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
π
of the integral group ring
π
It is named after the mathematician C. T. C. Wall.
By work of John Milnor[1] 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
By a construction due to Milnor it is possible to extend r to a homotopy equivalence
is a CW-complex obtained from K by attaching cells to kill the relative homotopy groups
π
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
Wall then showed that the cellular chain complex
is chain-homotopy equivalent to a chain complex
of finite type of projective
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