I've written some introductory notes on different topics within algebraic topology and included them here. Do tell me if you find errors or have suggestions for improvements.
- An introduction to fiber bundles.
- An introduction to principal bundles.
- An introduction to characteristic classes.
- An introduction to systems of local (twisted) coefficients.
- Homotopy groups of spheres.
- Introduction to Algebraic K-theory.
- Introduction to the Serre Spectral Sequence.
- Classification of vector bundles over the projective line.
- Stability theorems for persistent homology.
- The main theorems of (co)homology theory proved. (Still being revised—I will update as I write more.)
Here are some fun theorems of topology. I've written up their proofs, and made a problem seet that would be great for good topology students of different levels. These problem sheets are meant to guide the student through the proof, leaving the key parts as problems. (More are coming!)
- The Banach-Tarski Paradox for S^2 (I hope to update it to B^3, and in full generality, soon.)
- The Banach-Tarski Paradox Problem Set
- Borsuk-Ulam Problem Sheet (by my esteemed professor from Moscow, Alexei Gorinov)
- The Ham Sandwich Theorem Problem Sheet (by my esteemed professor from Moscow, Alexei Gorinov)
I've also included my MSc dissertation below.