# Finite domination and compact ENRs

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.

• @Tyrone but what about being a compact ENR? Mar 27 at 19:15
• I completely missed the word compact. Mar 27 at 19:17
• No, this is the Borsuk conjecture, which was solved positively by West. Every compact ANR has the homotopy type of a finite complex. Mar 27 at 19:21

It was originally conjectured by Borsuk that every compact ANR should be homotopy equivalent to a finite CW complex. While it was known that every separable ANR has the homotopy type of a countable CW complex, the finer statement was an open problem for some time in the '60s and '70s.

It was J. West

Mapping Hilbert Cube Manifolds to ANR's: A Solution of a Conjecture of Borsuk, Ann. Math., 106, (1977), 1-18.

who finally arrived at a positive solution to Borsuk's conjecture (see Corollary 5.2).

T. Chapman had previously shown that every compact Hilbert Cube manifold has the homotopy type of a finite CW complex. In the paper linked above, West showed that every compact ANR has the homotopy type of compact Hilbert cube manifold, thus concluding;

Theorem: [West] Every compact ANR has the homotopy type of a finite CW complex.

Since Wall has shown that there are finitely dominated spaces not of the homotopy type of a finite complex, it must be that there are fintiely dominated spaces not of the homotopy type of any compact ANR.