Definition:Linear Homotopy

Definition

Let $X$ be a topological space.

Let $Y$ be a subspace of the Euclidean space $\R^n$.

Let $f: X \to Y$, $g: X \to Y$ be continuous mappings.

For all $x \in X$, let the line segment in $\R^n$ from $\map f x$ to $\map g x$ be a subset of $Y$.

Let $f \simeq g$ by means of a homotopy $H: X \times \closedint 0 1 \to Y$, where $H$ is defined as:

$\map H {x, t} = \paren {1 - t} \map f x + t \map g x$

This is known as a linear homotopy.

Also see

• Results about linear homotopies can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): homotopy
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): homotopy