Celebratio Mathematica

[1] A. A. Ranicki: “Algebraic \( L \)-theory, III: Twisted Laurent extensions,” pp. 412–463 in Algebraic \( K \)-theory III (Seattle, WA, 28 August–8 September 1972). Edited by H. Bass. Lecture Notes In Mathematics 343. Springer (Berlin), 1973. Parts I and II were published in Proc. Lond. Math. Soc. 27 (1973). Part IV was published in Comment. Math. Helv. 49 (1974). MR 414663 Zbl 0278.18008 incollection

[2] A. A. Ranicki: “Algebraic \( L \)-theory, I: Foundations,” Proc. Lond. Math. Soc. (3) 27 : 1 (July 1973), pp. 101–125. Part III was published in Algebraic \( K \)-theory III (1973). Part IV was published in Comment. Math. Helv. 49 (1974). MR 414661 Zbl 0269.18009 article

Where algebraic \( K \)-theory deals with modules, \( L \)-theory considers modules with quadratic forms. The \( L \)-groups are of interest to topologists because they are the surgery obstruction groups, as described by Wall [1970a]. Although isomorphism groups of quadratic forms have been studied before, by Witt and others, the topological applications require new algebraic methods (cf. [Wall 1970b, 1973]).

The \( L \)-groups \( L_n(\pi) \) were obtained in [Wall 1970a] as the solutions to a specific topological problem, leaving open the question of the algebraic framework best suited for an ‘\( L \)-theory’. In [1] Novikov used algebraic \( K \)-theory and the formalism of hamiltonian physics to provide such machinery, though not as coherently as might be desired.

In this paper we shall use the ideas of [Novikov 1970] to give the foundations of \( L \)-theory over a ring with involution, \( A \). We shall define \( L \)-groups \( U_n(A) \), \( V_n(A) \), and \( W_n(A) \) as stable isomorphism groups of ‘\( \pm \) forms’ and ‘\( \pm \) formations’ involving finitely generated (f.g.) projective, stably f.g. free, and based \( A \)-modules respectively, depending on \( n \pmod 4 \) only.

[3] A. A. Ranicki: “Algebraic \( L \)-theory, II: Laurent extensions,” Proc. Lond. Math. Soc. (3) 27 : 1 (July 1973), pp. 126–158. Part III was published in Algebraic \( K \)-theory III (1973). Part IV was published in Comment. Math. Helv. 49 (1974). MR 414662 article

[4] A. A. Ranicki: Algebraic \( L \)-theory. Ph.D. thesis, University of Cambridge, 1973. Advised by A. Casson and J. F. Adams. This appears to have been published in four parts. Parts I and II were published in Proc. Lond. Math. Soc. 27 (1973); Part III was published in Algebraic \( K \)-theory III (1973); Part IV was published in Comment. Math. Helv. 49 (1974). phdthesis

Andrew J. Casson	Related
John Frank Adams	Related

[5] A. A. Ranicki: “Algebraic \( L \)-theory, IV: Polynomial extension rings,” Comment. Math. Helv. 49 (1974), pp. 137–167. Parts I and II were published in Proc. Lond. Math. Soc. 27 (1973). Part III was published in Algebraic \( K \)-theory III (1973). MR 414664 Zbl 0293.18021 article

[6] A. Ranicki and D. Sullivan: “A semi-local combinatorial formula for the signature of a \( 4k \) manifold,” J. Diff. Geom. 11 : 1 (1976), pp. 23–29. MR 423366 Zbl 0328.57003 article

[7] A. Ranicki: “On the algebraic \( L \)-theory of semisimple rings,” J. Algebra 50 : 1 (January 1978), pp. 242–243. MR 479891 Zbl 0372.18005 article

[8] A. Ranicki: “Localization in quadratic \( L \)-theory,” pp. 102–157 in Algebraic topology (Waterloo, ON, 15 May–19 June 1978). Edited by P. Hoffman and V. Snaith. Lecture Notes In Mathematics 741. Springer (Berlin), 1979. MR 557165 Zbl 0418.18010 incollection

Peter Hoffman	Related
Victor Percy Snaith	Related

[9] A. Ranicki: “The surgery obstruction of a disjoint union,” J. Lond. Math. Soc., II. Ser. 20 : 3 (1979), pp. 559–566. MR 561148 Zbl 0421.57010 article

The surgery obstruction \[ \sigma_*(f,b) \in L_n(\pi_1(X)) \] of an \( n \)-dimensional degree 1 normal map \[ (f,b): M \to X \] (in the sense of Browder [1972] and Wall [1970]) was formulated in [Ranicki 1980] as the quadratic Poincaré cobordism class of a pair \( (C,\psi) \) consisting of an \( n \)-dimensional \( \mathbb{Z}[\pi_1(X)] \)-module chain complex \( C \) and a chain level quadratic structure \( \psi \) inducing Poincaré duality \[ H^{n-*}(C) \stackrel{\sim}{\to} H_*(C) .\] If the manifold \[ M = \bigcup_{i=1}^N M_i \] is the disjoint union of manifolds \( M_i \) it is natural to seek an expression for the surgery obstruction of \( (f,b) \) in terms of quadratic structures defined by the restrictions \[ (f_i,b_i) = (f,b)|:M_i\to X ,\] which are normal maps of degree \( d_i \) with \[ f_{i^*}[M_i] = d_i[X] \in H_n(X), \] \( d_i\in\mathbb{Z} \), \( \sum_{i=1}^N d_i = 1 \). The algebraic theory of surgery of [Ranicki 1980] is here used to provide such an expression, describing the pair \( (C,\psi) \) in terms of similar pairs \( (C_i,\psi_i) \) which are associated to \( (f_i,b_i) \). For the sake of simplicity we shall be working with the oriented case — the unoriented case is exactly the same, but with more complicated terminology.

[10] A. Ranicki: “The total surgery obstruction,” pp. 275–316 in Algebraic topology (Aarhus, Denmark, 7–12 August 1978). Edited by J. L. Dupont and I. H. Madsen. Lecture Notes In Mathematics 763. Springer (Berlin), 1979. MR 561227 Zbl 0428.57012 incollection

Johan L. Dupont	Related
Ib Henning Madsen	Related

[11] A. Ranicki: “The algebraic theory of surgery, I: Foundations,” Proc. Lond. Math. Soc. (3) 40 : 1 (January 1980), pp. 87–192. MR 560997 Zbl 0471.57010 article

[12] A. Ranicki: “The algebraic theory of surgery, II: Applications to topology,” Proc. Lond. Math. Soc. (3) 40 : 2 (1980), pp. 193–283. MR 566491 Zbl 0471.57011 article

[13] 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].

[14] A. Ranicki: Exact sequences in the algebraic theory of surgery. Mathematical Notes 26. Princeton University Press, 1981. MR 620795 Zbl 0471.57012 book

[15] A. A. Ranicki: “Book review: A. Bak, ‘\( K \)-theory of forms’,” Bull. Amer. Math. Soc. (N.S.) 7 : 1 (1982), pp. 279–280. MR 1567369 article

[16] H. J. Munkholm and A. A. Ranicki: “The projective class group transfer induced by an \( S^1 \)-bundle,” pp. 461–484 in Current trends in algebraic topology (London, ON, 29 June–10 July 1981), part 2. Edited by R. M. Kane, S. O. Kochman, P. S. Selick, and V. P. Snaith. CMS Conference Proceedings 2. American Mathematical Society (Providence, RI), 1982. MR 686160 Zbl 0564.57018 incollection

Stanley Oscar Kochman	Related
Hans Jørgen Munkholm	Related
Paul Sydney Selick	Related
Victor Percy Snaith	Related
Richard Michael Kane	Related

[17] 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

Frank Stringfellow Quinn, III	Related
Norman J. Levitt	Related

[18] 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

Frank Stringfellow Quinn, III	Related
Norman J. Levitt	Related

[19] A. Ranicki: “The algebraic theory of finiteness obstruction,” Math. Scand. 57 : 1 (1985), pp. 105–126. MR 815431 Zbl 0589.57018 article

The finiteness obstruction \[ [X]\in \tilde{K}_0(\mathbf{Z}[\pi_1(X)]) \] of Wall [1965, 1966] is an algebraic \( K \)-theory invariant of a finitely dominated \( \mathrm{CW} \) complex \( X \) such that \( [X] = 0 \) if and only if \( X \) is homotopy equivalent to a finite \( \mathrm{CW} \) complex. We develop here an algebraic theory of finiteness obstruction for chain complexes in an additive category. The theory helps to clarify the passage \( X \to [X] \) from topology to algebra. Such a clarification may be of interest in its own right, but in any case the new theory is necessary for some recent generalizations of the original obstruction theory to more complicated topological finiteness problems.

