Dates: 02/06/03, 02/13/03
Speaker: Anna Davis

Title: A GEOMETRIC APPROACH TO FINITENESS OBSTRUCTION THEORY OF WALL

Abstract:
Wall's finiteness obstruction theory arises from the question: If Y is a finitely dominated CW complex, when is Y homotopy equivalent to some finite CW complex? Wall defines the finiteness obstruction of a finitely dominated CW complex Y to be an element \sigma(Y) of the reduced projective class group \tilde K_0(Z\pi_1(Y)). Wall shows that \sigma(Y)=0 iff Y is homotopy equivalent to a finite complex. Wall also shows that \sigma(Y) is an invariant of homotopy type. Product and sum theorems for Wall's obstruction were obtained by Siebenmann. Relative finiteness obstruction theory, also introduced by Wall, arises from the question: If Y is finitely dominated rel. X, when is Y homotopy equivalent to a finite complex rel. X? Wall defines the relative finiteness obstruction of a finitely dominated pair (Y, X) to be an element \sigma(Y, X) of \tilde K_0(Z\pi_1(Y)). The obstruction vanishes iff Y is homotopy equivalent rel. X to some X\cup K where K is a finite CW complex. Wall's approach to obstruction theory is purely algebraic. A geometric construction of ordinary obstruction theory was carried out by Ferry. A geometric construction of relative finiteness obstruction together with sum, product and invariance theorems was introduced by Chapman and myself. In a series of two talks, I will give an outline of the algebraic constructions of Wall and a more detailed description of geometric constructions carried out by Ferry, Chapman and myself.