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Can someone explain me in descriptive words or maybe with an image, what homotopic actually means and what its relevance is?

Thank you for your time,


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Which parts of the wikipedia article do you find vague? – user17794 Jun 17 '12 at 2:37
up vote 1 down vote accepted

Let $X$ and $Y$ be topological spaces, and let $F_0, F_1:X\rightarrow Y$ be continuous maps. We say that $F_0$ and $F_1$ are homotopic and denoted by $F_0 \simeq F_1$ if there exists a continuous map $H: X \times I \rightarrow Y$ (which is called a homotopy from $F_0$ to $F_1$) such that for all $x \in X$,

$$H(x,0)=F_0(x),$$ $$H(x,1)=F_1(x).$$

If we think of the parameter $t$ as the time parameter which changes from 0 to 1, then the homotopy $H$ represents a continuous deformation of the map $F_0$ to the map $F_1.$

We know that a path in a topological space $X$ is a continuous map $f:I \rightarrow X$ such that $f(0)=x_0$ and $f(1)=x_1$. The points $x_0$ and $x_1$ are called initial point and terminal point of $f$, respectively. We say that $f$ is a path from $x_0$ to $x_1.$ For any two paths in the topological space $X$, we need a stronger relation between the paths in order to find the holes of the space $X.$

Let $f_0, f_1:I\rightarrow X$ be two paths in $X$. We say that $f_0$ and $f_1$ are path-homotopic and denoted by $F_0 \sim F_1$ if $f_0$ and $f_1$ have the same initial point $x_0$ and the same terminal point $x_1$ and there exists a continuous map $H: I \times I \rightarrow X$ (which is called a path-homotopy from $f_0$ to $f_1$) such that

$$ H(s,0)=f_0(s) $$ $$ H(s,1)=f_1(s) $$ $$ H(0,t)=x_0 $$ $$ H(1,t)=x_1 $$

for all $s \in I$ and for all $t \in I.$

In other words, the path-homotopy $H$ represents a continuous deformation of the path $f_0$ to the path $f_1$. Also end points remain fixed during the deformation.

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