[20] A. Ranicki: Algebraic and geometric splittings of the \( K \)- and \( L \)-groups of polynomial extensions. Preprint 34, Sonderforschungsbereich für Geometrie und Analysis, University of Göttingen, 1985. A version of this was published in Transformation groups (1986). Zbl 0585.57019 techreport

[21] R. J. Milgram and A. Ranicki: “Some product formulae for nonsimply connected surgery problems,” Trans. Am. Math. Soc. 297 : 2 (October 1986), pp. 383–413. MR 854074 Zbl 0604.57023 article

[22] A. Ranicki: “Algebraic and geometric splittings of the \( K \)- and \( L \)-groups of polynomial extensions,” pp. 321–363 in Transformation groups (Poznań, Poland 5–9 July 1985). Edited by S. Jackowski and K. Pawałowski. Lecture Notes In Mathematics 1217. Springer (Berlin), 1986. A preprint appeared as Mathematica Göttingensis no. 34 (1985). MR 874187 Zbl 0615.57017 incollection

Krzysztof Marian Pawałowski	Related
Stefan Jackowski	Related

[23] W. Lück and A. Ranicki: Chain homotopy projections. Preprint 73, Sonderforschungsbereich für Geometrie und Analysis, University of Göttingen, 1986. A version of this was published in J. Algebra 120:2 (1989). Zbl 0616.57011 techreport

[24] A. Ranicki: “The \( L \)-theory of twisted quadratic extensions,” Can. J. Math. 39 : 2 (1987), pp. 345–364. MR 899842 Zbl 0635.57017 article

For surgery on codimension 1 submanifolds with nontrivial normal bundle the theory of Wall [1970a, § 12C] has obstruction groups \[ LN_*(\pi^{\prime}{\to}\pi) ,\] with \( \pi \) a group and \( \pi^{\prime} \) a subgroup of index 2, such that there is defined an exact sequence involving the ordinary \( L \)-groups of rings with involution \begin{multline*} \dots \to LN_n(\pi^{\prime}{\to}\pi) \to L_n(\mathbf{Z}[\pi]) \\ \to L_{n+1}(\mathbf{Z}[\pi^{\prime}] {\to} \mathbf{Z}[\pi]^w) \to LN_{n-1}(\pi^{\prime}{\to}\pi) \to \dots \end{multline*} with the superscript \( w \) signifying a different involution on \( \mathbf{Z}[\pi] \). Geometry was used in [13] to identify \[ LN_n(\pi^{\prime}\to\pi) = L_n(\mathbf{Z}[\pi^{\prime}],\alpha,u), \] with \( (\alpha,u) \) an antistructure on \( \mathbf{Z}[\pi^{\prime}] \) in the sense of Wall [1970b]. The main result of this paper is a purely algebraic version of this identification, for any twisted quadratic extension of a ring with antistructure.

[25] J. F. Davis and A. A. Ranicki: “Semi-invariants in surgery,” \( K \)-Theory 1 : 1 (1987), pp. 83–109. MR 899918 Zbl 0639.57015 article

[26] A. Ranicki: “The algebraic theory of torsion, II: Products,” \( K \)-Theory 1 : 2 (1987), pp. 115–170. Part I was published in Algebraic and geometric topology (1985). MR 899919 Zbl 0591.18007 article

[27] 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.

[28] I. Hambleton, A. Ranicki, and L. Taylor: “Round \( L \)-theory,” J. Pure Appl. Algebra 47 : 2 (August 1987), pp. 131–154. MR 906966 Zbl 0638.18003 article

Laurence Robert Taylor	Related
Ian Hambleton	Related

[29] A. Ranicki and M. Weiss: Chain complexes and assembly. Preprint 28, Sonderforschungsbereich für Geometrie und Analysis, University of Göttingen, 1987. A version of this was published in Math. Z. 204:2 (1990). Zbl 0616.55016 techreport

[30] K. Y. Lam, A. Ranicki, and L. Smith: “Jordan normal form projections,” Arch. Math. 50 : 2 (1988), pp. 113–117. MR 930110 Zbl 0636.15005 article

Let \( K \) be a field, and let \( T:V \to V \) be an endomorphism of a finite-dimensional vector space \( V \) over \( K \) such that \( K \) contains the roots \( \lambda_1 \), \( \lambda_2,\dots \), \( \lambda_m \) of the characteristic polynomial \begin{align*} \chi(t) &= \det(tI - T)\\ &= (t - \lambda_1)^{n_1}(t - \lambda_2)^{n_2}\dots (t - \lambda_m)^{n_m}. \end{align*} \( V \) is a direct sum of \( T \)-invariant subspaces \[ V_j = \ker(T - \lambda_j I)^{n_j} ,\] one for each eigenvalue \( \lambda_j \), such that \[ T - \lambda_j I: V_k \to V_k \] is nilpotent for \( j = k \) and an automorphism for \( j \neq k \). We obtain in this paper an explicit formula for the projection \[ p_j(T) = p_j(T)^2: V \to V \] onto the subspace \( V_j \), as a polynomial in \( T \).

Lawrence Smith	Related
Kee Yuen Lam	Related

[31] W. Lück and A. Ranicki: “Surgery transfer,” pp. 167–246 in Algebraic topology and transformation groups (Göttingen, Germany, 23–29 August 1987). Edited by T. tom Dieck. Lecture Notes In Mathematics 1361. Springer (Berlin), 1988. MR 979509 Zbl 0677.57012 incollection

Wolfgang Lück	Related
Tammo tom Dieck	Related

[32] A. Ranicki: Additive \( L \)-theory. Preprint 12, Sonderforschungsbereich für Geometrie und Analysis, University of Göttingen, 1988. A version of this was later published in K-Theory 3:2 (1989). Zbl 0646.18007 techreport

[33] A. Ranicki: “Additive \( L \)-theory,” \( K \)-Theory 3 : 2 (1989), pp. 163–195. A preprint appeared as Mathematica Göttingensis no. 12 (1988). MR 1029957 Zbl 0686.57017 article

[34] W. Lück and A. Ranicki: “Chain homotopy projections,” J. Algebra 120 : 2 (February 1989), pp. 361–391. A preprint appeared as Mathematica Göttingensis no. 73 (1986). MR 989903 Zbl 0671.18005 article

[35] R. J. Milgram and A. A. Ranicki: “The \( L \)-theory of Laurent extensions and genus 0 function fields,” J. Reine Angew. Math. 406 (1990), pp. 121–166. MR 1048238 Zbl 0692.13010 article

[36] A. Ranicki and M. Weiss: “Chain complexes and assembly,” Math. Z. 204 : 2 (1990), pp. 157–185. A preprint appeared as Mathematica Göttingensis no. 28 (1987). MR 1055984 Zbl 0669.55010 article

[37] W. Lück and A. Ranicki: “Surgery obstructions of fibre bundles,” J. Pure Appl. Algebra 81 : 2 (1992), pp. 139–189. MR 1176019 Zbl 0755.57013 article

