The Geometric Finiteness Obstruction

Abstract

The purpose of this paper is to develop a geometric approach to Wall's finiteness obstruction. We will do this for equivariant CW-complexes. The main advantage will be that we can derive all the formal properties of the equivariant finiteness obstruction easily from this geometric description. Namely, the obstruction property, homotopy invariance, the sum and product formulas, and the restriction formula can be stated and proved in a simple manner. Also a characterization of the finiteness obstruction by a universal property is quickly available. This geometric approach is similar to the treatment of Whitehead torsion by Cohen in [3]. In the first section we define a functor Wa from the category of G-spaces to the category of abelian groups. We assign to a finitely dominated G-CW-complex X an element w(X) e Wa(X) called its finiteness obstruction. The finiteness obstruction vanishes if and only if X is G-homotopic to a finite G-CW-complex and satisfies a sum formula and is homotopy invariant. The notion of a universal functorial additive invariant is introduced in the second section where its existence and uniqueness are proved. Product and restriction formulas for the universal additive invariant are obtained by abstract nonsense. We define equivariant Euler characteristics in the third section generalizing the notion of the Euler characteristic of a finite CW-complex. The goal of the fourth section is to prove that the equivariant Euler characteristic and finiteness obstruction determine the universal functorial additive invariant for finite, respectively finitely dominated, G-CW-complexes. The fifth section contains some algebraic computations of Wa in terms of reduced projective class groups of certain integral group rings. In the nonequivariant case Wall's algebraic approach and our geometric one agree. Finally, in the sixth section, the results of the second and fourth sections are used to state an abstract product formula, a restriction formula, and a diagonal product formula. We make some remarks about the simple-homotopy approach to the finiteness obstruction due to Ferry. The treatment by Ferry in [7] is extended by Kwasik in [14] to the equivariant case. In § 6 we construct geometrically an injection I(Y): W(Y)^>Wh(YxS) into the equivariant Whitehead group of Y x S sending our geometric finiteness obstruction to that of Kwasik. A compact Lie group is denoted by G.