Wall's finiteness obstruction explained

\widetilde{K} _{0(Z[\pi} _{1(X)])} of the integral group ring Z [\pi _{1(X)]} . 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

r:K\to X and i\colon X\to K such that r\circ i\simeq 1 _{X} . By a construction due to Milnor it is possible to extend r to a homotopy equivalence \bar{r}\colon \bar{K}\to X where \bar{K} is a CW-complex obtained from K by attaching cells to kill the relative homotopy groups \pi _{n(r)} .

The space

\bar{K} 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 \pi _{n(r)} with H _{n(\widetilde{X},\widetilde{K})} . Wall then showed that the cellular chain complex C _{*(\widetilde{X})} is chain-homotopy equivalent to a chain complex A _{*} of finite type of projective Z [\pi _{1(X)]} -modules, and that H _{n(\widetilde{X},\widetilde{K})\cong} H _{n(A} _{*)} will be finitely generated if and only if these modules are stably-free. Stably-free modules vanish in reduced K-theory. This motivates the definitionw(X) = \sum ^{i[A} _{i]\in\widetilde{K}} _{0(Z[\pi} _{1(X)])} .

See also

• Algebraic K-theory
• Whitehead torsion

References

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