Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Some product formulae for nonsimply connected surgery problems

Authors: R. J. Milgram and Andrew Ranicki
Journal: Trans. Amer. Math. Soc. 297 (1986), 383-413
MSC: Primary 57R67
MathSciNet review: 854074
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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})$.

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Article copyright: © Copyright 1986 American Mathematical Society