Linear homotopy

Definition

Suppose $U$ is a subset of ${\mathbb{R}}^n$ and $f,g:X\to U$ are continuous maps. Suppose further that for every $x\in X$, the line segment joining $f(x)$ to $g(x)$ lies completely inside $U$. The linear homotopy between $f$ and $g$ is a homotopy defined as follows:

$x\mapsto (1-t)f(x)+tg(x)$

where the computation on the right side is in ${\mathbb{R}}^n$. Essentially we are moving from $f(x)$ to $g(x)$ along a straight line.

A composite of several linear homotopies is termed a piecewise linear homotopy. If there exists a piecewise linear homotopy between two functions $f,g:X\to U$ then we say that $f$ and $g$ are piecewise linearly homotopic maps.

Further information: Linear homotopy theorem

Facts

One nice thing about linear homotopies is that they do not unnecessarily move points. In other words, if $f(x)=g(x)$ for some point $x$, the linear homotopy from $f$ to $g$ fixes $x$ at every point. Linear homotopies are thus useful for showing that given retracts are deformation retracts.

Linear homotopies are commonly seen in the following kinds of sets:

• Convex subset of Euclidean space: Any two functions to such a set are linearly homotopic
• Star-like subset of Euclidean space: Any two functions to such a set are homotopic via a composite of at most two linear homotopies
• Compact retract of open subset of Euclidean space