Why does a torus have first Betti number 2?
Conditions
- The object is a torus.
- The first Betti number counts 1-dimensional holes.
Reasoning, step by step
- Recall that the first Betti number counts 1-dimensional circular holes.
- Identify the independent circular paths on the surface of the torus.
- Note one path goes along the longitude (around the main ring).
- Note the second path goes along the meridian (through the hole).
- Conclude that there are two independent 1-dimensional holes, so .
Example
The video highlights two distinct circular paths on the torus, one along the longitude and one along the meridian, and displays the label .
Common misconceptions
- Counting the 2-dimensional hole of the torus towards the first Betti number.
- Assuming the two loops on the torus are topologically dependent and count as one hole.
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