TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 297. Number 2, October 1986 ■ SOME PRODUCT FORMULAE FOR NONSIMPLY CONNECTEDSURGERYPROBLEMS R. J. MILGRAM AND ANDREW RANICKI Abstract. For an n-dimensional normal map /: M" -» N" with finite fundamental group w,(Ar) = v7 and PL 1 torsion kernel Z[7r]-modules Kt(M) the surgery obstruction »,(/)6tJ(Z[»]) is expressed in terms of the projective classes [Kt(M)] s K0(Z[tr]), assuming K,(M) — 0 if n = 2i. This expression is used to evaluate in certain cases the surgery obstruction o,(g) s L'm + „(Z[iTx X tt]) of the (m + n )-dimensional normal map g = 1 X /: M, X M -» Mx X N defined by product with an m-dimensional manifold A/,, where w, = irx(Mx). A key problem in surgery theory is to understand how to calculate the surgery obstructions for surgery problems (*) /: M" -■> N" where / is a degree 1 normal map and M", N" are closed n-dimensional manifolds. C. T. C. Wall [20] has pointed out that the problem (*) determines an element «(/) e fl„K(*> x G/TOP> bmn) x (Pt-)) and there is a map e: Q*(B„ilN) X G/TOP, BWi{N) X {pt.}) - L*{vx(N)) so that the surgery obstruction ct*(/) is e(a(f)). Wall also pointed out that if trx(N) is finite, then a(f) is already determined by restriction to the 2-Sylow subgroup of irx, and the groups L+(tt) have been extensively studied when tt is a finite 2-group. (See e.g. Pardon [12], Carlsson- Milgram [3, 5], Hambleton-Milgram [9], Wall [21] and Bak-Kolster [1].) Indeed the projective L-groups L+(tt) are completely known, and the groups Lhm(ir) and L%(m) are effectively computable in tems of certain additional facts about K0(Z[tr]) and Wh(tr). So further progress depends on studying the map e. For the projective L groups L%(tt) with it a finite 2-group, this was done by L. Taylor-B. Williams [18] and independently by I. Hambleton [8]. But for the more basic case of L+(it) our information is much more limited. There are some key examples (Cappell-Shaneson [2], Taylor-Williams [18]) which show that this problem is much harder, but general information is hard to come by. Received by the editors June 3, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 57R67. ©1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page 383 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use