In a previous paper we obtained an algebraic description of the transfer maps \[ p^*: L_n(\mathbb{Z}[\pi_1(B)]) \to L_{n+d}(\mathbb{Z}[\pi_1(E)]) \] induced in the Wall surgery obstruction groups by a fibration \[ F \to E \stackrel{p}{\longrightarrow} B \] with the fibre \( F \) a \( d \)-dimensional Poincaré complex. In this paper we define a \( \pi_1(B) \)-equivariant symmetric signature \[ \sigma^*(F,\omega) \in L^d(\pi_1(B),\mathbb{Z}) \] depending only on the fibre transport \[ \omega: \pi_1(B)\to [F,F] ,\] and prove that the composite \[ p_*\,p^*: L_n(\mathbb{Z}[\pi_1(B)]) \to L_{n+d}(\mathbb{Z}[\pi_1(B)]) \] is the evaluation \( \sigma^*(F,\omega)\,\otimes\,? \) of the product \[ \otimes: L^d(\pi_1(B),\mathbb{Z})\otimes L_n(\mathbb{Z}[\pi_1(B)]) \to L_{n+d}(\mathbb{Z}[\pi_1(B)]) .\] This is applied to prove vanishing results for the surgery transfer, such as \( p^* = 0 \) if \( F = G \) is a compact connected \( d \)-dimensional Lie group which is not a torus, and \[ F \to E \stackrel{p}{\longrightarrow} B \] is a \( G \)-principal bundle. An appendix relates this expression for \( p^*\,p_* \) to the twisted signature formula of Atiyah, Lusztig and Meyer.

[38] A. Ranicki: Lower \( K \)- and \( L \)-theory. London Mathematical Society Lecture Note Series 178. Cambridge University Press, 1992. MR 1208729 Zbl 0752.57002 book

[39] A. A. Ranicki: Algebraic \( L \)-theory and topological manifolds. Cambridge Tracts in Mathematics 102. Cambridge University Press, 1992. Reprinted in paperpack in 2008. MR 1211640 Zbl 0767.57002 book

[40] A. A. Ranicki and M. Yamasaki: “Symmetric and quadratic complexes with geometric control,” Proceedings of the Topology and Geometry Research Center 3 (1993), pp. 139–152. Dedicated to Professor Frank Raymond on his 60th birthday. article

Masayuki Yamasaki	Related
Frank Albert Raymond	Related

[41] A. Ranicki and M. Yamasaki: “Controlled \( K \)-theory,” Topology Appl. 61 : 1 (January 1995), pp. 1–59. MR 1311017 Zbl 0835.57013 article

[42] A. Ranicki: “Finite domination and Novikov rings,” Topology 34 : 3 (July 1995), pp. 619–632. MR 1341811 Zbl 0859.57024 article

[43] A. Ranicki: “Bordism of automorphisms of manifolds from the algebraic \( L \)-theory point of view,” pp. 314–327 in Prospects in topology: Proceedings of a conference in honor of William Browder (Princeton, NJ, March 1994). Edited by F. Quinn. Annals of Mathematics Studies 138. Princeton University Press, 1995. MR 1368666 Zbl 0933.57033 incollection

Frank Stringfellow Quinn, III	Related
William Browder	Related

[44] Novikov conjectures, index theorems and rigidity (Oberwolfach, Germany, 6–10 September 1993), vol. 1. Edited by S. C. Ferry, A. Ranicki, and J. Rosenberg. London Mathematical Society Lecture Note Series 226. Cambridge University Press, 1995. MR 1388294 Zbl 0829.00027 book

Steven Charles Ferry	Related
Jonathan Micah Rosenberg	Related

[45] S. C. Ferry, A. Ranicki, and J. Rosenberg: “A history and survey of the Novikov conjecture,” pp. 7–66 in Novikov conjectures, index theorems and rigidity (Oberwolfach, Germany, September 1993), vol. 1. Edited by S. C. Ferry, A. Ranicki, and J. Rosenberg. London Mathematical Society Lecture Note Series 226. Cambridge University Press, 1995. MR 1388295 Zbl 0954.57018 incollection

Steven Charles Ferry	Related
Jonathan Micah Rosenberg	Related

[46] A. Ranicki: “On the Novikov conjecture,” pp. 272–337 in Novikov conjectures, index theorems and rigidity (Oberwolfach, Germany, September 1993), vol. 1. Edited by S. C. Ferry, A. Ranicki, and J. Rosenberg. London Mathematical Society Lecture Note Series 226. Cambridge University Press, 1995. MR 1388304 Zbl 0954.57017 incollection

Steven Charles Ferry	Related
Jonathan Micah Rosenberg	Related

[47] Novikov conjectures, index theorems and rigidity (Oberwolfach, Germany, 6–10 September 1993), vol. 2. Edited by S. C. Ferry, A. Ranicki, and J. Rosenberg. London Mathematical Society Lecture Note Series 227. Cambridge University Press, 1995. MR 1388306 Zbl 0829.00028 book

Steven Charles Ferry	Related
Jonathan Micah Rosenberg	Related

[48] B. Hughes and A. Ranicki: Ends of complexes. Cambridge Tracts in Mathematics 123. Cambridge University Press, 1996. Reprinted in paperback in 2008. MR 1410261 Zbl 0876.57001 book

[49] A. A. Ranicki, A. J. Casson, D. P. Sullivan, M. A. Armstrong, C. P. Rourke, and G. E. Cooke: The Hauptvermutung book: A collection of papers of the topology of manifolds. \( K \)-Monographs in Mathematics 1. Kluwer Academic Publishers (Dordrecht), 1996. MR 1434100 Zbl 0853.00012 book

Andrew J. Casson	Related
Dennis P. Sullivan	Related
Mark Anthony Armstrong	Related
George E. Cooke	Related
Colin P. Rourke	Related	Volume

[50] A. A. Ranicki: “On the Hauptvermutung,” pp. 3–31 in A. A. Ranicki, A. J. Casson, D. P. Sullivan, M. A. Armstrong, C. P. Rourke, and G. E. Cooke: The Hauptvermutung book: A collection of papers of the topology of manifolds. Edited by A. A. Ranicki. \( K \)-Monographs in Mathematics 1. Kluwer Academic Publishers (Dordrecht), 1996. MR 1434101 Zbl 0871.57023 incollection

Andrew J. Casson	Related
Dennis P. Sullivan	Related
Mark Anthony Armstrong	Related
George E. Cooke	Related
Colin P. Rourke	Related	Volume

[51] Surgery and geometric topology (Sokado, Japan, 17–20 September 1996). Edited by A. Ranicki and M. Yamasaki. Science Bulletin of Josai University 2. Josai University (Sokado, Japan), 1997. MR 1443701 Zbl 0870.00035 book

[52] A. Ranicki: “45 slides on chain duality,” pp. 105–118 in Surgery and geometric topology (Sokado, Japan, 17–20 September 1996). Edited by A. Ranicki and M. Yamasaki. Science Bulletin of Josai University 2. Josai University (Sokado, Japan), 1997. MR 1443712 Zbl 0879.57016 incollection

[53] A. Ranicki and M. Yamasaki: “Controlled \( L \)-theory (preliminary announcement),” pp. 119–136 in Surgery and geometric topology (Sakado, Japan, 17–20 September 1996). Edited by A. Ranicki and M. Yamasaki. Science Bulletin of Josai University 2. Josai University (Sakado, Japan), 1997. MR 1443713 Zbl 0905.57021 incollection

[54] A. Ranicki: High-dimensional knot theory: Algebraic surgery in codimension 2. Springer Monographs in Mathematics. Springer (Berlin), 1998. With appendix by Elmar Winkelnkemper. MR 1713074 Zbl 0910.57001 book

