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# Homology, Homotopy and Applications

## Volume 11 (2009)

### Number 2

### A relative version of the finiteness obstruction theory of C. T. C. Wall

Pages: 381 – 404

DOI: https://dx.doi.org/10.4310/HHA.2009.v11.n2.a17

#### Author

#### Abstract

In his 1965 paper C. T. C. Wall demonstrated that if a CW complex Y is finitely dominated, then the reduced projective class group of Y contains an obstruction which vanishes if and only if Y is homotopy equivalent to a finite CW complex. Wall also demonstrated that such an obstruction is invariant under homotopy equivalences. Subsequently Sum and Product Theorems for this obstruction were proved by L. C. Siebenmann.

In his second paper on the subject Wall gives an algebraic definition of the relative finiteness obstruction. If a CW complex Y is finitely dominated rel. a subcomplex X, then the reduced projective class group of Y contains an obstruction which vanishes if and only if Y is homotopy equivalent to a finite complex rel. X.

In this paper we will use a geometric construction to reduce the relative finiteness obstruction to the non-relative version. We will demonstrate that the relative finiteness obstruction is invariant under certain types of homotopy equivalences. We will also prove the relative versions of the Sum and the Product Theorems.

#### Keywords

CW complex, finiteness obstruction, relative finiteness obstruction

#### 2010 Mathematics Subject Classification

57Q12

Published 1 January 2009