By A. A. Ranicki

This booklet offers the definitive account of the functions of this algebra to the surgical procedure class of topological manifolds. The primary result's the identity 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 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 worldwide quadratic duality buildings on chain complexes. The algebraic L-theory meeting map is used to offer a simply algebraic formula of the Novikov conjectures at the homotopy invariance of the better signatures; the other formula unavoidably elements via this one.

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The integral element of P+XY becomes when expressed in differential quotients of PX : àœK : -yai V H l % Σ =ai ~VH dux ÔXa VH 9ft 'ρ+Ι du, du. da* 9i àxa àx« ax* dwp+i àup+\ du, +\> Ψ p+l ) The Dutch original contains a few errors (see Erratum at the end of Verslagen 31 Juni 1906), which have been rectified in this translation. g. X 1 2 3 . /> dwx . . VH d/fy+2 dux d^j 9^ dwj dux dXi123... p + du1 . . dup+\ àXp+2 d^-(_2 dup+\ + θ^ d^j dup+\ dup+\ -f . . (n — p terms). If we add to these the following terms with the value 0: ΟΛ^ dœx dœx dux dux dux ÖX123...

The orthogonal transformations of 4-space that E. Jahnke had found in Caspary 1883, and reconsidered, look in quaternion form as x-^ax. Jahnke mistook them for rotations. Since conjugation interchanges left and right multiplications, every rotation of 4-space can be obtained as a product of two transformations of this kind (instead of a left and a right multiplication). This is the theorem Jahnke claimed against Brouwer on the strength of Jahnke, 1896, 1897, 1902, though in fact it is at least Caspary's result, if it is not older.

Is thus, entirely filled with these representing lines which are in (1,1 ^correspondence with the represented systems of planes. We shall call that Sz i. 0 W regarded as a complex of the rays representing the equiangular to the right systems of planes, "the representing Sz to the right of S4" or shorter "the Sr ofSA". In the same way we form the "Si of S4". Each pair of planes in S4 is then unequivocally determined by its represen-tants in Sr and St and reversely the pair of planes determines unequivocally its représentants.