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homotopy with a contractible domain (Theorem)

Theorem. Assume that $ Y$ is an arbitrary topological space and $ X$ is a contractible topological space. Then all maps $ f:X\rightarrow Y$ are homotopic if and only if $ Y$ is path connected.

Proof: Assume that all maps are homotopic. In particular constant maps are homotopic, so if $ y_{1},y_{2}\in Y$, then there exists a continous map $ H:I\times Y\rightarrow Y$ such that $ H(0,y)=y_1$ and $ H(1,y)=y_2$ for all $ y\in Y$. Thus the map $ \alpha:I\rightarrow Y$ defined by the formula $ \alpha(t)=H(t,y_0)$ for a fixed $ y_0\in Y$ is the wanted path.

On the other hand assume that $ Y$ is path connected. Since $ X$ is contractible, then for any $ c\in X$ there exists a continous homotopy $ H:I\times X\rightarrow X$ connecting the identity map and a constant map $ c$. Let $ f:X\rightarrow Y$ be an arbitrary map. Define a map $ F:I\times X\rightarrow Y$ by the formula: $ F(t,x)=f(H(t,x))$. This map is a homotopy from $ f$ to a constant map $ f(c)$. Thus every map is homotopic to some constant map.

The space $ Y$ is path connected, so for all $ y_1,y_2\in Y$ there exists a path $ \alpha:I\rightarrow Y$ from $ y_1$ to $ y_2$. Therefore constant maps are homotopic via the homotopy $ H(t,x)=\alpha(t)$.

Finaly for any continous maps $ f,g:X\rightarrow Y$ and any point $ c\in X$ we get:

$\displaystyle f\simeq f(c)\simeq g(c)\simeq g,$
which completes the proof. $ \square$


Corollary. If $ X$ is a contractible space, then for any topological space $ Y$ there exists a bijection between the set $ [X,Y]$ of homotopy classes of maps from $ X$ to $ Y$ and the set $ \pi_0(Y)$ of path components of $ Y$.

Proof: Assume that $ Y=\bigcup Y_{i}$, where $ Y_i$ are path components of $ Y$. It is well known that contractible spaces are path connected, thus the image of any continous map $ f:X\rightarrow Y$ is contained in $ Y_i$ for some $ i$. It follows from the theorem that two maps from $ X$ to $ Y$ are homotopic if and only if their images are contained in the same $ Y_i$. Thus we have a well defined, injective map
$\displaystyle \psi:[X,Y]\rightarrow\pi_0(Y)$
$\displaystyle \psi([f])=Y_i,$
where $ i$ is such that $ f(X)\subseteq Y_i$. This map is also surjective, since for any $ i $ there exists $ y\in Y_i$, so the class of the constant map $ f(x)=y$ is mapped into $ Y_i$. $ \square$



"homotopy with a contractible domain" is owned by joking.
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See Also: homotopy, contractible

Keywords:  contractible domain, homotopy
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Cross-references: surjective, injective, well defined, contained, image, path components, classes, bijection, completes, point, identity map, homotopy, path, fixed, constant maps, path connected, homotopic, maps, contractible, topological space

This is version 23 of homotopy with a contractible domain, born on 2008-04-30, modified 2008-04-30.
Object id is 10557, canonical name is HomotopyWithAContractibleDomain.
Accessed 200 times total.

Classification:
AMS MSC: 55P99 (Algebraic topology :: Homotopy theory :: Miscellaneous)
Pending Errata and Addenda
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