Timeline for On the existence, for $\langle X,R\rangle$ a finite presentation of a group $G$, of an exact sequence of $\mathbb{Z}G$ modules
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| Dec 21, 2023 at 12:20 | comment | added | HJRW | In this case, the “monodromy action” isn’t very complicated. Just notice that $G$ acts freely on the cells of the universal cover. Choosing orbit representatives identifies the cells of the universal cover with a disjoint union of copies of $G$, by orbit-stabiliser. The chain groups are formal sums of cells, i.e. direct sums of copies of $\mathbb{Z}G$. | |
| Dec 21, 2023 at 10:14 | history | edited | William Thomas | CC BY-SA 4.0 |
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| Dec 21, 2023 at 10:13 | comment | added | William Thomas | On your second point you are of course correct this was a typo, it should be $H_2= ker \delta_2$, will edit accordingly, the argument should still be correct. In regards to the monodromy action in homology, this also goes more generally by the name of ‘homology with local coefficients ‘, a reference in page 328 of Hatcher where they give general construction, which we apply with $M=\mathbb{Z}G$. | |
| Dec 21, 2023 at 6:06 | comment | added | Souvik Mandal | Thanks for ur reply. But I have one doubt. I have read cellular chain complex (for calculating homology) but I couldn't understand monodromy action since as per i know monodromy action comes in homotopy (not homology as per my little knowledge). So would u please tell some references? It will be of great help. 2nd doubt is that-since universal cover $\tilde{X}$ is 2-complex then shouldn't $C_{3}(\tilde X)$ be zero ? (Since theres no 3-cell) | |
| Dec 20, 2023 at 14:17 | history | edited | William Thomas | CC BY-SA 4.0 |
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| S Dec 20, 2023 at 13:55 | review | First answers | |||
| Dec 20, 2023 at 15:02 | |||||
| S Dec 20, 2023 at 13:55 | history | answered | William Thomas | CC BY-SA 4.0 |