We develop an epsilon-controlled algebraic L–theory, extending our earlier work on epsilon-controlled algebraic K –theory. The controlled L–theory is very close to being a generalized homology theory; we study analogues of the homology exact sequence of a pair, excision properties, and the Mayer–Vietoris exact sequence. As an application we give a controlled L–theory proof of the classic theorem of Novikov on the topological invariance of the rational Pontrjagin classes. 57R67; 18F25 1

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It is well-known that an n-dimensional Poincaré complex X n, n ≥ 5, has the homotopy type of a compact topological n–manifold if the total surgery obstruction s(Xn) vanishes. The present paper discusses recent attempts to prove analogous result in dimension 4. We begin by reviewing the necessary algebraic and controlled surgery theory. Next, we discuss the key idea of Quinn’s approach. Finally, we present some cases of special fundamental groups, due to the authors and to Yamasaki.

We develop an epsilon-controlled algebraic L–theory, extending our earlier work on epsilon-controlled algebraic K –theory. The controlled L–theory is very close to being a generalized homology theory; we study analogues of the homology exact sequence of a pair, excision properties, and the Mayer–Vietoris exact sequence. As an application we give a controlled L–theory proof of the classic theorem of Novikov on the topological invariance of the rational Pontrjagin classes. 57R67; 18F25 1

Abstract. This article sketches a modification of the constructions of ANR homology manifolds pioneered by Bryant, Ferry, Mio and Weinberger. The original constructions were limited to dimensions 6 and greater while the version described here seems to work in dimension 4. If so it is not only the lowest expected dimension for exotic homology manifolds, but also gives new results for manifolds. In particular the 4-dimensional surgery conjecture for arbitrary fundamental groups is a consequence. This is a sketch for experts and is missing a lot of detail. Consequently it should be considered a “conjecture with hints ” rather than a theorem. This article describes a modification of the constructions of ANR homology manifolds pioneered by Bryant, Ferry, Mio and Weinberger [BFMW1, BFMW2]. The original constructions were limited to dimensions 6 and greater. Ferry and Johnston have announced an extension to dimension 5. The version described here uses a different approach to the construction of “controlled Poincaré spaces ” that seems to work in dimension 4. This is robust in that it does not depend on special low-dimensional tricks, but rather the approach has a different calculation of the