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# 数学代写|拓扑学代写TOPOLOGY代考|Computing the Fundamental Group of a Circle

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## 数学代写|拓扑学代写TOPOLOGY代考|Computing the Fundamental Group of a Circle

So far, it is not yet clear whether the fundamental group is an interesting invariantthat is, does it ever distinguish spaces? Are there any spaces at all with nontrivial fundamental group? In case the name didn’t give it away, here’s a spoiler: yes! We will show that the circle has nontrivial fundamental group.

This loop appears not to be homotopic to the trivial loop: it seems that this loop goes around once, and the trivial loop goes around 0 times. But how can we prove that, by doing some clever homotopy, we can’t shrink it down to a point?

There are several ways of proving this, and the different techniques highlight different properties of fundamental groups. In this section, we’ll see a way to do it using a first example of covering spaces, while in the next chapter we’ll see a different proof. We won’t talk more about covering spaces in general in this book, but the procedure we employ here to compute fundamental groups is very general and can be used to compute the fundamental group of any reasonably nice space.
The outline of the proof is the following: We want to start with a loop on the circle, lift it up to some other space, and see what the lifted version of the loop looks like.

## 数学代写|拓扑学代写TOPOLOGY代考|More Fundamental Groups

We have worked quite hard to find a space whose fundamental group is non-trivial. We should capitalize on this result and see if we can find other, related spaces whose fundamental groups can now be computed easily as a result of our hard work. An example where this approach is successful is for product spaces.

We must first make a small digression and attempt to put the notion of the product of two topological spaces $X$ and $Y$ on a slightly more rigorous footing. Just as we defined the product of two groups, let us define the product space as
$$X \times Y:={(x, y): x \in X \text { and } y \in Y} .$$
So far, this just defines $X \times Y$ as a set of points. To really turn $X \times Y$ into a topological space, we have to extend the topological notions from $X$ and $Y$ to $X \times Y$. We gave the precise mathematical definition of a topological space earlier, in Chapter 3 , but let us repeat it once more.

A topological space $X$ is a set of points together with a topology, which we’ll loosely take to mean “a way of defining an open set.” If $X \subseteq \mathbb{R}^3$ then we said that a subset $U \subseteq X$ is open if and only if, for every $x \in U$, we can find $\varepsilon>0$ so that the open ball $B_{\varepsilon}(x) \subseteq \mathbb{R}^3$ satisfies $B_{\varepsilon}(x) \cap X \subseteq U$. Thus we use the relatively open balls $B_{\varepsilon}(x) \cap X$ for all $x \in X$ and $\varepsilon>0$ to prove the openness of any subset of $X$. We say that the relatively open balls of $X$ constitute a “basis” for $X$.

More generally, we may have some space $X$ that is not a subset of $\mathbb{R}^3$-or any $\mathbb{R}^n$ for that matter-yet we still wish to consider it to be a topological space. What this means is that we need a way of deciding whether a subset of $X$ is open or not. We allow ourselves flexibility in how this is done, but we require that certain natural properties of open sets that we have seen before still hold.

## 数学代写|拓扑学代写TOPOLOGY代考|More Fundamental Groups

$$X \times Y:=(x, y): x \in X \text { and } y \in Y$$

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