User:Alec/Bitchin

Dispute

$$\xymatrix{ & C_n(X) \ar@{-->}[drr]^{\pi_3\circ f} \ar[rr]^f \ar[d]_{\pi_1} \ar[dl]_{\pi_2} & & C_{n-1}(X) \ar[d]^{\pi_3} & \\ \frac{C_n(X)}{\text{Ker}(f)} \ar@{-->}@/_4.5ex/[rrr]_{\pi_3\circ f_2'}\ar@{.>}@/^9.5ex/[urrr]^{f_2'} & \frac{C_n(X)}{C_n(A)} & & \frac{C_{n-1}(X)}{C_{n-1}(A)} & }$$
Set up

If $f$ is say $\partial_n:C_n(X)\rightarrow C_{n-1}(Y)$ in a chain complex (found in this case in singular homology theory) it is claimed that for any topological subspace of $X$ that $f$ induces a map:

• $f':\frac{C_n(X)}{C_n(A)}\rightarrow\frac{C_{n-1}(X)}{C_{n-1}(A)}$

Specifically it is claimed that:

• $\text{Ker}(\pi_1)\subseteq \text{Ker}(f)$

But I don't yet see how that helps. Alec (talk) 22:23, 14 February 2017 (UTC)