Timeline for Example of a finitely-dominated CW complex which is not finite?
Current License: CC BY-SA 4.0
6 events
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| Apr 20, 2019 at 3:52 | comment | added | Tim Campion | @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. | |
| Apr 20, 2019 at 3:02 | comment | added | Kapil | 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. | |
| Apr 19, 2019 at 20:40 | comment | added | Tim Campion | Thanks, both of you, maybe I should have noticed this. I don't suppose there are examples out there which arise more naturally? | |
| Apr 19, 2019 at 20:35 | comment | added | Denis Nardin | This is also discussed in Wall's original paper ("Finiteness conditions for CW complexes") | |
| Apr 19, 2019 at 20:27 | comment | added | Andy Putman | 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. | |
| Apr 19, 2019 at 19:33 | history | asked | Tim Campion | CC BY-SA 4.0 |