BY-SA 4.0 |