June 14, 2025 A bit of rough remarks on maximal atlases and smooth structure, but basically done with W1L2.
June 14, 2025 Herculean Task: Properties of Topological manifold.
June 14, 2025 Potential look back: to show the following is not a topological manifold. (A non example from W1L2).
June 14, 2025 Skipping proof for Double origin problem since it involves demands significant involvement which might disrupt the flow of my terry tao read and also the whole notes in general. So for now I note such places and move on. The same might happen for mobius strip and klein bottle proofs most. Until I fall into a rut of writing these notes and thinking about differential geometry, I better skip herculean tasks. Once I get back enough headspace and routine, I can work on them. For now I move on, go ahead and build those notes. The other parts are important too, and once I feel a sense of progress, I can come back!
June 14, 2025 The document now has some week 1 material and then skips to W4L2 stopping finally with the content of W10L1. Week 1 materials would feel more complete if I include the proof for the double origin problem (that it’s not a topological manifold due to its non-Hausdorffness). But as of W1L1, only the question was posed. I did prove it in the other dg notes document in goodnotes escaping the concept of quotient topology. All the examples and non examples of topological manifolds ranging from sphere, circle, cone, cusp, open set, Euclidean space to uncountable discrete spaces (and product of such space with another), double origin line are TeXed. An appendix exploring some of the tips of the icebergs seen during the lectures in included. As of now it includes stuff form W1L1 only - connected components of manifolds with different dimensions, homology groups and open mapping theorem in R^n (which I first heard from Dr. Vamsi in his complex geometry lectures at IISER Pune). The content of these three tips of icebergs is yet to be included. TLDR: W1L1 is formally complete. Next up: Double origin Problem → W1L2 and Tips of Iceberg.