Milestone map
Milestone map
2 milestones
Do the problem sets and write them up
7–14 weeks (10–12 hrs/week)
Take Algebraic topology for free and do the exercises in chapters 0 and 1 (homotopy, the fundamental group, covering spaces) and the start of chapter 2 (homology). Write every solution up in LaTeX and keep the source and the PDF in one public repo as you go.
Proof required
Link your public repo with the LaTeX source and the PDF of the exercises in chapters 0 and 1 (homotopy, the fundamental group, covering spaces) and the start of chapter 2 (homology), and show your self-check (Self-check without an answer key).
What gets checked
- Every proof is complete and correct.
- Each computed group is checked by a second method where one exists.
- Pictures match the arguments.
Common mistakes
- Copying posted solutions: verifiers read the reasoning and the history.
- A proof that skips the hard step.
- Asking for review before the self-check.
Powstik Guide
A public repo with your LaTeX write-ups of the exercises in chapters 0 and 1 (homotopy, the fundamental group, covering spaces) and the start of chapter 2 (homology).
Steps
- Make a public repo for this course.
- Do one problem set at a time; typeset it in LaTeX and commit.
- Self-check: The book has no answer key. For each problem, name the definition or theorem every step uses; a step you cannot name is a gap. Then swap write-ups with a study partner before you ask for review.
- Fix what the self-check shows, and say what you fixed.
- Ask three people who know the topic to check it.
Template
README Course: Algebraic topology (Algebraic Topology by Allen Hatcher (free book)) What is here (problem sets, PDF): Self-check result: The hardest proof and why:
What gets sent back
- Answers with no reasoning.
- Photos of handwriting instead of typeset work.
- Self-check not shown.
Resources
Foundationstart here
Depthgo deeper
What a verifier looks for
- Every proof is complete and correct.
- Each computed group is checked by a second method where one exists.
- Pictures match the arguments.
- Pick two proofs and check every step: each argument must be correct and complete, not just present.
- Ask about one step; the answer should be clear.
You'll sign in first, then come straight back here.
Compute fundamental groups
1–3 weeks (5–8 hrs/week)
Use what you learned. Make a proof set that computes the fundamental group of four spaces (circle, torus, figure eight, projective plane) with full proofs. Typeset it in LaTeX (or a notebook with LaTeX maths) in your public repo.
Proof required
Link a proof set that computes the fundamental group of four spaces (circle, torus, figure eight, projective plane) with full proofs, in your public repo.
What gets checked
- Each computation is proved, not quoted.
- Van Kampen or covering arguments are used correctly.
- Every step is complete.
Common mistakes
- A gap in the main argument.
- A result stated but not proved.
Powstik Guide
A proof set that computes the fundamental group of four spaces (circle, torus, figure eight, projective plane) with full proofs.
Steps
- Write one line: what you will prove or show.
- Draft it, then read it cold the next day.
- Ask someone to read it and mark the first step they doubt.
- Fix it and typeset the final version.
- Ask three people to check it.
Template
Title: What this shows: Main result (statement): Proof: What I would look at next:
What gets sent back
- A main step left unproved.
- Not clearly the person's own work.
- Claims more than it proves.
Resources
Foundationstart here
Depthgo deeper
What a verifier looks for
- Each computation is proved, not quoted.
- Van Kampen or covering arguments are used correctly.
- Every step is complete.
- Every argument must be correct and complete; mark the first gap you find.
- Check that the work is the person's own.
You'll sign in first, then come straight back here.