[55] A. Ranicki: “Singularities, double points, controlled topology and chain duality,” Doc. Math. 4 (1999), pp. 1–59. MR 1677659 Zbl 0917.57018 article

[56] C. T. C. Wall: Surgery on compact manifolds, 2nd edition. Edited by A. A. Ranicki. Mathematical Surveys and Monographs 69. American Mathematical Society (Providence, RI), 1999. Edited and with a foreword by A. A. Ranicki. MR 1687388 Zbl 0935.57003 book

[57] M. Farber and A. Ranicki: “The Morse–Novikov theory of circle-valued functions and noncommutative localization,” pp. 381–388 in Solitons, geometry, and topology: On the crossroads (Moscow, 25–29 May 1998). Edited by V. M. Bukhshtaber and A. A. Mal’tsev. Trudy Matematicheskogo Instituta Imeni V. A. Steklova 225. MAIK Nauka/Interperiodica (Moscow), 1999. Conference dedicated to the 60th birthday of Sergei Petrovich Novikov. An English translation was published as Solitons, geometry, and topology: On the crossroads (1999). Zbl 0983.58007 incollection

Sergey Petrovich Novikov	Related
Michael S. Farber	Related
Anatoliĭ Ivanovic Maltsev	Related
Victor Matveevich Buchstaber	Related

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[58] M. Farber and A. Ranicki: “The Morse–Novikov theory of circle-valued functions and noncommutative localization,” pp. 363–371 in Solitons, geometry, and topology: On the crossroads (Moscow, 25–29 May 1998). Edited by V. M. Bukhshtaber and A. A. Mal’tsev. Proceedings of the Steklov Institute of Mathematics 225. MAIK Nauka/Interperiodica (Moscow), 1999. Conference dedicated to the 60th birthday of Sergei Petrovich Novikov. Translation of Russian original published as Tr. Mat. Inst. Steklova no. 225 (1999). MR 1725953 ArXiv math/9812122 incollection

Sergey Petrovich Novikov	Related
Michael S. Farber	Related
Anatoliĭ Ivanovic Maltsev	Related
Victor Matveevich Buchstaber	Related

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	AUTHOR = {Farber, M. and Ranicki, A.},
	TITLE = {The {M}orse--{N}ovikov theory of circle-valued
		 functions and noncommutative localization},
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	ISBN = {9785020024045},
}

[59] Surveys on surgery theory: Papers dedicated to C. T. C. Wall on the occasion to his 60th birthday, vol. 1. Edited by S. Cappell, A. Ranicki, and J. Rosenberg. Annals of Mathematics Studies 145. Princeton University Press, 2000. MR 1746325 Zbl 0933.00057 book

Sylvain Edward Cappell	Related
C. T. C. Wall	Related
Jonathan Micah Rosenberg	Related

[60] S. Cappell, A. Ranicki, and J. Rosenberg: “C. T. C. Wall’s contributions to the topology of manifolds,” pp. 3–15 in Surveys on surgery theory: Papers dedicated to C. T. C. Wall on the occasion to his 60th birthday, vol. 1. Edited by S. Cappell, A. Ranicki, and J. Rosenberg. Annals of Mathematics Studies 145. Princeton University Press, 2000. MR 1747526 Zbl 0940.57001 incollection

Sylvain Edward Cappell	Related
C. T. C. Wall	Related
Jonathan Micah Rosenberg	Related

[61] H. Johnston and A. Ranicki: “Homology manifold bordism,” Trans. Am. Math. Soc. 352 : 11 (2000), pp. 5093–5137. MR 1778506 Zbl 0958.57024 ArXiv math/9909130 article

The Bryant–Ferry–Mio–Weinberger surgery exact sequence for compact \( ANR \) homology manifolds of dimension \( \geq 6 \) is used to obtain transversality, splitting and bordism results for homology manifolds, generalizing previous work of Johnston.

First, we establish homology manifold transversality for submanifolds of dimension \( \geq 7 \): if \( f:M \to P \) is a map from an \( m \)-dimensional homology manifold \( M \) to a space \( P \), and \( Q\subset P \) is a subspace with a topological \( q \)-block bundle neighborhood, and \( m - q \geq 7 \), then \( f \) is homology manifold \( s \)-cobordant to a map which is transverse to \( Q \), with \( f^{-1}(Q)\subset M \) an \( (m{-}q) \)-dimensional homology submanifold.

Second, we obtain a codimension \( q \) splitting obstruction \[ s_Q(f) \in LS_{m-q}(\Phi) \] in the Wall \( LS \)-group for a simple homotopy equivalence \( f:M \to P \) from an \( m \)-dimensional homology manifold \( M \) to an \( m \)-dimensional Poincaré space \( P \) with a codimension \( q \) Poincaré subspace \( Q\subset P \) with a topological normal bundle, such that \( s_Q(f) = 0 \) if (and for \( m - q \geq 7 \) only if) \( f \) splits at \( Q \) up to homology manifold \( s \)-cobordism.

Third, we obtain the multiplicative structure of the homology manifold bordism groups \[ \Omega_*^H \cong \Omega_*^{TOP}[L_0(\mathbb{Z})] .\]

[62] Quadratic forms and their applications (Dublin, 5–9 July 1999). Edited by E. Bayer-Fluckiger, D. Lewis, and A. Ranicki. Contemporary Mathematics 272. American Mathematical Society (Providence, RI), 2000. MR 1803355 Zbl 0956.00036 book

Eva Bayer-Fluckiger	Related
David W. Lewis	Related

[63] A. V. Pajitnov and A. A. Ranicki: “The Whitehead group of the Novikov ring,” \( K \)-Theory 21 : 4 (2000), pp. 325–365. MR 1828181 Zbl 0996.19002 ArXiv math/0012031 article

The Bass–Heller–Swan–Farrell–Hsiang–Siebenmann decomposition of the Whitehead group \[ K_1(A_{\rho}[z,z^{-1}]) \] of a twisted Laurent polynomial extension \( A_{\rho}[z,z^{-1}] \) of a ring \( A \) is generalized to a decomposition of the Whitehead group \( K_1(A_{\rho}((z))) \) of a twisted Novikov ring of power series \[ A_{\rho}((z)) = A_{\rho}[[z]]\,[z^{-1}] .\] The decomposition involves a summand \( W_1(A,\rho) \) which is an Abelian quotient of the multiplicative group \( W(A,\rho) \) of Witt vectors \[ 1 + a_1z + a_2z^2 + \dots \in A_{\rho}[[z]] .\] An example is constructed to show that in general the natural surjection \[ W(A,\rho)^{ab}\to W_1(A,\rho) \] is not an isomorphism.

