So i am really a novice in terms of homology theory, and i just recently read about the relative homology groups. I have a question i couln't answer myself and i hope someone can help me getting a better understanding.
Let $X$ be a top. space and $A \subset X$ a subspace. What exactly does $$H_*(X,A) = 0$$ tell me intuitively?
I know that if we have $H_*(X,A) = 0$, the inclusion $\iota: A \to X$ induces an isomorphism $\iota_*:H_*(A) \to H_*(X)$ from which i conclude that $A$ and $X$ are homotopy equivalent spaces.
Is there anything else i should be able to conclude from the given information that $H_*(X,A) = 0$ ?
Or asked differently, let's say i've been given a top. space $X$ and a subspace $A \subset X$ and let's assume $A$ is a deformation retract of $X$.
Can i immediately conclude that $X_*(X,A) = 0$? Or does this not hold in general?
Highly appreciating any help. Thanks a lot.
When the inclusion of $A$ in $X$ is nice enough*, then outside degree zero, the relative homology is the same as the homology of the quotient $X/A$. In this case, the vanishing just means that $X/A$ is acyclic.
*there needs to be an open neighborhood of $A$ in $X$ which deformation retracts to $A$. I don't know an intuitive way of thinking about it if that's not the case!