Making live Commutative diagrams in Latex while listening and trying to understand critical constructive proofs is not recommended. Especially when new concepts to define Homology theory abstractly are being introduced. So I shall type out the notes later. For now I will shortly upload the handwritten notes.
\[H_n(X)= \frac{Z_n(X)}{B_n(X)}\]
Proved the theorem that Homology of convex sets is zero, ie. \(X\) is convex \(\implies H_n(X)=0\). This is easy to see once the next theorem is stated and proved (identity map induces identity homomorphism at the homology level but by computation it turns out to be the zero map which is only possible in the zero group) but otherwise the proof basically involves computing the Quotient space, using the fact that we can express points in a convex domain by a neat parametrization. This shows that every cycle is also a boundary in a convex set (remember that in an exercise previously we proved that \(d^2=0\) which corresponds to the fact that every boundary is a cycle and further singular chains form a chain complex).
Next, proved that if two maps are homotopic then they induce the same homomorphism at the level of homology. Proof involves constrcuting a map which is derived as a natural map on functors. We want all the relevant composite diagrams to commute. Divide \(\Delta_n \times I\) into \(\Delta_{n+1}\) simplices (prism type structures) and then interpolate. This is used to define Chain homotopy.
All of this exemplifies the notion prevalent in Category theory: that of studying objects via maps associated with it. Here, we are trying to study the category of Topological spaces via Maps from simplexes that are easier to handle. This is why singular n-simplexes are of critical importance in the study of homology theory.