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Homotopy equivalence
Suppose that $X$ and $Y$ are topological spaces and $f,g: X \to Y$ are continuous functions such that $fg$ and $gf$ are homotopic to the identity maps on $Y$ and $X$ respectively: $$fg \simeq id_{Y},$$ $$gf \simeq id_{X},$$ then $f$ is called a homotopy equivalence. In this case $X$ and $Y$ are called homotopy equivalent, or are of same homotopy type: $$X\simeq Y.$$
Examples. The ring is homotopy equivalent to the circle: $${\bf D}^n\simeq \{ 0\}$$
This sequence is often used to illustrate homeomorphisms but, in reality, it's homotopy equivalence:
Properties.
- Homotopy equivalence is an equivalence relation for topological spaces.
- A homeomorphism is a homotopy equivalence.
- Homology groups are preserved under homotopy equivalence.
A topological space $X$ is called contractible if $X$ is homotopy equivalent to a point, i.e., $X\simeq \{x_0\}$.
A non-trivial example of a contractible space is a two-room space:
Another, the dunce cap:
