I’m reading hatcher’s algebraic topology, chapter 2 on homology. After defined the relative homology group $H_n(X, A)$, he gives a remark claiming that: (https://pi.math.cornell.edu/~hatcher/AT/AT.pdf page124 at the bottom)
The quotient $C_n(X)/C_n(A)$ could also be viewed as a subgroup of $C_n(X)$, the subgroup with basis the singular n simplices $σ: ∆n→X $ whose image is not contained in $A$. However, the boundary map does not take this subgroup of $C_n(X)$ to the corresponding subgroup of $C_{n−1}(X)$, so it is usually better to regard $C_n(X, A)$ as a quotient rather than a subgroup of $C_n(X)$.
I understand the part that “ this subgroup of $C_n(X)$ to the corresponding subgroup of $C_{n−1}(X)$”, since some singular n-simplex do have a part of boundary contained in $A$:
But I don’t understand the statement “The quotient $C_n(X)/C_n(A)$ could also be viewed as a subgroup of $C_n(X)$, the subgroup with basis the singular n simplices $σ: ∆n→X $ whose image is not contained in $A$”
. How can it be also viewed as this subgroup? They seems not the same. One is a quotient group, the other is a subgroup. And the boundary operator behaves different. What does hatcher’s “viewed as” means?
Thanks!
Best Answer
$C_n(X)$ is the group of linear combinations of singular simplices in $X$, and $C_n(X)/C_n(A)$ is (isomorphic to) the group of linear combinations of simplices where the ones from $C_n(A)$ are not present. (Strictly speaking: every class in $[\sigma] \in C_n(X)/C_n(A)$ has a representative (a sum) without elements of $C_n(A)$ in it.) Put another way: $C_n(X; G)$ is a direct sum of $G$ over all singular simplices and factoring out $C_n(A; G)$ just kills some of the direct summands; the result, the smaller sum, trivially injects into the big one.
As he says, that's not the best way to think of it anyway: taking the boundary may get you from $\operatorname{Sing}(X) \setminus \operatorname{Sing}(A)$ back to $\operatorname{Sing}(A).$