Exercises

1. Verify that the diagram (37.5) commutes.
2. Determine $H_n(X, A)$ when $A = \emptyset$, and when $A$ is a singleton and $n\geq 1$. What happens if $n = 0$?
3. Compute $H_n(S^1\times S^1, S^1\vee S^1)$ and compare it with the absolute homology $H_n((S^1\times S^1)/(S^1\vee S^1))$.
4. Compute $H_k(E^n, S^{n-1})$ and compare it with $H_k(E^n/S^{n-1})$.
5. In example (35.1), prove that $X/A$ is homeomorphic to $\mathbb{R}P^2$. Compare the groups $H_n(X, A)$ with the groups $H_n(X/A)$. Hint: To set up the homeomorphism note that $(x, y) \mapsto (x\sqrt{1-y^2}, y)$ maps each $[-1, 1]\times \{y\}$ homeomorphically onto the chord at height $y$.

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Lecture - XXXVIII Excision theorem

in In this lecture we prove the most important theorem homology theory known as the excision theorem. We shall conclude the lecture with the definition of local homology groups that play an important role in the theory of orientability of topological manifolds. We begin with the ubiquitous five lemma.