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Algebraic Topology

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Ferry, Andrew Ranicki, and Jonathan Rosenberg and Ferry [ChF] and Quinn [Q3] to obtain a foliated control theorem. A full discussion of their results would occupy too much space for this survey, but we include a few precise statements to give the flavor of their work. Definition. A path γ in the unit sphere bundle SM is said to be (β, )controlled if there is a second path φ in SM such that (1) The image of φ is contained in an arc of length β inside a flow line of the geodesic flow. (2) d(γ(t), φ(t)) < for all t ∈ [0, 1].

The twisted products M ×Γ M and N ×Γ N contain copies of the tangent microbundle so, by Kister’s theorem [Kis], they are isomorphic to the tangent microbundles of M and N and unstable tangentiality follows. For details, see [FW1]. We shall now show that the above assembly map is split. As we have seen in the section on surgery spectra (Quinn) the surgery exact sequence is the long exact homotopy sequence of a (co)fibration of spectra: A H • (M ; L• ) −→ L• (Z[π1 (M )]) → S• (M ) . A spectrum-level version of the vanishing result above shows that the map S• (M ) → ΣH • (M ; L• ) is nullhomotopic.

The symmetric Poincar´e complex theory of Mishchenko [Mis2] was extended by Ranicki ([Ran4], [Ran5], [Ran 7]) to a comprehensive theory of chain complexes with Poincar´e duality, including the quadratic structures required for the Wall surgery obstruction. The surgery obstruction of an n-dimensional normal map (f, b) : M → X was 15 See [Ni1] and [CaW1] for a correction. 28 Steven C. Ferry, Andrew Ranicki, and Jonathan Rosenberg expressed as a quadratic Poincar´e cobordism class σ∗ (f, b) ∈ Ln (Z[π1 (X)]) of the quadratic Poincar´e duality on the algebraic mapping cone C(f !