Daniel Gray Quillen	Related
Andrei Vladimirovich Pajitnov	Related

[64] Surveys on surgery theory: Papers dedicated to C. T. C. Wall on the occasion of his 60th birthday, vol. 2. Edited by S. Cappell, A. Ranicki, and J. Rosenberg. Annals of Mathematics Studies 149. Princeton University Press, 2001. MR 1818769 Zbl 0957.00062 book

Sylvain Edward Cappell	Related
C. T. C. Wall	Related
Jonathan Micah Rosenberg	Related

[65] S. Ferry and A. Ranicki: “A survey of Wall’s finiteness obstruction,” pp. 63–79 in Surveys on surgery theory: Papers dedicated to C. T. C. Wall on the occasion of his 60th birthday, vol. 2. Edited by S. Cappell, A. Ranicki, and J. Rosenberg. Annals of Mathematics Studies 149. Princeton University Press, 2001. MR 1818772 Zbl 0967.57003 ArXiv math/0008070 incollection

Sylvain Edward Cappell	Related
Jonathan Micah Rosenberg	Related
Steven Charles Ferry	Related
C. T. C. Wall	Related

[66] A. Ranicki: “An introduction to algebraic surgery,” pp. 81–163 in Surveys on surgery theory: Papers dedicated to C. T. C. Wall on the occasion of his 60th birthday, vol. 2. Edited by S. Cappell, A. Ranicki, and J. Rosenberg. Annals of Mathematics Studies 149. Princeton University Press, 2001. MR 1818773 Zbl 0974.57001 ArXiv math/0008071 incollection

Sylvain Edward Cappell	Related
C. T. C. Wall	Related
Jonathan Micah Rosenberg	Related

[67] A. Ranicki: “Algebraic Poincaré cobordism,” pp. 213–255 in Topology, geometry, and algebra: Interactions and new directions (Stanford, CA, 17–21 August 1999). Edited by A. Adem, G. Carlsson, and R. Cohen. Contemporary Mathematics 279. American Mathematical Society (Providence, RI), 2001. Conference in honor of R. James Milgram. MR 1850750 Zbl 0996.57015 ArXiv math/0008228 incollection

Alejandro Adem	Related
Gunnar Erik Carlsson	Related
Ralph Louis Cohen	Related
Richard James Milgram	Related

[68] A. Neeman and A. Ranicki: Noncommutative localization and chain complexes, I: Algebraic \( K \)- and \( L \)-theory. Preprint, September 2001. ArXiv math/0109118 techreport

The noncommutative (Cohn) localization \( \sigma^{-1}R \) of a ring \( R \) is defined for any collection \( \sigma \) of morphisms of f.g. projective left \( R \)-modules. We exhibit \( \sigma^{-1}R \) as the endomorphism ring of \( R \) in an appropriate triangulated category. We use this expression to prove that if \( \sigma^{-1}R \) is “stably flat over R” (meaning that \[ \mathrm{Tor}^R_i(\sigma^{-1}R,\sigma^{-1}R) = 0 \] for \( i > 0 \)) then every bounded f.g. projective \( \sigma^{-1}R \)-module chain complex \( D \) with \[ [D] \in \operatorname{im}\bigl(K_0(R) \to K_0(\sigma^{-1}R)\bigr) \] is chain equivalent to \( \sigma^{-1}C \) for a bounded f.g. projective \( R \)-module chain complex \( C \), and that there is a localization exact sequence in higher algebraic \( K \)-theory \[ \dots \to K_n(R) \to K_n(\sigma^{-1}R)\to K_n(R,\sigma) \to K_{n-1}(R) \to \dots, \] extending to the left the sequence obtained for \( n \leq 1 \) by Schofield. For a noncommutative localization \( \sigma^{-1}R \) of a ring with involution \( R \) there are analogous results for algebraic \( L \)-theory, extending the results of Vogel from quadratic to symmetric \( L \)-theory.

[69] A. Ranicki: “The algebraic construction of the Novikov complex of a circle-valued Morse function,” Math. Ann. 322 : 4 (2002), pp. 745–785. MR 1905105 Zbl 1001.58006 ArXiv math/9903090 article

[70] A. Ranicki: “Foundations of algebraic surgery,” pp. 491–514 in Topology of high-dimensional manifolds (Trieste, Italy 21 May–8 June 2001). Edited by F. T. Farrell, L. Göttsche, and W. Lück. ICTP Lecture Notes 9. The Abdus Salam International Centre for Theoretical Physics (Trieste), 2002. MR 1937022 Zbl 1063.57027 ArXiv math/0111315 incollection

Wolfgang Lück	Related
Lothar Göttsche	Related
Francis Thomas Farrell	Related

[71] A. Ranicki: “The structure set of an arbitrary space, the algebraic surgery exact sequence and the total surgery obstruction,” pp. 515–538 in Topology of high-dimensional manifolds (Trieste, Italy 21 May–8 June 2001). Edited by F. T. Farrell, L. Göttsche, and W. Lück. ICTP Lecture Notes 9. The Abdus Salam International Centre for Theoretical Physics (Trieste, Italy), 2002. MR 1937023 Zbl 1069.57019 ArXiv math/0111316 incollection

Wolfgang Lück	Related
Lothar Göttsche	Related
Francis Thomas Farrell	Related

[72] A. Ranicki: “Circle valued Morse theory and Novikov homology,” pp. 539–569 in Topology of high-dimensional manifolds (Trieste, Italy, 21 May–8 June 2001). Edited by F. T. Farrell, L. Göttsche, and W. Lück. ICTP Lecture Notes 9. The Abdus Salam International Centre for Theoretical Physics (Trieste, Italy), 2002. MR 1937024 Zbl 1068.57031 ArXiv math/0111317 incollection

Wolfgang Lück	Related
Lothar Göttsche	Related
Francis Thomas Farrell	Related

[73] A. Ranicki: Algebraic and geometric surgery. Oxford Mathematical Monographs. Oxford University Press, 2002. MR 2061749 Zbl 1003.57001 book

[74] O. Cornea and A. Ranicki: “Rigidity and gluing for Morse and Novikov complexes,” J. Eur. Math. Soc. 5 : 4 (2003), pp. 343–394. MR 2017851 Zbl 1052.57052 ArXiv math/0107221 article

We obtain rigidity and gluing results for the Morse complex of a real-valued Morse function as well as for the Novikov complex of a circle-valued Morse function. A rigidity result is also proved for the Floer complex of a hamiltonian defined on a closed symplectic manifold \( (M,\omega) \) with \[ c_1\vert_{\pi_2(M)} = [\omega]\vert_{\pi_2(M)} = 0 .\] The rigidity results for these complexes show that the complex of a fixed generic function/hamiltonian is a retract of the Morse (respectively Novikov or Floer) complex of any other sufficiently \( C^0 \) close generic function/hamiltonian. The gluing result is a type of Mayer–Vietoris formula for the Morse complex. It is used to express algebraically the Novikov complex up to isomorphism in terms of the Morse complex of a fundamental domain. Morse cobordisms are used to compare various Morse-type complexes without the need of bifurcation theory.

[75] 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

[76] A. Ranicki: “Blanchfield and Seifert algebra in high-dimensional knot theory,” Mosc. Math. J. 3 : 4 (2003), pp. 1333–1367. MR 2058802 Zbl 1059.19003 ArXiv math/0212187 article

[77] A. Neeman, A. Ranicki, and A. Schofield: “Representations of algebras as universal localizations,” Math. Proc. Camb. Philos. Soc. 136 : 1 (2004), pp. 105–117. MR 2034017 Zbl 1088.16020 ArXiv math/0205034 article

Given a presentation of a finitely presented group, there is a natural way to represent the group as the fundamental group of a 2-complex. The first part of this paper demonstrates one possible way to represent a finitely presented algebra \( S \) in a similarly compact form. From a presentation of the algebra, we construct a quiver with relations whose path algebra is finite dimensional. When we adjoin inverses to some of the arrows in the quiver, we show that the path algebra of the new quiver with relations is \( M_n(S) \) where \( n \) is the number of vertices in our quiver. The slogan would be that every finitely presented algebra is Morita equivalent to a universal localization of a finite dimensional algebra.

Aidan H. Schofield	Related
Amnon Neeman	Related

[78] A. Neeman and A. Ranicki: “Noncommutative localisation in algebraic \( K \)-theory, I,” Geom. Topol. 8 (2004), pp. 1385–1425. MR 2119300 Zbl 1083.18007 ArXiv math/0410620 article

This article establishes, for an appropriate localisation of associative rings, a long exact sequence in algebraic \( K \)-theory. The main result goes as follows. Let \( A \) be an associative ring and let \( A \to B \) be the localisation with respect to a set \( \sigma \) of maps between finitely generated projective \( A \)-modules. Suppose that \( \mathrm{Tor}_n^A(B,B) \) vanishes for all \( n > 0 \). View each map in \( \sigma \) as a complex (of length 1, meaning one non-zero map between two non-zero objects) in the category of perfect complexes \( D\operatorname{perf}(A) \). Denote by \( \langle\sigma\rangle \) the thick subcategory generated by these complexes. Then the canonical functor \[ D\operatorname{perf}(A) \to D\operatorname{perf}(B) \] induces (up to direct factors) an equivalence \[ D\operatorname{perf}(A)/\langle\sigma\rangle \to D\operatorname{perf}(B) .\] As a consequence, one obtains a homotopy fibre sequence \[ K(A,\sigma) \to K(A) \to K(B) \] (up to surjectivity of \( K_0(A) \to K_0(B) \)) of Waldhausen \( K \)-theory spectra.

