the same question I asked chatGPT and deepseek returns unsatisfying answer was well addressed by Gemini Pro 2.5: “why torus’s fundamental group is abelian, it’s hard to comprehend why go through loop A then loop B is same if reverse order?”
Okay, let’s break down why the fundamental group of the torus is abelian (commutative), meaning path ab is homotopic (can be continuously deformed into) path ba. The square model of the torus is the key visualization tool here.
- The Torus and its Generators
Torus: Imagine a donut shape.
Square Model: A more convenient way to think about it is a square where opposite edges are identified (glued together).
Top edge is glued to the bottom edge.
Left edge is glued to the right edge.
All four corners become the same point on the torus. Let’s use this corner as our base point.
Generating Loops: From this base point (corner):
Loop a: Traverse along the bottom edge from left to right. When you reach the right corner, you “teleport” back to the left corner (because left and right edges are glued). This is one fundamental loop.
Loop b: Traverse along the left edge from bottom to top. When you reach the top corner, you “teleport” back to the bottom corner (because top and bottom edges are glued). This is the second fundamental loop.
- Visualizing ab and ba on the Square
Path ab:
Start at the bottom-left corner.
Traverse a: Go right along the bottom edge to the bottom-right corner.
Traverse b: From the bottom-right corner (which is the same point as the bottom-left), go up along the right edge (because right is glued to left) to the top-right corner.
So, ab traverses the bottom edge, then the right edge. It starts at the bottom-left and ends at the top-right (which is the same as the base point after identification).
Path ba:
Start at the bottom-left corner.
Traverse b: Go up along the left edge to the top-left corner.
Traverse a: From the top-left corner (which is the same point as the bottom-left), go right along the top edge (because top is glued to bottom) to the top-right corner.
So, ba traverses the left edge, then the top edge. It also starts at the bottom-left and ends at the top-right (the base point).
- The Homotopy: Why ab is the same as ba
Now, the crucial part: can we continuously deform the path ab (bottom edge -> right edge) into the path ba (left edge -> top edge) without breaking the path and keeping the start/end points fixed?
Yes! Imagine the path ab drawn on the square. Imagine the path ba drawn on the same square.
The path ab goes around two sides of the square boundary.
The path ba goes around the other two sides of the square boundary.
Because the entire interior of the square is part of the torus surface, we can continuously “slide” or “push” the path ab across the diagonal of the square until it becomes the path ba.
Think of it like this: You can smoothly push the corner where a meets b diagonally across the square towards the corner where b meets a. The interior of the square provides the “room” needed for this deformation.
Formal Argument (The Commutator):
Another way to see this is to consider the loop aba⁻¹b⁻¹ (go a, then b, then a backwards, then b backwards).
On the square:
a: Bottom edge L->R
b: Right edge B->T
a⁻¹: Top edge R->L
b⁻¹: Left edge T->B
This combined path aba⁻¹b⁻¹ traces the entire boundary of the square. Since the entire square (including its interior) represents the torus surface, and we can continuously shrink any loop drawn inside a 2D region down to a single point, the path aba⁻¹b⁻¹ can be continuously shrunk down to the base point (the corner).
If aba⁻¹b⁻¹ is homotopic to the identity (constant) loop, then by “multiplying” by ba on the right (in the group sense):
(aba⁻¹b⁻¹) (ba) is homotopic to (identity) (ba)
aba⁻¹(b⁻¹b)a is homotopic to ba
aba⁻¹a is homotopic to ba
ab(a⁻¹a) is homotopic to ba
ab is homotopic to ba
Contrast with Figure Eight:
In the figure eight (S¹ ∨ S¹), let the loops be a and b, meeting at a single point.
ab: Go around loop A, then loop B.
ba: Go around loop B, then loop A.
