By A. A. Ranicki

ISBN-10: 0521055210

ISBN-13: 9780521055215

ISBN-10: 0521420245

ISBN-13: 9780521420242

This ebook offers the definitive account of the functions of this algebra to the surgical procedure class of topological manifolds. The vital result's the id of a manifold constitution within the homotopy form of a Poincaré duality house with a neighborhood quadratic constitution within the chain homotopy kind of the common conceal. the variation among the homotopy sorts of manifolds and Poincaré duality areas is pointed out with the fibre of the algebraic L-theory meeting map, which passes from neighborhood to worldwide quadratic duality buildings on chain complexes. The algebraic L-theory meeting map is used to provide a in basic terms algebraic formula of the Novikov conjectures at the homotopy invariance of the better signatures; the other formula inevitably components via this one.

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**Additional info for Algebraic L-theory and Topological Manifolds**

**Example text**

Ln (Λ+ (R)) −−→ Ln (Λ(R)) = Ln (A (R)) = Ln+4∗ (R) 60 Algebraic L-theory and topological manifolds The symmetric L-groups of Λ+ (R) are the symmetric L-groups of R as originally deﬁned by Mishchenko [115] L∗ (Λ+ (R)) = L∗ (R) . { ∗ L (Λ+ (R))−−→L∗ (Λ(R)) It was shown in Ranicki [144] that the maps L∗ (Λ+ (R))−−→L∗ (Λ(R)) { are not isomorphisms in general, and also that are L∗ (Λ+ (R)) = L∗ (R) with L∗ (R) the original 4-periodic quadratic L-groups of Wall [180]. Call Ln (R) the connective symmetric L-groups of R, to distinguish them from the 4-periodic symmetric L-groups Ln+4∗ (R).

C and D) are highly B-connected.

Ln (A) −−→ Ln (A) −−→ N Ln (A) −−→ Ln−1 (A) −−→ . . 9) of normal complexes. 8 (i) and its relative version relating (symmetric, quadratic) Poincar´e triads and normal pairs. 1] for algebraic Poincar´e triads. In the case A = Aq (R) (q = h, p) for a ring with involution R write the normal L-groups as N L∗ (Aq (R)) = N L∗q (R) . 12 The hyperquadratic L-groups L∗q (R) (q = h, p) of Ranicki [146, p. 137] are the cobordism groups of (symmetric, quadratic) Poincar´e pairs over a ring with involution R which ﬁt into an exact sequence 1+T J ∂ .

### Algebraic L-theory and Topological Manifolds by A. A. Ranicki

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