long exact sequence of homology groups

Question

In the long exact sequence $\cdots \to \tilde{H} (A) \to \tilde{H}(X) \to H(X, A) \to \cdots$, why isn’t the third homology in the sequence given in a reduced homology?

Also I am wondering why the order is specifically $A\to X\to (X, A)$. Is there any intuitive motivation behind it?

Answer 1

The answer to your second question is an algebraic one. Let $X$ be a top. space and $A\subset X$ a subspace. Let $(C_\bullet(X),\partial_\bullet)$ be a chain complex. The relative chaingroup $C_n(X,A)$ is defined as the quotient $$ C_n(X,A):= C_n(X)\big/C_n(A)$$

Observe that the differential $\partial_\bullet$ takes $$C_\bullet(A)\to C_{\bullet-1}(A)$$ which then induces a complex of quotients $$\cdots \to C_n(X,A)\to C_{n-1}(X,A)\to C_{n-2}(X,A) \to \cdots$$

Which gives rise to a short exact sequence $$0\to C_n(A)\xrightarrow{i} C_n(X)\xrightarrow{p} C_n(X,A) \to 0 \quad (*)$$ where $i$ is the inclusion and $p$ the quotient map. Since the differential $\partial_\bullet$ is part of each of the chain complexes $$C_\bullet(A), C_\bullet(X), C_\bullet(X,A)$$ we can interpret $(*)$ as a short exact sequence of chain complexes for which you'll get the homology groups $H_n(-)$ of the respective complexes $C_\bullet(A), C_\bullet(X), C_\bullet(X,A)$ by the snake lemma.

The snake lemma provides the long exact homology sequence you are looking at $$\cdots H_{n}(A) \to H_{n}(X) \to H_{n}(X,A) \xrightarrow{\rho} H_{n-1}(A)\to \cdots$$ where $\rho$ denotes the connecting map.

So generally speaking, the sequence $$H_n(A)\to H_n(X)\to H_n(X,A)$$ comes from the short exact sequence $(*)$ and applying the snake lemma to get the long exact sequence of the respective homology groups.

Answer 2

The answer of both questions are given by the natural isomorphism (when (X,A) is a good pair, it means $A$ is a closed subspace which is a deformation retract of some neighbourhood in $X$.) $$H_n(X,A)\cong {\overset{\sim}{H}}_n(X/A).$$ The proof of this fact is a consequence of excision, the proof can be found in Hatcher's book Prop. 2.22.

Thus relative homology groups are really reduced homology groups, and the long exact sequence can be obtained applying the reduced homology functor to the sequence

$$A\to X\to X/A.$$