# Example of a finitely-dominated CW complex which is not finite?

It's easy to find sources discussing Wall's finiteness obstruction, but harder to find discussion of examples. What's an example of a finitely-dominated CW complex which is not homotopy equivalent to a finite CW complex?

• Wall's theorem not only gives a complete obstruction, but also shows how to realize any nonzero element of $\tilde{K}_0(\mathbb{Z}[\pi])$ with $\pi$ a finitely presentable group as the obstruction for some finitely-dominated CW complex (see a sketch of the construction in Theorem 3.1 of Ferry-Ranicki's survey). This gives a rich store of examples. – Andy Putman Apr 19 at 20:27
• This is also discussed in Wall's original paper ("Finiteness conditions for CW complexes") – Denis Nardin Apr 19 at 20:35
• Thanks, both of you, maybe I should have noticed this. I don't suppose there are examples out there which arise more naturally? – Tim Campion Apr 19 at 20:40
• Does a finitely dominated CW complex mean a space $X$ with a surjective continuous map from a finite CW complex $Y$? If so, then the following may provide an example. Consider the space $X$ that is the union (in the plane) of circles $C_n$ of radius $1/n$ tangent at a point $p$. Now divide a circle $Y$ into disjoint arcs $A_n$ of positive length that add up to the full circle. Wrap each $A_n$ onto $C_n$ starting and ending at the point $p$. This gives a map from $Y$ to $X$ which is continuous and surjective. – Kapil Apr 20 at 3:02
• @Kapil By a finitely-dominated space I mean a space of the homotopy type of a CW complex which is homotopy equivalent to a retract of a finite CW complex. – Tim Campion Apr 20 at 3:52