# Applying the Borsuk-Ulam theorem to the torus: A helpful hint

• qinglong.1397
In summary, the Borsuk-Ulam theorem does not hold for the torus, as shown by Hatcher's proof for the map f: S^2\rightarrow \mathbb{R}^2. The method used for S^2 may not be applicable for this new problem, but it is still worth exploring. A helpful hint is to consider whether the theorem holds for S^1, which can provide some insight into the solution for the torus.

## Homework Statement

Does the Borsuk-Ulam theorem hold for the torus? In other words, for every map $$f: S^1\times S^1\rightarrow \mathbb{R}^2$$ must there exist $$(x, y) \in S^1\times S^1$$ such that $$f(x,y)=f(-x, -y)$$?

## The Attempt at a Solution

In Hatcher's book, he gave the proof for the map $$f: S^2\rightarrow \mathbb{R}^2$$.

For this new problem, I really do not know what to do. It seems that Hatcher's method is useful, but I just do not know how to use it. So, please, tell me some hint. Thank you very much!

Here's a hint. It doesn't hold for S^1, does it?

Dick said:
Here's a hint. It doesn't hold for S^1, does it?

## 1. What does "hint" mean in a scientific context?

In science, a hint refers to a small piece of information or clue that can help guide or direct research or experiments. It is often used to suggest possible avenues for further investigation.

## 2. How can a hint be helpful in scientific research?

A hint can be helpful in scientific research as it can provide a starting point or guide for further investigation. It can also help researchers make connections between different pieces of information and come up with new ideas or hypotheses.

## 3. How can I ask for a hint without sounding too needy or incompetent?

When asking for a hint in a scientific context, it is important to be specific and provide background information on your research or experiment. This shows that you have put effort into your work and are genuinely seeking guidance. You can also ask for a hint in a respectful and humble manner, such as "I would appreciate any suggestions or insights you may have to offer."

## 4. What is the difference between a hint and a solution in science?

A hint is a small piece of information or clue that can guide research or experiments, while a solution is a definitive answer or resolution to a problem. Hints are often used as stepping stones towards finding a solution, but they do not provide a complete answer.

## 5. Are hints considered cheating in science?

No, hints are not considered cheating in science. In fact, seeking guidance and advice from others is a common practice in the scientific community. However, it is important to give credit to the source if a hint leads to a breakthrough in research or experiments.

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