An explicit projection, by Andrew A. Ranicki

The Wall finiteness obstruction is the principal application of the projective class group $K_0(\Lambda)$ to topology, with $\Lambda=Z[G]$ the group ring of the fundamental group $G$. The finiteness obstruction is an element of the reduced projective class group $\widetilde K_0(\Lambda)={\rm coker}(K_0(Z) \sa K_0(\Lambda))$. This paper uses the Rim cartesian square and the Milnor Mayer-Vietoris exact sequence to provide an explicit construction of a f.g. projective $Z[G]$-module $P={\rm im}(p)$ such that $[P] \ne 0\in \widetilde K_0(Z[G])$, with $G=Q_8$ the quaternion group of order 8 and $p=p^2$ a $2 \times 2$ projection matrix with entries in $Z[Q_8]$. This example is well-known to the experts, but an explicit construction of $p$ might be of interest to students of algebraic $K$-theory.


Andrew A. Ranicki <a.ranicki@edinburgh.ac.uk>