Definition:Homotopy Equivalence
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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