In subsequent articles we will present the \( K \)- and \( L \)-theoretic consequences of the main theorem in a form more suitable for the applications to surgery. For example if, in addition to the vanishing of \( \mathrm{Tor}_n^A(B,B) \), we also assume that every map in \( \sigma \) is a monomorphism, then there is a description of the homotopy fiber of the map \( K(A) \to K(B) \) as the Quillen \( K \)-theory of a suitable exact category of torsion modules.

[79] A. Ranicki: “Algebraic and combinatorial codimension-1 transversality,” pp. 145–180 in Proceedings of the Casson Fest (Fayetteville, AR, 10–12 April 2003 and Austin, TX, 19–21 May 2003). Edited by C. Gordon and Y. Rieck. Geometry & Topology Monographs 7. Geometry & Topology Publications (Coventry, UK), 2004. MR 2172482 Zbl 1080.57031 ArXiv math/0308111 incollection

Andrew J. Casson	Related
Yo'av Rieck	Related
Cameron McAllan Gordon	Related	Volume

[80] F. Connolly and A. Ranicki: “On the calculation of UNil,” Adv. Math. 195 : 1 (2005), pp. 205–258. MR 2145796 Zbl 1080.57025 ArXiv math/0304016 article

[81] D. P. Sullivan: Geometric topology: Localization, periodicity and Galois symmetry. Edited by A. Ranicki. \( K \)-Monographs in Mathematics 8. Springer (Dordrecht), 2005. 1970 MIT notes. MR 2162361 Zbl 1078.55001 book

[82] M. Banagl and A. Ranicki: “Generalized Arf invariants in algebraic \( L \)-theory,” Adv. Math. 199 : 2 (January 2006), pp. 542–668. MR 2189218 Zbl 1213.57034 ArXiv math/0304362 article

[83] Exotic homology manifolds: Proceedings of the mini-workshop (Oberwolfach, Germany, 29 June–5 July 2003). Edited by F. Quinn and A. Ranicki. Geometry and Topology Monographs 9. Geometry & Topology Publications (Coventry, UK), 2006. MR 2216238 Zbl 1104.57001 book

[84] A. Ranicki: “Dedicated to the memory of Desmond Sheiham (13th November 1974–25th March 2005),” pp. vii–viii in Noncommutative localization in algebra and topology (Edinburgh, 29–30 April 2002). Edited by A. Ranicki. London Mathematical Society Lecture Note Series 330. Cambridge University Press, 2006. MR 2222477 Zbl 1119.01320 incollection

[85] A. Ranicki: “Noncommutative localization in topology,” pp. 81–102 in Noncommutative localization in algebra and topology (Edinburgh, 29–30 April 2002). Edited by A. Ranicki. London Mathematical Society Lecture Note Series 330. Cambridge University Press, 2006. MR 2222483 Zbl 1125.55004 ArXiv math/0303046 incollection

[86] W. Mio and A. Ranicki: “The quadratic form \( E_8 \) and exotic homology manifolds,” pp. 33–66 in Exotic homology manifolds: Proceedings of the mini-workshop (Oberwolfach, Germany, 29 June–5 July 2003). Edited by F. Quinn and A. Ranicki. Geometry and Topology Monographs 9. Geometry & Topology Publications (Coventry, UK), 2006. Dedicated to John Bryant on his 60th birthday. MR 2222490 Zbl 1109.57016 ArXiv math/0403261 incollection

Washington Mio	Related
Frank Stringfellow Quinn, III	Related
John Logan Bryant	Related

[87] A. Ranicki and M. Yamasaki: “Controlled \( L \)-theory,” pp. 105–153 in Exotic homology manifolds: Proceedings of the mini-workshop (Oberwolfach, Germany, 29 June–5 July 2003). Edited by F. Quinn and A. Ranicki. Geometry & Topology Monographs 9. Geometry & Topology Publications (Coventry, UK), 2006. MR 2222493 Zbl 1127.57014 ArXiv math/0402217 incollection

Frank Stringfellow Quinn, III	Related
Masayuki Yamasaki	Related

[88] Noncommutative localization in algebra and topology (Edinburgh, 29–30 April 2002). Edited by A. Ranicki. London Mathematical Society Lecture Note Series 330. Cambridge University Press, 2006. MR 2222649 Zbl 1108.13001 book

[89] A. Ranicki: “Preface,” pp. ix in Non-commutative localization in algebra and topology (Edinburgh, 29–30 April 2002). Edited by A. Ranicki. London Mathematical Society Lecture Note Series 330. Cambridge University Press, 2006. MR 2222650 incollection

[90] D. Sheiham: “Universal localization of triangular matrix rings,” Proc. Am. Math. Soc. 134 : 12 (2006), pp. 3465–3474. Prepared posthumously by Andrew Ranicki and Aidan Schofield. MR 2240657 Zbl 1115.16010 article

Desmond Sheiham	Related
Aidan H. Schofield	Related

[91] A. Ranicki and D. Sheiham: “Blanchfield and Seifert algebra in high-dimensional boundary link theory, I: Algebraic \( K \)-theory,” Geom. Topol. 10 : 3 (2006), pp. 1761–1853. Desmond Sheiham died 25 March 2005. This paper is dedicated to the memory of Paul Cohn and Jerry Levine. MR 2284050 Zbl 1125.19003 ArXiv math/0508405 article

The classification of high-dimensional \( \mu \)-component boundary links motivates decomposition theorems for the algebraic \( K \)-groups of the group ring \( A[F_{\mu}] \) and the noncommutative Cohn localization \( \Sigma^{-1}A[F_{\mu}] \), for any \( \mu \geq 1 \) and an arbitrary ring \( A \), with \( F_{\mu} \) the free group on \( \mu \) generators and \( \Sigma \) the set of matrices over \( A[F_{\mu}] \) which become invertible over \( A \) under the augmentation \( A[F_{\mu}] \to A \). Blanchfield \( A[F_{\mu}] \)-modules and Seifert \( A \)-modules are abstract algebraic analogues of the exteriors and Seifert surfaces of boundary links. Algebraic transversality for \( A[F_{\mu}] \)-module chain complexes is used to establish a long exact sequence relating the algebraic \( K \)-groups of the Blanchfield and Seifert modules, and to obtain the decompositions of \( K_*(A[F_{\mu}]) \) and \( K_*(\Sigma^{-1}A[F_{\mu}]) \) subject to a stable flatness condition on \( \Sigma^{-1}A[F_{\mu}] \) for the higher \( K \)-groups.

Desmond Sheiham	Related
Paul Moritz Cohn	Related
Jerome Paul Levine	Related

[92] 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

[93] I. Hambleton, A. Korzeniewski, and A. Ranicki: “The signature of a fibre bundle is multiplicative \( \mod 4 \),” Geom. Topol. 11 : 1 (2007), pp. 251–314. MR 2302493 Zbl 1136.55013 ArXiv math/0502353 article

We express the signature modulo 4 of a closed, oriented, \( 4k \)-dimensional \( PL \) manifold as a linear combination of its Euler characteristic and the new absolute torsion invariant defined by Korzeniewski [Absolute Whitehead torsion, Geom. Topol. 11 (2007) 215–249]. Let \( F \to E \to B \) be a \( PL \) fibre bundle, where \( F \), \( E \) and \( B \) are closed, connected, and compatibly oriented \( PL \) manifolds. We give a formula for the absolute torsion of the total space \( E \) in terms of the absolute torsion of the base and fibre, and then combine these two results to prove that the signature of \( E \) is congruent modulo 4 to the product of the signatures of \( F \) and \( B \).

