**Edit:** In the comments, Tyrone points out that West's positive answer to Borsuk's conjecture implies that every compact ENR is homotopy equivalent to a finite CW complex. It follows that the only finitely dominated spaces which are homotopy equivalent to compact ENRs are those finitely dominated spaces whose Wall finiteness obstruction is trivial. This completely settles the question.

A space $X$ is said to be *finitely dominated* if there is a finite CW complex $K$ and maps $r: X \to K$, $s: K \to X$ such that $r \circ s: K \to K$ is homotopic to the identity map, i.e., $X$ is a homotopy retract of $K$.

Equivalently, $X$ is finitely dominated if there is a space $K'$ homotopy equivalent to a finite CW complex and maps $r: X \to K'$, $s: K' \to X$ such that $r' \circ s': K' \to K'$ is the identity, i.e., $X$ is a (strict) retract of $K'$.

A space $X$ is said to be an *ENR* (Euclidean neighborhood retract) if there is an embedding $i: X\to \Bbb R^n$ such that
$i(X)$ is a retract of some open neighborhood $U \subset \Bbb R^n$.

It is known that $X$ is a compact ENR if and only if $X$ is a (strict) retract of a finite CW complex (cf. Hatcher's book App. A).

The above leads to the following question:

**Question** *Is every finitely dominated space homotopy equivalent to a compact ENR? If not, what are the obstructions?*

**Notes:** (1) The question asks whether or not the property of being a homotopy retract of a finite complex is the same as that of being homotopy equivalent to a strict retract of a finite complex.

(2) If $X$ is simply connected and finitely dominated, then Wall shows that X is homotopy equivalent to a finite CW complex. It follows that X is homotopy equivalent to a compact ENR. So if there a counterexample, if it exists, is necessarily not 1-connected.

compactENR? $\endgroup$