| Version 9 |
Version 8 |
| Let $(X,x_{0})$ be a pointed topological space (that is, a topological space with a chosen basepoint $x_{0}$). |
Let $(X,x_{0})$ be a pointed topological space (i.e. a topological space with a chosen basepoint |
| Denote by $[(S^1,1),(X,x_{0})]$ |
$x_{0}$). Denote by $[(S^{1},1),(X,x_{0})]$ the set of homotopy classes of maps $\sigma:S^{1} \rightarrow X$ |
| the set of homotopy classes of maps $\sigma\colon S^{1} \to X$ |
such that $\sigma(1)=x_{0}$. Here, $1$ denotes the basepoint $(1,0) \in S^{1}$. Define a product |
| such that $\sigma(1)=x_{0}$. |
$[(S^{1},1),(X,x_{0})] \times [(S^{1},1),(X,x_{0})] \rightarrow [(S^{1},1),(X,x_{0})]$ |
| Here, $1$ denotes the basepoint $(1,0) \in S^{1}$. |
by $[\sigma][\tau]=[\sigma\tau]$, where $\sigma\tau$ means ``travel along $\sigma$ and then $\tau$''. |
| Define a product |
This gives $[(S^{1},1),(X,x_{0})]$ a group structure and we define the \emph{fundamental group} of $X$ |
| $[(S^1,1),(X,x_{0})] \times [(S^1,1),(X,x_{0})] \to [(S^1,1),(X,x_{0})]$ |
to be $\pi_{1}(X,x_{0})= [(S^{1},1),(X,x_{0})]$. The fundamental group of a topological space is an example of a homotopy group. |
| by $[\sigma][\tau]=[\sigma\tau]$, |
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| where $\sigma\tau$ means ``travel along $\sigma$ and then $\tau$''. |
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| This gives $[(S^1,1),(X,x_{0})]$ a group structure |
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| and we define the \emph{fundamental group} of $(X,x_0)$ |
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| to be $\pi_1(X,x_{0})= [(S^1,1),(X,x_{0})]$. |
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| In general, the fundamental group of a topological space |
Two homotopically equivalent spaces have the same fundamental group. Moreover, it can be shown that $\pi_{1}$ is a |
| depends upon the choice of basepoint. |
functor from the category of (small) pointed topological spaces to the category of (small) groups. |
| However, basepoints in the same path-component of the space |
Thus the fundamental group is a topological invariant in the sense that if $X$ is homeomorphic |
| will give isomorphic groups. |
to $Y$ via a basepoint preserving map, $\pi_{1}(X,x_0)$ is isomorphic to $\pi_{1}(Y,y_{0})$. |
| In particular, this means that the fundamental group of a (non-empty) path-connected space is well-defined, up to isomorphism, |
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| without the need to specify a basepoint. |
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| Two homotopically equivalent path-connected spaces have the same fundamental group. |
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| It can be shown that $\pi_1$ is a functor |
Examples of the fundamental groups of some familiar spaces are: $\pi_{1}(\mathbb{R}^n)\cong\{0\}$ for each $n$, $\pi_{1}(S^1)\cong\mathbb{Z}$ and $\pi_{1}(T)\cong \mathbb{Z}\oplus\mathbb{Z}$ where $T$ is the torus. |
| from the category of pointed topological spaces to the category of groups. |
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| In particular, the fundamental group is a topological invariant, |
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| in the sense that |
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| if $(X,x_0)$ is homeomorphic to $(Y,y_0)$ via a basepoint preserving map, |
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| then $\pi_1(X,x_0)$ is isomorphic to $\pi_1(Y,y_{0})$. |
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| Examples of the fundamental groups of some familiar spaces are: $\pi_{1}(\mathbb{R}^n)\cong\{0\}$ for each $n$, |
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| $\pi_{1}(S^1)\cong\mathbb{Z}$ |
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| and $\pi_{1}(T)\cong \mathbb{Z}\oplus\mathbb{Z}$, where $T$ is the torus. |
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| The concept of the fundamental group can be generalized to higher dimensions, giving the series of homotopy groups. |
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| The fundamental group is the first homotopy group. |
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