By A. A. Ranicki

This publication offers the definitive account of the functions of this algebra to the surgical procedure class of topological manifolds. The relevant result's the id of a manifold constitution within the homotopy kind of a Poincaré duality house with an area quadratic constitution within the chain homotopy form of the common disguise. the adaptation among the homotopy different types of manifolds and Poincaré duality areas is pointed out with the fibre of the algebraic L-theory meeting map, which passes from neighborhood to international quadratic duality buildings on chain complexes. The algebraic L-theory meeting map is used to offer a basically algebraic formula of the Novikov conjectures at the homotopy invariance of the better signatures; the other formula unavoidably components via this one.

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The meridian circles {x} × ∂D2 in such attached S 1 × D2 ’s are not isotopic in ∂N to circle fibers of N , otherwise A would be compressible in M (recall that A is vertical in N ) . 38 Torus Decomposition Ch. 3 Thus the circle fibers wind around the attached S 1 × D2 ’s a non-zero number of times in the S 1 direction. Hence the circle bundle structure on N extends to model Seifert fiberings of these S 1 × D2 ’s, and so M is Seifert fibered. 3: Only the uniqueness statement remains to be proved.

Therefore α cuts off a disk D from S , meeting ∂M in an arc γ , say. The existence of D implies that the two ends of γ lie on the same side of the vertical arc β = D ∩ ∂M in ∂M . But this is impossible since γ is horizontal and therefore proceeds monotonically through the circle fibers in this component of ∂M . ” Now we may assume the components of S ∩ A are either vertical circles or horizontal arcs. Thus if we let S1 = S ∩ M1 , it follows that ∂S1 consists entirely of horizontal or vertical circles in the torus ∂M1 .

If S is ∂ compressible, it must have ∂ -parallel annulus components. These can be eliminated by pushing them across T0 , decreasing the number of circles of S ∩ T0 . So we may assume S is incompressible and ∂ -incompressible in T × I . , fibers of Mϕ , or annuli whose boundary circles have the same slope in both ends T × {0} and T × {1}. In the latter case, ϕ must preserve this slope in order for S to glue together to form the original surface S . This means ϕ has an eigenvector in Z2 , so is conjugate to ± n1 01 .