Munkres §51: Homotopy of Paths

◄ Munkres §52: The Fundamental Group

A basic problem of topology is determining whether two spaces are homeomorphic. If so, then a continuous mapping between them with continuous inverse exists. Otherwise, if one can find a topological property that holds for one but not the other, then they cannot be homeomorphic:

• $(0,1)$ and $[0,1]$ are not homeomorphic because the latter is compact and the former is not.
• $\mathbb{R}$ and the long line are not homeomorphic because the former has a countable basis and the latter does not.
• $\mathbb{R}$ and ${\mathbb{R}}^2$ are not homeomorphic because deleting a point from the latter leaves a connected space while doing so from the former does not.

However, distinguishing between ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$ is more involved: deleting a point from the latter leaves a simply connected space while doing so from the former does not.

To distinguish between more spaces, a more general tool, their fundamental groups, can be used: two spaces are homeomorphic if their fundamental groups are isomorphic. For example, if a space is simply connected, then its fundamental group is trivial.

These writings build the tools necessary to talk about the fundamental group.

Claim: If $h,h^{\prime }:X\to Y$; $k,k^{\prime }:Y\to Z$; $h\simeq h^{\prime }$; and $k\simeq k^{\prime }$, then $k\circ h\simeq k^{\prime }\circ h^{\prime }$.

Proof: $(x,t)\mapsto G(F(x,t),t)$ is a homotopy.

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Claim: $[X,I]$ is a singleton.

Proof: $(x,t)\mapsto (1-t)f(x)+tg(x)$ is a homotopy.

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Claim: If $Y$ is path-connected, then $[I,Y]$ is a singleton.

Proof: If $h$ is a path from $f(1)$ to $g(0)$, then $f,g\in [I,Y]$ are closed subpaths of and thus homotopic to $f\ast h\ast g$.

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Claim: $I$ and $\mathbb{R}$ are contractible.

Proof: $(x,t)\mapsto (1-t)x$ is a homotopy, i.e., ${\text{id}}_I$ and ${\text{id}}_{\mathbb{R}}$ are nulhomotopic to $0$.

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Claim: If $X$ is contractible, then $X$ is path-connected.

Proof: $F(x,\cdot )$ is a path from $x$ to $x_0$, where $F$ is a homotopy between ${\text{id}}_X$ and $x_0$.

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Claim: If $Y$ is contractible, then $[X,Y]$ is a singleton.

Proof: ${\text{id}}_Y\circ f=f$ is homotopic to $y_0\circ f=y_0$ since ${\text{id}}_Y$ is homotopic to $y_0$.

$$\begin{array}{}◼& \end{array}$$

Claim: If $X$ is contractible and $Y$ is path-connected, then $[X,Y]$ is a singleton.

Proof: $f\circ {\text{id}}_X=f$ is homotopic to $f\circ x_0=f(x_0)$ since ${\text{id}}_X$ is homotopic to $x_0$ and all constant maps are nulhomotopic since $Y$ is path-connected.

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