Homotopy equivalence?

Question

Can someone explaine what this means mathematicaly :

"Let us denote by $h: X\rightarrow Y$ a homotopic equivalence map for which $h|_{Y}$ is the identity "

• Remark: $Y$ is include in $X$

Please

Thank you.

Answer 1

This makes sense when $Y$ is a subspace of $X$ (otherwise we cannot restrict $X$ to $Y$). Then it just means that $h: X \to Y$ is a homotopy equivalence satisfying $h(y)=y$ for all $y \in Y$.

A homotopy equivalence is a map which admits an inverse up to homotopy.