nLab delayed homotopy

Idea

A homotopy between continuous functions between topological spaces is called delayed if it starts out being constant near one boundary of the interval.

(If it is constant near both boundaries we say it has sitting instants).

Definition

For $I = [0,1]$ the unit interval and $X$ and $Y$ any topological spaces, a continuous map $F: X\times I\to Y$ is a delayed homotopy (between $F(-,0)$ and $F(-1))$ if there exist $t_0\gt 0$ such that $F(x,t)=F(x,0)$ for all $0\leq t\leq t_0$.

Properties

In Dold-fibrations

Delayed homotopies appear in an alternative characterization of Dold fibrations. See there for details.

Smoothing of delayed homotopies

If a continuous homotopy between two smooth functions is delayed at both ends of the inerval it may be approximated by a smooth homotopy . See Steenrod-Wockel approximation theorem.