CHAPTER 2 Outline of proof In the introduction, we gave a rough description of certain invariants of type signature and Euler characteristic for manifolds and Poincar´ e duality spaces. This led us to a map of the form (1.1). We wish to show that the map is highly connected. The main tools in the proof are (i) a controlled version of the Casson-Sullivan-Wall-Quinn-Ranicki (CSWQR) theorem in surgery theory (ii) more invariants of type signature and Euler characteristic for manifolds and Poincar´ e duality spaces in a controlled setting (iii) a simple downward induction, where the induction beginning relies on (i) while (ii) enables us to do the induction steps. Let S(M ×Ri c) be the controlled structure space of M ×Ri here we view M ×Ri as an open dense subset of the join M Si−1. An element of S(M × Ri c) should be thought of as a pair (N, f) where N is a manifold of dimension m + i, without boundary, and f : N M × Ri is a controlled homotopy equivalence [3]. There is also a controlled block structure space Scs(M × Ri c) where the decoration cs (controlled simple) indicates that we allow only structures with vanishing controlled Whitehead torsion. The homotopy type of Scs(M × Ri c) can be described by a formula which combines the CSWQR ideas with controlled algebra [3]: namely, (2.1) Scs(M × Ri c) fiber Ω∞+m+iL• cs % (M × Ri,ν c) −→ 8Z where L• cs (M × Ri,ν c) is the controlled quadratic L-theory (with vanishing con- trolled Whitehead torsion) of the control space (M Si−1,M × Ri). Taking i to the limit we have colim i≥0 Scs(M × Ri c) fiber colim i≥0 Ω∞+m+iLcs%(M × Ri,ν c) −→ 8Z where the colimits are formed using product with R in various shapes. Moreover, it is well-known [36] that the inclusions colim i≥0 S(M × Ri c) ←− colim i≥0 Scs(M × Ri c) −→ colim i≥0 Scs(M × Ri c) are homotopy equivalences. Therefore we have (2.2) colim i≥0 S(M × Ri c) fiber colim i≥0 Ω∞+m+iL• cs % (M × Ri,ν c) −→ 8Z and this is the starting point for our downward induction. 7
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