Georgia Institute of Technology College of Sciences

In the third talk of this seminar series, we continue to follow the manuscript of Carl Miller. We will begin with a quick review of chain complexes and simplicial isomorphisms and then we will detour to discuss the geometric interpretation of homology groups in lower dimensions. This work can help us understand the structure of simplicial complexes with boundary maps and their homology groups. Then we go back to abstract homological algebra which is the study of homology groups without reference to simplicial complexes. We will introduce the Snake Lemma without proof. Finally, we will apply this lemma to prove the goal of this chapter: collapsibility for a simplicial complex implies its homology groups are trivial which is called acyclicity.