Homotopies

Entry conditions

Use homotopy only when you have topological spaces and continuous maps between them.

Definitions

Two continuous maps $f,g:X\to Y$ are homotopic if there exists a continuous map

$$H:X\times [0,1]\to Y$$

with $H(x,0)=f(x)$ and $H(x,1)=g(x)$.

Vocabulary (plain language)

• Deformation: a continuous transformation over time.
• Parameter: the time variable in $[0,1]$.

Symbols used

• $H$: the homotopy between $f$ and $g$.

Intuition

Homotopy says that two maps are the same if one can be continuously deformed into the other.

Worked example

On the circle $S^1$, the identity map is not homotopic to a constant map, because the deformation would have to “collapse” the circle.

How to recognize the structure

• You can define $H$ as a continuous map on $X\times [0,1]$.
• The endpoints of $H$ are the given maps.

Common mistakes

• Using homotopy without specifying the parameter interval.