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Paths and Homotopy

$\X$ topological space
$\px,\,\y\in \X$

A path from $\px$ to $\y$ is

\htmlClass{var-alpha}\alpha\colon I\to \htmlClass{var-X}X

\htmlClass{var-alpha}\alpha(0)=\htmlClass{var-x}{x},\quad\htmlClass{var-alpha}\alpha(1)=\htmlClass{var-y}y

$\X$ is path-connected if, for all $\px,\y\in \X,$ there exists a path from $\px$ to $\y.$

Adjust the slider to move along the path

\htmlClass{var-alpha}\alpha.

\{z^2 = xy - 1\}

Homotopies

Definition. (Homotopy)

Let \(\xymatrix{\X \ar@<1ex>[r]^-\f \ar@<-1ex>[r]_-\g & \Y}\) be maps between spaces $\X$ and $\Y.\$

A homotopy from

\f

to

\g

is a map

\H\colon \X\times I \to \Y

with

\H(-,0)=\f,\quad \H(-,1)=\g.

We say that $\f$ and $\g$ are homotopic if there exists a homotopy between them, and write $\f\simeq \g.$

Paths are homotopies

\px,\,\y\in\X

\(\xymatrix{{*} \ar@<1ex>[r]^-\px \ar@<-1ex>[r]_-\y & \X}\)

A homotopy between these functions is the same as a path from $\px$ to $\y\text!$

Homotopies are paths

\X\times I\to \Y

map from the cylinder on $\X$ to $\Y$

\X\to\Map(I,\Y)

map from $\X$ to the space of paths in $\Y$

I\to \Map(\X,\Y)

path in the space of maps $\X\to\Y$

(Need to be careful with point-set topology here)

compact-open topology

\(\xymatrix{ \X \ar@<1ex>[r]^-\f \ar@<-1ex>[r]_-\g & Y }\)

\X\sqcup \X \xrightarrow{\<\f,\g\>} \Y

\(\xymatrix{ \X \times S^0 \ar[r]^-{\<\f,\g\>} \ar@[class:rewrite-arrow][d] & \Y\\ \class{rewrite-arrow}{\X \times D^1} \ar@{..>}@[class:rewrite-fill][ur]_-{\class{rewrite-fill}\H} }\)

Drag the orange circle along the cylinder to see the homotopy.

Subspaces of mapping spaces

Often want to impose conditions on a homotopy

e.g. \(\xymatrix{(\X,\px) \ar@<1ex>[r]^-\f \ar@<-1ex>[r]_-\g & (\Y,\y)}\) based maps, i.e. $f(x)=y,\ g(x) = y$

a based homotopy is a homotopy $H$ such that $H(x,t) = y$ for all $t$

I \to \Map(X, Y)

I \to A \subset \Map(X, Y)

A = \{f\colon X\to Y \mid f(x) = y\}