Some product formulae for nonsimply connected surgery problems

Abstract:

For an $n$-dimensional normal map $f :{M^n} \to {N^n}$ with finite fundamental group ${\pi _1}(N) = \pi$ and PL $1$ torsion kernel $Z[\pi ]$-modules ${K_{\ast }}(M)$ the surgery obstruction ${\sigma _{\ast }}(f) \in L_n^h(Z[\pi ])$ is expressed in terms of the projective classes $[{K_{\ast }}(M)] \in {\tilde K_0}(Z[\pi ])$, assuming ${K_i}(M) = 0$ if $n = 2i$. This expression is used to evaluate in certain cases the surgery obstruction ${\sigma _ {\ast } }(g) \in L_{m + n}^h(Z[{\pi _1} \times \pi ])$ of the $(m + n)$-dimensional normal map $g = 1 \times f:{M_1} \times M \to {M_1} \times N$ defined by product with an $m$-dimensional manifold ${M_1}$, where ${\pi _1} = {\pi _1}({M_1})$.