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Teaching video
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Graduate chapters and video availability01 · Read and understand
What you will learn
- Use boundary maps to distinguish cycles from boundaries.
- Justify the conclusion "H₀=Z, H₁=Z², H₂=Z" using the stated assumptions.
Before you start
Cellular chain complexes and integer groups.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Use the torus cell structure with one vertex, two edges a,b, and one face attached by aba⁻¹b⁻¹.
Why this math matters
Use boundary maps to distinguish cycles from boundaries. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- Homology uses integer coefficients.
- The stated cell attachment is the standard torus model.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Compute torus homology from a cell model
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Use the torus cell structure with one vertex, two edges a,b, and one face attached by aba⁻¹b⁻¹.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
C₂=Z, C₁=Z², C₀=Z; ∂₁=0
Each edge starts and ends at the same vertex.
Work through the mathematics
∂₂=0 because the face has net coefficients 1−1 for a and b
Cellular homology counts signed edge traversals, so the commutator attachment has zero abelian boundary.
Check the conclusion
H₀=Z, H₁=Z², H₂=Z
Kernels modulo images give one connected component, two independent one-dimensional holes, and one oriented surface class.
The result
H₀=Z, H₁=Z², H₂=Z
Kernels modulo images give one connected component, two independent one-dimensional holes, and one oriented surface class.
Common mistakes to catch
- Homology records less information than the full fundamental group.
- A chain boundary map is not simply the geometric perimeter length.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
What is the Euler characteristic from these ranks?
Show a hint
Alternate the Betti numbers.
Reveal answer and explanation
1−2+1=0
This agrees with the cell count.
Practice 2
Does ∂₂=0 mean the attaching loop is trivial in the wedge of circles?
Show a hint
Homology forgets commutator order.
Reveal answer and explanation
No
The commutator is nontrivial in the free fundamental group but has zero abelianized boundary.
Take the idea with you
Compare the topological information retained by loops before and after abelianization.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Find a closed differential form with nonzero circulation
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