• Find the \(4^{th}\) lecture notes linked: Algebraic Topology MA 842 Lecture 2
• Using Snake Lemma or otherwise, prove theorem on long exact sequences based on short exact sequences, apply it to the specific case to get the Mayer-Vietoris Sequence.
• \(\textbf{Mayer-Vietoris Sequence}\) provides a very useful tool to practically compute homology and cohomology groups.
• Based on the above results, compute the Homology Group of \(S^1\). It turns out that \(H_n(S^1)\) is isomporphic to \(\mathbb{Z}\) for \(n=0,1\) and \(H_n(S^1)=0\) for \(n \ge 2\).
• Relevant references are the Wikipedia article on Seifert–van Kampen theorem.