CHAPTER 1 Introduction and Statement of Results 1.1. INTRODUCTION The Fundamental Theorem of Algebraic K-Theory relates the K-theory of the Laurent polynomial extension R[t, £_1] of a ring R to the K-theory of the ring (see [2]). For the functor i^i, the result is due to Bass, Heller and Swan [3]: Ki{R[t, r1}) ^ K^R) 0 K0(R) 0 NU(iJ) 0 Nil(i2). When R is the integral group ring ZTX\(X) of the fundamental group of a space X, then i?[£,£_1] = ZTTI(X X S1) and the Bass-Heller-Swan decomposition gives a calculation of the Whitehead group of X x S1: Wh(7n(X x S1)) ** Wh(7n(X)) 0Ko(Z7r1(X)) 0 Nil(Z7n(X)) 0Nil(Z7n(X)). For recent expositions of these results as well as for applications to topology, see Ranicki [42] and Rosenberg [43]. Whitehead groups measure the difference between homotopy equivalences and simple homotopy equivalences. That is, every homotopy equivalence / : Y X between finite CW complexes determines an element r(f) Wh(7Ti(X)) which vanishes if and only if / is simple (see [13]). For a finite CW complex X all homotopy equivalences to X from other finite CW complexes can be organized into a moduli space, the Whitehead space W/i(X), which is the domain of higher simple homotopy theory (see [23], [24], [27], [29]). One of the new results of this paper is a geometrically defined Nil space based on the earlier work of Prassidis [41]. Our main result is a moduli space version of the Bass-Heller-Swan decomposi- tion where the moduli spaces involved in the decomposition are Whitehead spaces and Nil spaces. More specifically, we prove THEOREM (Main Theorem). If X is a finite CW-complex, then there is a ho- motopy equivalence Wh(XxSl) ~ Wh(X) x Sl^WhiX) x Afil(X) x Afil(X). The classical Bass-Heller-Swan decomposition follows from this homotopy equiv- alence by considering the set of path components of the spaces involved. That is, there are isomorphisms TToWhiXxS1) = Wh(7Ti(X X S1)), 7r0W/l(X) = Wh(7T!(X)), 7r0n-lWh(X) = K0(Z7n(X)), and 7r0A/T/(X) = NTI(Z7TI(X)). However, the present paper does not represent a new proof of the Bass-Heller-Swan decomposition. The Fundamental Theorem of Algebraic K-Theory was extended to higher K- theory by Quillen (see [22], [45]). More recently, Klein, Vogell, Waldhausen and Williams have established a Fundamental Theorem in the A-theory of spaces [38]. l
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