Celebratio Mathematica

E. K. Pedersen and A. Ranicki: “Projective surgery theory,” Topology 19 : 3 (1980), pp. 239–254. MR 579574 Zbl 0477.57020 article

A simple (resp. finite) \( n \)-dimensional Poincaré complex \( X \) (\( n\geq 5 \)) is simple homotopy (resp. homotopy) equivalent to a compact \( n \)-dimensional CAT (= DIFF, PL or TOP) manifold if and only if the Spivak normal fibration \( \nu_x \) admits a CAT reduction for which the corresponding normal map \( (f,b):M \to X \) from a compact CAT manifold \( M \) has Wall surgery obstruction \( \sigma_*^s(f, b) = 0 \) in \( L_n^s(\pi_1(X)) \) (resp. \( \sigma_*^h(f, b) = 0 \) in \( L_n^h(\pi_1(X)) \)). The surgery obstruction groups \( L_*^s(\pi) \) (resp. \( L_*^h(\pi) \)) of a group \( \pi \) are defined algebraically as Witt groups of quadratic structures on finitely based (resp. f.g. free) \( \mathbb{Z}[\pi] \)-modules, and geometrically as bordism groups of normal maps to simple (resp. finite) Poincaré complexes \( X \) with fundamental group \( \pi_1(X) = \pi \).

The object of this paper is to extend the above theory to finitely dominated Poincaré complexes, that i Poincaré complexes in the sense of Wall [1970], and to the Witt groups \( L_*^p(\pi) \) of quadratic structures on f.g. projective \( \mathbb{Z}[\pi] \)-modules introduced by Novikov [1970], the groups denoted by \( U_*(\mathbb{Z}[\pi]) \) in Ranicki [1973].

E. K. Pedersen, F. Quinn, and A. Ranicki: “Controlled surgery with trivial local fundamental groups,” pp. 421–426 in High-dimensional manifold topology (Trieste, Italy, 21 May–8 June 2001). Edited by F. T. Farrell and W. Lück. World Scientific (River Edge, NJ), 2003. MR 2048731 Zbl 1050.57025 ArXiv math/0111269 incollection

Erik Kjær Pedersen	Related
Wolfgang Lück	Related
Francis Thomas Farrell	Related
Frank Stringfellow Quinn, III	Related

E. K. Pedersen and A. Ranicki: “Mini-workshop: The Hauptvermutung for high-dimensional manifolds,” Oberwolfach Rep. 3 : 3 (2006), pp. 2195–2225. Workshop held in Oberwolfach, Germany, 13–19 August 2006. Zbl 1109.57300 article

I. Hambleton, E. K. Pedersen, A. Ranicki, and H. Reich: “Manifold perspectives,” Oberwolfach Rep. 6 : 2 (2009), pp. 1487–1546. Workshop held in Oberwolfach, Germany, 24–30 May 2009. MR 2648932 Zbl 1177.57003 article

Ian Hambleton	Related
Erik Kjær Pedersen	Related
Holger Reich	Related

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Andrew A. Ranicki

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