You cannot deform ab into ba. Why? Because there’s no 2D surface connecting the loops like the interior of the square for the torus. The space “pinches” at the intersection point. Trying to slide path a past path b (or vice-versa) is blocked by this pinch point. The commutator aba⁻¹b⁻¹ is not shrinkable to a point; it represents a non-trivial loop itself.
Conclusion: The commutativity of the torus’s fundamental group arises because the loops (a and b) can be thought of as traversing the sides of a 2D patch (the square), and this patch allows the paths ab and ba to be continuously deformed into one another across its surface. This 2D “room to maneuver” is absent in the figure eight, leading to its non-abelian fundamental group.
2. Visualizing ab and ba on the Square
- Path ab:
- Start at the bottom-left corner.
- Traverse a: Go right along the bottom edge to the bottom-right corner.
- Traverse b: From the bottom-right corner (which is the same point as the bottom-left), go up along the right edge (because right is glued to left) to the top-right corner.
- So, ab traverses the bottom edge, then the right edge. It starts at the bottom-left and ends at the top-right (which is the same as the base point after identification).
- Path ba:
- Start at the bottom-left corner.
- Traverse b: Go up along the left edge to the top-left corner.
- Traverse a: From the top-left corner (which is the same point as the bottom-left), go right along the top edge (because top is glued to bottom) to the top-right corner.
- So, ba traverses the left edge, then the top edge. It also starts at the bottom-left and ends at the top-right (the base point).
3. The Homotopy: Why ab is the same as ba
Now, the crucial part: can we continuously deform the path ab (bottom edge -> right edge) into the path ba (left edge -> top edge) without breaking the path and keeping the start/end points fixed?
Yes! Imagine the path ab drawn on the square. Imagine the path ba drawn on the same square.
The path ab goes around two sides of the square boundary.
The path ba goes around the other two sides of the square boundary.
Because the entire interior of the square is part of the torus surface, we can continuously “slide” or “push” the path ab across the diagonal of the square until it becomes the path ba.
Think of it like this: You can smoothly push the corner where a meets b diagonally across the square towards the corner where b meets a. The interior of the square provides the “room” needed for this deformation.
Formal Argument (The Commutator):
Another way to see this is to consider the loop aba⁻¹b⁻¹ (go a, then b, then a backwards, then b backwards).
On the square:
a: Bottom edge L->R
b: Right edge B->T
a⁻¹: Top edge R->L
b⁻¹: Left edge T->B
This combined path aba⁻¹b⁻¹ traces the entire boundary of the square. Since the entire square (including its interior) represents the torus surface, and we can continuously shrink any loop drawn inside a 2D region down to a single point, the path aba⁻¹b⁻¹ can be continuously shrunk down to the base point (the corner).
If aba⁻¹b⁻¹ is homotopic to the identity (constant) loop, then by “multiplying” by ba on the right (in the group sense):
(aba⁻¹b⁻¹) (ba) is homotopic to (identity) (ba)
aba⁻¹(b⁻¹b)a is homotopic to ba
aba⁻¹a is homotopic to ba
ab(a⁻¹a) is homotopic to ba
ab is homotopic to ba
Contrast with Figure Eight:
In the figure eight (S¹ ∨ S¹), let the loops be a and b, meeting at a single point.
ab: Go around loop A, then loop B.
ba: Go around loop B, then loop A.
You cannot deform ab into ba. Why? Because there’s no 2D surface connecting the loops like the interior of the square for the torus. The space “pinches” at the intersection point. Trying to slide path a past path b (or vice-versa) is blocked by this pinch point. The commutator aba⁻¹b⁻¹ is not shrinkable to a point; it represents a non-trivial loop itself.
Conclusion: The commutativity of the torus’s fundamental group arises because the loops (a and b) can be thought of as traversing the sides of a 2D patch (the square), and this patch allows the paths ab and ba to be continuously deformed into one another across its surface. This 2D “room to maneuver” is absent in the figure eight, leading to its non-abelian fundamental group.
Naixian Zhang 