These notes are my attempt to organize the knowledge I have acquired systematically and coherently in a way that can also be used for self-study by students. I have tried to gather the essentials from all the books and references I have read in my journey studying this beautiful subject, so the beginner doesn't get lost in such a vast literature. I intend to give enough detail so the student can gain a deep understanding without losing the big picture in the technicalities.
I started this project in December 2022 and I want to keep improving these notes, so please email me if you have suggestions or notice any errors/typos. Finally, I'm grateful to my friends, who encouraged me to start these notes. In particular, with my friend Esteban Saldarriaga, who taught me a lot about Latex.
The order of the notes reflects the order in which I think it's more natural to read them. The prerequisite for a set of notes is contained in the previous ones.
Fundamental Group, (Co)Homology and Spectral Sequences (Last version Jul/2024)
I recommend studying these notes together with a book on smooth manifolds. The book ''Introduction to Smooth Manifolds" by John M. Lee is an excellent choice. If the reader already knows smooth manifolds (or if they have enough time for a third book) I recommend studying the basics of category theory (A good goal is to understand limits and colimits). The book ''Category Theory in Context" by Emily Riehl is great.
Fiber Bundles and Characteristic Classes (Not finished/ Last version Oct 2024)
Homotopy Theory (Not finished/Last version Oct 2024)