Algebraic Topology

模块目标

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

课程大纲

1. 01 Homotopy and path homotopy 20h

   学习结果 Prove that path homotopy is an equivalence relation on paths with fixed endpoints.

   形成性检查 书面推导

2. 02 The fundamental group 25h

   学习结果 Compute π₁ of a circle or a wedge of circles at synopsis grain.

   形成性检查 书面推导

3. 03 Covering spaces (taught grain) 25h

   学习结果 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.

   形成性检查 四选一检查

4. 04 Homology (taught grain) 30h

   学习结果 Compute a homology group of a standard complex and interpret a non-zero class.

   形成性检查 书面推导

5. 05 Distinguishing spaces 20h

   学习结果 Give two spaces that a chosen invariant distinguishes, and a pair it does not.

   形成性检查 反例

6. 06 Synthesis 30h

   学习结果 Combine a computation with a classification or non-homeomorphism claim the invariant supports.

   形成性检查 书面推导

学习结果

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

学科技能

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

可迁移技能

Keep basepoints and homeomorphism versus homotopy equivalence distinct.