Algebraic L theory and topological manifolds by Ranicki

By Ranicki

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The cobordism groups Ln (B, β) (n ≥ 0) of ndimensional symmetric Poincar´e complexes (C, φ, γ, χ) in A (R) with a chain bundle map (f, b): (C, γ)−−→(B, β) fit into an exact sequence ∂ . . −−→ Ln (R) −−→ Ln (B, β) −−→ Qn (B, β) −−→ Ln−1 (R) −−→ . . with Ln (R) −−→ Ln (B, β) ; (C, ψ) −−→ ((C, (1 + T )ψ, 0, ψ), 0) , Ln (B, β) −−→ Qn (B, β) ; ((C, φ, γ, χ), (f, b)) −−→ (f, b)% (γ, χ) , ¯ β, ¯ χ) ¯ φ, ∂ : Qn (B, β) −−→ Ln−1 (R) ; (φ, χ) −−→ ∂(B, ¯ , ¯ β, ¯ χ) ¯ φ, where (B, ¯ the restriction of (B, φ, β, χ) to any finite subcomplex ¯ B ⊂ B supporting (φ, χ) ∈ Qn (B, β).

N (K) − The hyperquadratic signature of Ranicki [146, p. 619] defines a map from geometric to algebraic normal bordism σ ∗ : ΩN −→ Ln (Z[π]) ; X −−→ σ ∗ (X) = (C(X), φ, γ, χ) . n (K) − The signature maps fit together to define a map of exact sequences ... wΩ N n+1 (K) σ ˆ∗ ... wL n+1 u (Z[π]) σ∗ w L (Z[π]) wΩ σ∗ ∂ w L (Z[π]) n wΩ P n (K) n 1+T N n (K) u σ ˆ∗ w L (Z[π]) n J u w L (Z[π]) n w ... w ... The normal signature is the stable hyperquadratic signature −→ N Ln (Z[π]) = lim Ln+4k (Z[π]) .

Z[Z2 ] −−→ Z[Z2 ] −−→ Z[Z2 ] −−→ Z[Z2 ] to define for any finite chain complex C in A the Z-module chain complexes W % C = HomZ[Z2 ] (W, C ⊗A C) = HomZ[Z2 ] (W, HomA (T C, C)) W% C = W ⊗Z[Z2 ] (C ⊗A C) = W ⊗Z[Z2 ] HomA (T C, C) . 30 Algebraic L-theory and topological manifolds The boundary of the n-chain φ = {φs ∈ HomA (C r , Cn−r+s ) | r ∈ Z, s ≥ 0} ∈ (W % C)n ψ = {ψs ∈ HomA (C r , Cn−r−s ) | r ∈ Z, s ≥ 0} ∈ (W% C)n is the (n − 1)-chain with (∂φ)s = dC⊗A C (φs ) + (−)n+s−1 (φs−1 + (−)s T φs−1 ) (∂ψ)s = dC⊗A C (ψs ) + (−)n−s−1 (ψs+1 + (−)s+1 T ψs+1 ) for s ≥ 0, with φ−1 = 0.

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