Let $M$ and $N$ be $CW$-complexes.

Definition. (different from the isotopy notion in geometry of submanifolds). A (topological) isotopy is a fibre-wise continuous map $$ F: M\times [0,1]\longrightarrow N\times [0,1] $$ such that $F$ maps the fibre $M\times t$ homeomorphically onto a subset of the fibre $N\times t$ for each $t\in [0,1]$. Two injective continuous maps $$ f_1,f_2: M\longrightarrow N $$ are said isotopy equivalent, if there exists an isotopy $$ F: M\times [0,1]\longrightarrow N\times [0,1] $$ such that $$ f_1=F(\cdot,0),\\ f_2=F(\cdot,1). $$ The $CW$-complexes $M$, $N$ are said isotopy equivalent, if there exist injective continuous maps $$ f: M\longrightarrow N,\\ g: N\longrightarrow M $$ such that their compositions $fg$ is isotopy equivalent to $Id_N$ and $gf$ is isotopy equivalent to $Id_M$.

Question. Suppose $M$ is a compact manifold without boundary. Does there exist any non-compact $CW$-complex $N$ such that $M$ is isotopy equivalent to $N$?

Let $M$ and $N$ be $CW$-complexes.

Definition. (different from the isotopy notion in geometry of submanifolds). A (topological) isotopy is a fibre-wise continuous map $$ F: M\times [0,1]\longrightarrow N\times [0,1] $$ such that $F$ maps the fibre $M\times t$ homeomorphically onto a subset of the fibre $N\times t$ for each $t\in [0,1]$. Two injective continuous maps $$ f_1,f_2: M\longrightarrow N $$ are said isotopy equivalent, if there exists an isotopy $$ F: M\times [0,1]\longrightarrow N\times [0,1] $$ such that $$ f_1=F(\cdot,0),\\ f_2=F(\cdot,1). $$ The $CW$-complexes $M$, $N$ are said isotopy equivalent, if there exist injective continuous maps $$ f: M\longrightarrow N,\\ g: N\longrightarrow M $$ such that their compositions $fg$ is isotopy equivalent to $Id_N$ and $gf$ is isotopy equivalent to $Id_M$.

Question. Suppose $M$ is a compact manifold without boundary. Does there exist any non-compact $CW$-complex $N$ such that $M$ is isotopy equivalent to $N$? I cannot figure out any such example...