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Complements of codimension-two submanifolds. III. Cobordism theory

Justin Smith

1981
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Pacific Journal of Mathematics
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Introduction* This paper will study the relationship between the middle-dimensional complementary homology modules of a codimension-two imbedding of compact manifolds and its cobordism class. We will also obtain general results on the cobordism classification of codimension-two imbeddings that extend those of Kervaire ([14]) and Levine ([15]) on knots, Cappell and Shaneson (see [7] ) on local knots, parametrized knots and knotted lens spaces as well as results of Ocken (in [19] ) and Stoltzfus
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... 9] ) and Stoltzfus (in [37]). We also provide an algebraic formulation of general results of Cappell and Shaneson (in [7] and [8]) involving Poincare imbeddings and the codimensiontwo splitting problem. A major tool used in this paper is a type of surgery theory (developed in Chapter I of this paper and called dual surgery theory) that is dual to the homology surgery theory of Cappell and Shaneson in Chapter I of [7] in the following sense: whereas homology surgery theory measures the obstruction of a degree-1 normal map being normally cobordant to a simple homology equivalence, dual surgery theory starts with such a map and measures the obstruction to its being homology s-cobordant to a simple homotopy equivalence. This paper shows that dual surgery theory provides an algebraic formulation of the problem that solved in Theorem 3.3 of [7] geometrically. This theory is applied to show that, for many classes of codimensiontwo imbeddings-the set of cobordism classes of imbeddings has a natural group structure-in fact it is shown to be canonically isomorphic to a certain subgroup of a dual surgery obstruction group. The formulation of dual surgery theory in the present paper makes use of Ranicki's algebraic theory of surgery (see [25], [26], and [27]) so that we obtain an "instant dual surgery obstruction" whose computation doesn't require preliminary surgeries below the middle dimension. In many interesting cases the dual surgery obstruction is shown to be expressible as a linking form on a torsion module that directly generalizes the Blanchfield pairing (see [4] and [17]) in knot theory. This is applied in Chapter II of this paper to describe the manner in which the cobordism theory and the complementary homology of codimension-two imbeddings interact. This paper studies condimension-two imbeddings of compact

doi:10.2140/pjm.1981.94.423
fatcat:d3oq24dhjbcylksgzqnbkxdc5m