Another proof of generalized sum theorem for Whitehead torsion

Abstract

Introduction. Kwun and Szczarba [3.] established a sum theorem for Whitehead torsion of homotopy equivalence. Their theorem is stated as follows.Sum Theorem. Let f" X--.Y be the sum of cellular maps fl" X1 YI and f" X-Y., where X= XI U X2, and Y Y [3 Y. are finite cell complexes. Suppose tha$ Xo-X X is 1-connected, and that f , f and fo-f Xo (= f Xo) are homotopy equivalences. Then f is a homotopy equivalence andRecently, Hosokawa [1] has generalized this theorem to the case of X0 being non-simply connected. Indeed, for this general case he obtained the following equality r(f) ],r(f) + ]=.r( f =) ]o. r( f o) where ]0." Wh (zrY0)-Wh (Y). His proof is accomplished by a geometric idea.The purpose of this paper is to extend the definition of Whitehead torsion and to prove the generalized sum theorem by making use of the torsions in the extended sense.For details on the notions of Whitehead group and torsion, we refer the reader to Whitehead [5] and Milnor [4].1. Let (K, L) be a pair consisting of a finite, connected CWcomplex K, and a subcomplex L which is a deformation retract of K.