1.2. STATEMENT OF RESULTS 3 1.2. STATEMEN T O F RESULT S For a compact Hilbert cube manifold X the Whitehead space of X is a simplicial set yVh(X) whose homotopy groups are the higher Whitehead groups of any finite CW-complex homotopy equivalent to X. For example, 7roWh(X) = Wh(7Ti(X)), the classical algebraicly defined Whitehead group of the group 7Ti(X), and CtWh(X) is homotopy equivalent to the space of pseudoisotopies on X (see [27], [29]). For an arbitrary Hilbert cube manifold and a proper map p : X B the controlled Whitehead space is a simplicial set Wh(p : X B) whose homotopy groups are the domain of higher Whitehead torsion with control in B. The first result is a homotopy splitting of the controlled Whitehead group over a circle. THEOREM 1.2.1. If X is a compact Hilbert cube manifold and X x S1 S1 is projection, then there is a homotopy equivalence WhiXxS1 -+ S1) - Wh(X) x n^WhiX) with £l~lWh(X) a delooping ofWh(X), that is, a (non-connected) simplicial set whose loop space is homotopy equivalent to Wh(X), QQ~1)/Vh(X) ~ Wh(X). We will obtain this splitting as follows. From Hughes, Taylor and Williams [32] there is a homotopy equivalence * : WhiXxS1 - S1) - Map(S\ Wh(XxR - R)) (see Section 5.7). Evaluation at the basepoint of S1 yields a fibration QWh(X xR-R) i Map(S'1,V/i(XxR-+R)) - ^ Wh(XxR - R). In Chapter 5 we will define the unwrapping (or infinite transfer) map u : Wh(XxSx - S1) - Wh(XxR -* R) and the wrapping up map w : Wh(XxR -* R) - Wh(XxSl -+ S1) such that 1. uw ~ 1WM*XR-+R), and 2. EV ~ u. In fact, all of the Whitehead spaces considered here carry the structure of abelian monoid-like simplicial sets (see Section 5.2) which induce abelian group structures on TTQ of these spaces and the usual group structures on higher homotopy groups. The natural simplicial group structures on QWh(XxR -* R) and Map(S\ Wh(XxR - R)) induce abelian group structures on TTQ and the simplicial maps ^, /, E, u, and w induce group homomorphisms on homotopy groups including 7To. If i : QWh(XxR - R) - W^XxS1 - S1) is any simplicial map such that ty o i ~ J, then i* = ( ^ ) - 1 / * : 7TkQWh(XxR -+ R) - TTkWhiXxS1 -* S1) is a group homomorphism for each k 0. Thus, 0 - 7rkttWh(XxR - R) - ^ TTkWhiXxS1 -+ S1) - % nkWh{XxR -+ R) - 0
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