Algebraic Topology

Module aims

Compute a named invariant on standard spaces and use it to distinguish homotopy types when the computation applies.

Course outline

1. 01 Homotopy and path homotopy 20h

   Learning result Prove that path homotopy is an equivalence relation on paths with fixed endpoints.

   Formative check Written derivation

2. 02 The fundamental group 25h

   Learning result Compute π₁ of a circle or a wedge of circles at synopsis grain.

   Formative check Written derivation

3. 03 Covering spaces (taught grain) 25h

   Learning result For p:ℝ→S¹, p(t)=e^{2πit}, lift a loop based at 1 starting above it at 0, identify its integer endpoint and the integer-translation deck maps; use homotopy lifting to relate these to π₁(S¹,1) for this universal covering.

   Formative check Four-option check

4. 04 Homology (taught grain) 30h

   Learning result Compute a homology group of a standard complex and interpret a non-zero class.

   Formative check Written derivation

5. 05 Distinguishing spaces 20h

   Learning result Give two spaces that a chosen invariant distinguishes, and a pair it does not.

   Formative check Counterexample

6. 06 Synthesis 30h

   Learning result Combine a computation with a classification or non-homeomorphism claim the invariant supports.

   Formative check Written derivation

Learning results

Compute π₁ or a homology group of a standard space, citing the theorem that licenses the calculation.

Subject skills

Homotopy, fundamental groups, introductory covering methods and selected simplicial or singular homology.

Transferable skills

Keep basepoints and homeomorphism versus homotopy equivalence distinct.