Timeline for Simplicial chain complex with ordered simplices
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 5, 2019 at 21:59 | comment | added | APR | Could someone indicate how repetitions are required by Spanier? I don't see where his proof fails without them, but, as indicated, allowing repetitions is necessary for the result to be true. | |
| Oct 31, 2013 at 22:13 | comment | added | Russ Woodroofe | Indeed, the positive answer is strongly hinted at by the fact that degenerate mappings are allowed in the definition of singular homology. (Hatcher even says that the name "singular" came about because singularities are allowed in the mapping.) | |
| Apr 11, 2013 at 13:37 | comment | added | Benjamin Steinberg | You can also find this in Munkres book. | |
| Apr 10, 2013 at 22:58 | comment | added | Ricardo Andrade | By the way, let me remark that your $D_\ast(X)$ (called $\Delta(X)$ in Spanier's book) is simply the chain complex $C_\ast(S(X))$ associated with the simplicial set $S(X)$ naturally associated with the simplicial complex $X$. The $n$-simplices of $S(X)$ are your elements $[v_0,\ldots,v_n]_D$. | |
| Apr 10, 2013 at 22:57 | comment | added | Ricardo Andrade | @Albert: It seems your question is answered positively in theorem 4.3.8 of Spanier's book "Algebraic topology". It states, in your notation, that the natural map $D_\ast X\to C_\ast X$ is a quasi-isomorphism. The proof given by Spanier is a simple application of the method of acyclic models. | |
| Apr 10, 2013 at 21:32 | history | asked | Albert | CC BY-SA 3.0 |