Dispute
| [math]\xymatrix{ & C_n(X) \ar@{-->}[drr]^{\pi_3\circ f} \ar[rr]^f \ar[d]_{\pi_1} \ar[dl]_{\pi_2} & & C_{n-1}(X) \ar[d]^{\pi_3} & \\
\frac{C_n(X)}{\text{Ker}(f)} \ar@{-->}@/_4.5ex/[rrr]_{\pi_3\circ f_2'}\ar@{.>}@/^9.5ex/[urrr]^{f_2'} & \frac{C_n(X)}{C_n(A)} & & \frac{C_{n-1}(X)}{C_{n-1}(A)} &
}[/math]
|
| Set up
|
If [ilmath]f[/ilmath] is say [ilmath]\partial_n:C_n(X)\rightarrow C_{n-1}(Y)[/ilmath] in a
chain complex (found in this case in
singular homology theory) it is claimed that for any
topological subspace of [ilmath]X[/ilmath] that [ilmath]f[/ilmath] induces a
map:
- [ilmath]f':\frac{C_n(X)}{C_n(A)}\rightarrow\frac{C_{n-1}(X)}{C_{n-1}(A)} [/ilmath]
Specifically it is claimed that:
- [ilmath]\text{Ker}(\pi_1)\subseteq \text{Ker}(f)[/ilmath]
But I don't yet see how that helps. Alec (talk) 22:23, 14 February 2017 (UTC)
