Is there a rigorous way to describe $g(x)$ continuously deforming into $h(x)$ and could it be useful?

Question

One thing that bothers me about mappings is that they seem to instantaneously transport points from one space to another. I feel like there should be a space of unique, non-intersecting paths that each of the points travel on to get to their new destination. Does anyone agree?

For example consider a mapping $F:\Bbb R^2 \to \Bbb R^2$ with $F(x,y)=(e^x,e^y).$ Consider the function $g(x)=\frac{1}{x}$ embedded in the standard $x-y$ cartesian system. We start with $g$ and magically get $h(x)=e^{\frac{1}{\log(x)}}$ with no information about how $g$ was deformed into $h!$ Maybe it's just perspective but I feel like at every point in time we should be able to track the deformations as $g$ morphs into $h.$

I drew a picture with the paths that I think each of the points should follow as they start with $g$ and move to become $h.$

The upper bound path is $y=e^x$ and the lower bound path is $y=\log(x).$ The central path is $y=x.$

Obviously there's not enough rigor here, but I tried my best with what I know. My question is:

Is there a rigorous way to describe $g$ continuously deforming into $h$ and could it be useful?

Answer 1

It sounds like you're looking for the notion of homotopy.

Broadly speaking, the idea behind homotopy is that we're not just interested in individual continuous maps into a space $X$, but rather the manipulation of such maps. E.g. the picture you drew suggests that we should be able to take the map $$\alpha:\mathbb{R}\rightarrow\mathbb{R}: x\mapsto e^x$$ and "deform" it into the map $$\beta:\mathbb{R}\rightarrow\mathbb{R}: x\mapsto \ln(x),$$ for example hitting the map $\gamma:\mathbb{R}\rightarrow\mathbb{R}:x\mapsto x$ along the way.

The trick to making this intuition precise is to think about maps from a product space. Specifically, we're going to think of "a continuous map from $A$ to $B$ being deformed over time from $t=0$ to $t=1$" (say) as "a map from $A\times [0,1]$ to $B$." Conversely, given a continuous map $m:A\times [0,1]\rightarrow B$, for each $t\in[0,1]$ we get the "snapshot map" $m_t:A\rightarrow B: a\mapsto m(a,t)$, and we think of $m_0$ and $m_1$ as the "starting" and "finishing" maps.

We can then, for example, talk about when one continuous map $f:A\rightarrow B$ can be "deformed into" another continuous map $g:A\rightarrow B$ - namely, when there is a continuous map $m:A\times[0,1]\rightarrow B$ such that $m_0=f$ and $m_1=g$. When such an $m$ exists we say that $f$ and $g$ are homotopic.

• For example, the homotopy relation isn't very interesting in $\mathbb{R}^2$ (or more generally $\mathbb{R}^n$). Specifically, suppose we have two maps $f,g:X\rightarrow\mathbb{R}^2$. Then consider the map $$m: X\times[0,1]\rightarrow\mathbb{R}^2: m(x,t)=tg(x)+(1-t)f(x).$$ This $m$ is guaranteed to be a homotopy between $f$ and $g$. On the other hand, more complicated spaces make things more interesting. Consider the space $Y=\mathbb{R}^2\setminus\{(0,0)\}$, $f: S^1\rightarrow Y$ the usual map from the circle to the plane, and $g: S^1\rightarrow Y$ the constant map sending everything to $(17,42)$. Are $f$ and $g$ homotopic?

And there's a lot more to say. Of particular interest is the case when $A$ itself is the unit interval $[0,1]$ and we restrict attention to those homotopies $m:[0,1]\times[0,1]\rightarrow B$ which "keep the endpoints fixed," that is, which satisfy $m(0,0)=m(0,x)$ and $m(1,0)=m(1,x)$ for all $x\in[0,1]$ - these $m$s are the path homotopies and lead to the notion of the fundamental group(oid). The notion of homotopy also leads to a notion of similarity of spaces, namely homotopy equivalence. The wiki page has more information on the topic.