SURGERY TRANSFER by W.Luck and A.Ranicki Introduction Given a Hurewicz fibration F ,E P ,B with fibre an t n-dimensional geometric Poincare complex F we construct algebraic transfer maps in the Wall surgery obstruction groups ! p" : Lm(Z[~I(B) ] ) ~ Lm+n(Z[~l(E) ] ) (m~>0) and prove that they agree with the geometrically defined transfer maps. In subsequent work we shall obtain specific computations of the composites p p! , p!p with p! :Lm(Z[~I(E) ]) ~Lm(Z[~I(B) ]) the change of rings maps, and some vanishing results. ! The construction of p" is most straightforward in the case when F is finite, with L. the free L-groups ' L . In ~9 we shall extend the definition of p" to finitely dominated F and the projective L-groups L~, as S well as to simple F and the simple L-groups L,, and also to the intermediate cases. There are two main sources of applications of the surgery transfer. The equivariant surgery obstruction groups of Browder and Quinn [ I ] were defined in terms of the geometric surgery transfer maps of the normal sphere bundles of the fixed point sets. An algebraic version will necessarily involve the algebraic surgery transfer maps. (In this connection see LHck and Madsen [8].) The recent work of Hambleton, Milgram, Taylor and Williams [3] on the evaluation of the surgery obstructions of normal maps of closed manifolds with finite fundamental group depends on the factorization of the assembly map by twisted product formulae which are closely related to the algebraic surgery transfer. Our construction of the quadratic L-theory transfer maps is by a combination of the algebraic