Wall's finiteness obstruction
In geometric topology, a field within mathematics, the obstruction to a finitely dominated space "X" being homotopy-equivalent to a finite CW-complex is its Wall finiteness obstruction "w(X)" which is an element in the reduced zeroth algebraic K-theory formula_1 of the integral group ring formula_2. It is named after the mathematician C. T. C. Wall. By work of John Milnor on finitely dominated spaces, no generality is lost in letting "X" be a CW-complex. A "finite domination" of "X" is a finite CW-complex "K" together with maps formula_3 and formula_4 such that  formula_5. By a construction due to Milnor it is possible to extend "r" to a homotopy equivalence formula_6 where formula_7 is a CW-complex obtained from "K" by attaching cells to kill the relative homotopy groups formula_8. The space formula_7 will be "finite" if all relative homotopy groups are finitely generated. Wall showed that this will be the case if and only if his finiteness obstruction vanishes. More precisely, using covering space theory and the Hurewicz theorem one can identify formula_8 with formula_11. Wall then showed that the cellular chain complex formula_12 is chain-homotopy equivalent to a chain complex  formula_13 of finite type of projective formula_2-modules, and that formula_15 will be finitely generated if and only if these modules are stably-free. Stably-free modules vanish in reduced K-theory. This motivates the definition