Definition:Homotopy Equivalence

Definition

Let $X$ and $Y$ be topological spaces.

Let $f: X \to Y$ be a continuous mapping.

Let there exist a continuous mapping $g: Y \to X$ such that:

the composite mapping $g \circ f$ is homotopic to the identity mapping $I_X$ on $X$

the composite mapping $f \circ g$ is homotopic to the identity mapping $I_Y$ on $Y$.

Then $X$ and $Y$ are homotopy equivalent.

Also known as

Homotopy equivalence can also be seen hyphenated: homotopy-equivalence.

Also see

• Results about homotopy equivalences can be found here.

Sources

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