Andrew Korzeniewski	Related
Ian Hambleton	Related

[94] A. A. Ranicki: Algebraic \( L \)-theory and topological manifolds, Paperback reprint edition. Cambridge Tracts in Mathematics 102. Cambridge University Press, 2008. Reprint of 1992 original. Zbl 1143.57001 book

[95] B. Hughes and A. Ranicki: Ends of complexes, Paperback reprint edition. Cambridge Tracts in Mathematics 123. Cambridge University Press, 2008. Reprint of 1996 original. Zbl 1144.57001 book

[96] A. Ranicki: “Noncommutative localization in algebraic \( L \)-theory,” Adv. Math. 220 : 3 (February 2009), pp. 894–912. MR 2483230 Zbl 1155.19001 ArXiv 0810.2761 article

[97] A. Ranicki: “A composition formula for manifold structures,” pp. 701–727 in Special issue: In honor of Friedrich Hirzebruch, part 1, published as Pure Appl. Math. Q. 5 : 2. Issue edited by L. J. Ji, K. Liu, and S.-T. Yau. International Press (Somerville, MA), 2009. MR 2508900 Zbl 1193.57011 ArXiv math/0608705 incollection

Kefeng Liu	Related
Lizhen Ji	Related
Shing-Tung Yau	Related

[98] 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

[99] A. Ranicki: “Book review: M. Kreck and W. Lück, ‘The Novikov conjecture, geometry and algebra’,” Bull. Lond. Math. Soc. 42 : 1 (2010), pp. 181–183. MR 2586978 article

Matthias Kreck	Related
Wolfgang Lück	Related

[100] A. Ranicki and M. Weiss: “On the construction and topological invariance of the Pontryagin classes,” pp. 309–343 in Algebraic and geometric topology, in honor of Bruce Williams, published as Geom. Dedicata 148. Issue edited by B. Dwyer, J. Klein, and S. Weinberger. Springer Netherlands (Dordrecht), 2010. MR 2721630 Zbl 1204.57029 ArXiv 0901.0819 incollection

John Robert Klein	Related
Shmuel Aaron Weinberger	Related
Michael S. Weiss	Related
Edward Bruce Williams	Related
William Gerard Dwyer	Related

[101] M. Crabb and A. Ranicki: “The geometric Hopf invariant and double points,” J. Fixed Point Theory Appl. 7 : 2 (October 2010), pp. 325–350. MR 2729395 Zbl 1205.55011 ArXiv 1002.2907 article

[102]D. Burns, L. Ji, K. Liu, A. Ranicki, and S.-T. Yau: “Preface,” pp. 1–3 in Special issue: In honor of Michael Atiyah and Isadore Singer, published as Pure Appl. Math. Q. 6 : 2 (2010). MR 2761847 incollection

M. F. Atiyah	Related	Volume
Kefeng Liu	Related
Lizhen Ji	Related
Shing-Tung Yau	Related
Daniel Matthew Burns, Jr.	Related
Isadore M. Singer	Related	Volume

[103] A. Ranicki: “Commentary on ‘On the parallelizability of the spheres’ by R. Bott and J. Milnor and ‘On the nonexistence of elements of Hopf invariant one’ by J. F. Adams,” Bull. Am. Math. Soc., New Ser. 48 : 4 (2011), pp. 509–511. MR 2823019 Zbl 1277.57002 article

Raoul H. Bott	Related	Volume
John Frank Adams	Related
John W. Milnor	Related

[104] J. F. Davis, Q. Khan, and A. Ranicki: “Algebraic \( K \)-theory over the infinite dihedral group: An algebraic approach,” Algebr. Geom. Topol. 11 : 4 (2011), pp. 2391–2436. MR 2835234 Zbl 1236.19002 ArXiv 0803.1639 article

Two types of Nil-groups arise in the codimension 1 splitting obstruction theory for homotopy equivalences of finite CW-complexes: the Farrell–Bass Nil-groups in the nonseparating case when the fundamental group is an HNN extension and the Waldhausen Nil-groups in the separating case when the fundamental group is an amalgamated free product. We obtain a general Nil-Nil theorem in algebraic \( K \)-theory relating the two types of Nil-groups.

The infinite dihedral group is a free product and has an index 2 subgroup which is an HNN extension, so both cases arise if the fundamental group surjects onto the infinite dihedral group. The Nil-Nil theorem implies that the two types of the reduced \( \widetilde{\mathrm{Nil}} \)-groups arising from such a fundamental group are isomorphic. There is also a topological application: in the finite-index case of an amalgamated free product, a homotopy equivalence of finite CW-complexes is semisplit along a separating subcomplex.

James Frederic Davis	Related
Qayum Khan	Related

[105] J. F. Davis, C. Haesemeyer, A. Ranicki, and M. Schlichting: “New directions in algebraic \( K \)-theory,” Oberwolfach Rep. 8 : 2 (2011), pp. 1469–1509. Workshop held in Oberwolfach, Germany, 15–21 May 2011. MR 2978644 Zbl 1334.00091 article

Christian Haesemeyer	Related
James Frederic Davis	Related
Marco Schlichting	Related

[106] M. Borodzik, A. Némethi, and A. Ranicki: Codimension 2 embeddings, algebraic surgery and Seifert forms. Preprint, November 2012. ArXiv 1211.5964 techreport

We study the cobordism of manifolds with boundary, and its applications to codimension 2 embeddings \( M^m \subset N^{m+2} \), using the method of the algebraic theory of surgery. The first main result is a splitting theorem for cobordisms of algebraic Poincaré pairs, which is then applied to describe the behaviour on the chain level of Seifert surfaces of embeddings \( M^{2n-1} \subset S^{2n+1} \) under isotopy and cobordism. The second main result (update: which is false) is that the \( S \)-equivalence class of a Seifert form is an isotopy invariant of the embedding, generalizing the Murasugi–Levine result for knots and links. The third main result is a generalized Murasugi–Kawauchi inequality giving an upper bound on the difference of the Levine–Tristram signatures of cobordant embeddings.

András Némethi	Related
Maciej Borodzik	Related

[107] A. Ranicki and M. Weiss: “On the algebraic \( L \)-theory of \( \Delta \)-sets,” pp. 423–449 in Issue dedicated to F. Thomas Farrell and Lowell E. Jones, part 2, published as Pure Appl. Math. Q. 8 : 2. Issue edited by J. F. Davis, L. Ji, D. Juan-Pineda, J.-F. Lafont, and S. Prassidis. International Press (Somerville, MA), 2012. MR 2900173 Zbl 1256.19004 ArXiv math/0701833 incollection

Jean-François Rajesh Lafont	Related
Michael S. Weiss	Related
James Frederic Davis	Related
Efstratios Chrisostomos Prassidis	Related
Lizhen Ji	Related
Francis Thomas Farrell	Related
Lowell Edwin Jones	Related
Daniel Juan-Pineda	Related

[108] G. Segal, H. Bass, M. Atiyah, K. Brown, J. Cuntz, J. Duflot, J.-L. Loday, A. Ranicki, B. Mazur, D. Sullivan, U. Tillmann, and J. Quillen: “Daniel Quillen,” Notices Am. Math. Soc. 59 : 10 (2012), pp. 1392–1406. Eric Friedlander and Daniel Grayson were coordinating editors. MR 3025899 Zbl 1284.01060 article

