Celebratio Mathematica

Algebraic and geometric topology (New Brunswick, NJ, 6–13 July 1983). Edited by A. Ranicki, N. Levitt, and F. Quinn. Lecture Notes In Mathematics 1126. Springer (Cham, Switzerland), 1985. MR 802782 Zbl 0553.00007 book

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A. Ranicki: “The algebraic theory of torsion, I: Foundations,” pp. 199–237 in Algebraic and geometric topology (New Brunswick, NJ, 6–13 July 1983). Edited by A. Ranicki, N. Levitt, and F. Quinn. Lecture Notes In Mathematics 1126. Springer (Cham, Switzerland), 1985. Part II was published in \( K \)-Theory 1:2 (1987). MR 802792 Zbl 0567.57013 incollection

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Norman J. Levitt	Related

N. Levitt and A. Ranicki: “Intrinsic transversality structures,” Pac. J. Math. 129 : 1 (1987), pp. 85–144. MR 901259 Zbl 0661.57005 article

This paper introduces the notion of an intrinsic transversality structure on a Poincaré duality space \( X^n \). Such a space has an intrinsic transversality structure if the embedding of \( X^n \) into its regular neighborhood \( W^{n+k} \) in Euclidean space can be made “Poincaré transverse” to a triangulation of \( W^{n+k} \). This notion relates to earlier work concerning transversality structures on spherical fibrations, which are known to be essentially equivalent to topological bundle reductions. Thus, for \( n\geq 5 \), a Poincaré duality space \( X^n \) with a transversality structure on its Spivak normal fibration (i.e., with an “extrinsic” transversality structure) is, up to a surgery obstruction, realizable as a topological manifold. An intrinsic transversality structure, however, not only guarantees the existence of an extrinsic transversality structure but gives rise as well to a canonical solution of the resulting surgery problem. Thus, as our main result, an equivalence is obtained between intrinsic transversality structures and topological manifold structures. This yields a number of corollaries, among which the most important is a “local formula for the total surgery obstruction” which assembles this obstruction to the existence of a manifold structure on \( X^n \) from the local singularities of a realization of the simple homotopy type of \( X^n \) as a (non-manifold) simplicial complex.

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Andrew A. Ranicki

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