M. F. Atiyah	Related	Volume
Eric Mark Friedlander	Related
Dennis P. Sullivan	Related
Jean-Louis Loday	Related
Daniel Gray Quillen	Related
Ulrike Tillmann	Related
Barry C. Mazur	Related
Graeme Bryce Segal	Related
Joachim J. R. Cuntz	Related
Hyman Bass	Related
Daniel Richard Grayson	Related
Jeanne Duflot	Related
Kenneth Stephen Brown	Related

[109] M. Borodzik, A. Némethi, and A. Ranicki: “On the semicontinuity of the \( \mod 2 \) spectrum of hypersurface singularities,” J. Algebr. Geom. 24 : 2 (2015), pp. 379–398. MR 3311588 Zbl 1318.14005 ArXiv 1210.0798 article

András Némethi	Related
Maciej Borodzik	Related

[110] A. Ranicki and C. Weber: “Commentary on the Kervaire–Milnor correspondence 1958–1961,” Bull. Am. Math. Soc., New Ser. 52 : 4 (2015), pp. 603–609. MR 3393348 Zbl 1326.01037 article

The extant letters exchanged between Kervaire and Milnor during their collaboration from 1958–1961 concerned their work on the classification of exotic spheres, culminating in their 1963 Annals of Mathematics paper. Michel Kervaire died in 2007; for an account of his life, see the obituary by Shalom Eliahou, Pierre de la Harpe, Jean-Claude Hausmann, and Claude Weber in the September 2008 issue of the Notices of the American Mathematical Society. The letters were made public at the 2009 Kervaire Memorial Conference in Geneva. Their publication in this issue of the Bulletin of the American Mathematical Society is preceded by our commentary on these letters, providing some historical background.

Michel André Kervaire	Related
Charles Weber	Related
John W. Milnor	Related

[111] M. Borodzik, A. Némethi, and A. Ranicki: “Morse theory for manifolds with boundary,” Algebr. Geom. Topol. 16 : 2 (2016), pp. 971–1023. MR 3493413 Zbl 1342.57018 ArXiv 1207.3066 article

András Némethi	Related
Maciej Borodzik	Related

[112] É. Ghys, A. Ranicki, J.-M. Gambaudo, A. Cohen, J. van Wijk, J. Collins, M. Bourrigan, and C. Palmer: Six papers on signatures, braids and Seifert surfaces. Edited by É. Ghys and A. Ranicki. Ensaios Matemáticos 30. Sociedade Brasileira de Matemática (Rio de Janeiro), 2016. MR 3587804 Zbl 1420.57004 book

Julia Collins	Related
Albert Cohen	Related
Chris Palmer	Related
Maxime Bourrigan	Related
Jarke J. van Wijk	Related
Étienne Ghys	Related
Jean-Marc Gambaudo	Related

[113] É. Ghys and A. Ranicki: “Signatures in algebra, topology and dynamics,” pp. 1–173 in Six papers on signatures, braids and Seifert surfaces. Edited by É. Ghys and A. Ranicki. Ensaios Matematicos 30. Sociedade Brasileira de Matemática (Rio de Janeiro), 2016. MR 3617347 Zbl 1372.57001 ArXiv 1512.092582 incollection

[114] B. Farb, U. Hamenstädt, and A. Ranicki: “Surface bundles,” Oberwolfach Rep. 13 : 4 (2016), pp. 3149–3195. Workshop held in Oberwolfach, Germany, 4–10 December 2016. Zbl 1390.00088 article

Benson Stanley Farb	Related
Ursula Hamenstädt	Related

[115] M. Crabb and A. Ranicki: The geometric Hopf invariant and surgery theory. Springer Monographs in Mathematics. Springer (Cham, Switzerland), 2017. MR 3752161 Zbl 1404.55001 ArXiv 1602.08832 book

[116] D. Benson, C. Campagnolo, A. Ranicki, and C. Rovi: “Cohomology of symplectic groups and Meyer’s signature theorem,” Algebr. Geom. Topol. 18 : 7 (2018), pp. 4069–4091. MR 3892239 Zbl 1437.20046 ArXiv 1710.04851 article

Meyer showed that the signature of a closed oriented surface bundle over a surface is a multiple of 4, and can be computed using an element of \[ H^2(\mathrm{Sp}(2g,\mathbb{Z});\mathbb{Z}) .\] If we denote by \[ 1 \to \mathbb{Z} \to \widetilde{\mathrm{Sp}(2g,\mathbb{Z})} \to \mathrm{Sp}(2g,\mathbb{Z}) \to 1 \] the pullback of the universal cover of \( \mathrm{Sp}(2g,\mathbb{Z}) \), then by a theorem of Deligne, every finite index subgroup of \( \widetilde{\mathrm{Sp}(2g,\mathbb{Z})} \) contains \( 2\mathbb{Z} \). As a consequence, a class in the second cohomology of any finite quotient of \( \mathrm{Sp}(2g,\mathbb{Z}) \) can at most enable us to compute the signature of a surface bundle modulo 8. We show that this is in fact possible and investigate the smallest quotient of \( \mathrm{Sp}(2g,\mathbb{Z}) \) that contains this information. This quotient \( \mathfrak{h} \) is a nonsplit extension of \( \mathrm{Sp}(2g,2) \) by an elementary abelian group of order \( 2^{2g+1} \). There is a central extension \[ 1 \to \mathbb{Z}/2 \to \tilde{\mathfrak{h}} \to \mathfrak{h} \to 1 ,\] and \( \tilde{\mathfrak{h}} \) appears as a quotient of the metaplectic double cover \[ \mathrm{Mp}(2g,\mathbb{Z}) = \widetilde{\mathrm{Sp}(2g,\mathbb{Z})}/2\mathbb{Z} .\] It is an extension of \( \mathrm{Sp}(2g,2) \) by an almost extraspecial group of order \( 2^{2g+2} \), and has a faithful irreducible complex representation of dimension \( 2^g \) Provided \( g\geq 4 \), the extension \( \tilde{\mathfrak{h}} \) is the universal central extension of \( \mathfrak{h} \). Putting all this together, in Section 4 we provide a recipe for computing the signature modulo 8, and indicate some consequences.

Caterina Campagnolo	Related
David John Benson	Related
Carmen Rovi	Related

[117] M. Borodzik, S. Dangskul, and A. Ranicki: “Solid angles and Seifert hypersurfaces,” Ann. Global Anal. Geom. 57 : 3 (2020), pp. 415–454. MR 4080517 Zbl 1437.57021 ArXiv 1706.06405 article

Supreedee Dangskul	Related
Maciej Borodzik	Related

[118] D. Benson, C. Campagnolo, A. Ranicki, and C. Rovi: “Signature cocycles on the mapping class group and symplectic groups,” J. Pure Appl. Algebra 224 : 11 (2020). Article no. 106400, 49 pp. MR 4104487 Zbl 7206683 ArXiv 1811.09357 article

Werner Meyer constructed a cocycle in \( H^2(\mathrm{Sp}(2g,\mathbb{Z});\mathbb{Z}) \) which computes the signature of a closed oriented surface bundle over a surface. By studying properties of this cocycle, he also showed that the signature of such a surface bundle is a multiple of 4. In this paper, we study signature cocycles both from the geometric and algebraic points of view. We present geometric constructions which are relevant to the signature cocycle and provide an alternative to Meyer’s decomposition of a surface bundle. Furthermore, we discuss the precise relation between the Meyer and Wall–Maslov index. The main theorem of the paper provides the necessary group cohomology results to analyze the signature of a surface bundle modulo any integer \( N \). Using these results, we are able to give a complete answer for \( N = 2 \), 4, and 8, and based on a theorem of Deligne, we show that this is the best we can hope for using this method.

Caterina Campagnolo	Related
David John Benson	Related
Carmen Rovi	Related

Andrew A. Ranicki

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