Andrew Ranicki

The University of Edinburgh | UoE

Given a smooth closed oriented manifold M of dimension n embedded in Rn+2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^{n+2}$$\end{document}, we study p...

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 Poincar\'e pairs, which is then applied to describe the behaviour on the chain level of Seifert surfac...

We use purely topological methods to prove the semicontinuity of the mod 2 spectrum of local isolated hypersurface singularities in $\mathbb{C}^{n+1}$, using Seifert forms of high-dimensional non-spherical links, the Levine--Tristram signatures and the generalized Murasugi--Kawauchi inequality obtained in earlier work for cobordisms of links.

We develop Morse theory for manifolds with boundary. Besides standard and expected facts like the handle cancellation theorem and the Morse lemma for manifolds with boundary, we prove that, under a topological assumption, a critical point in the interior of a Morse function can be moved to the boundary, where it splits into a pair of boundary criti...

The geometric Hopf invariant of a stable map F is a stable Z_2-equivariant map h(F) such that the stable Z_2-equivariant homotopy class of h(F) is the primary obstruction to F being homotopic to an unstable map. In this paper we express the geometric Hopf invariant of the Umkehr map F of an immersion f:M^m \to N^n in terms of the double point set o...

We use sheaves and algebraic L-theory to construct the rational Pontryagin classes of fiber bundles with fiber R^n. This amounts to an alternative proof of Novikov's theorem on the topological invariance of the rational Pontryagin classes of vector bundles. Transversality arguments and torus tricks are avoided. Comment: 32 pages. Submitted for publ...

Given a noncommutative (Cohn) localization which is injective and stably flat we obtain a lifting theorem for induced f.g. projective -module chain complexes and localization exact sequences in algebraic L-theory, matching the algebraic K-theory localization exact sequence of Neeman-Ranicki [Amnon Neeman, Andrew Ranicki, Noncommutative localisation...

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 obta...

The algebraic $L$-groups $L_*(\A,X)$ are defined for an additive category $\A$ with chain duality and a $\Delta$-set $X$, and identified with the generalized homology groups $H_*(X;\LL_{\bullet}(\A))$ of $X$ with coefficients in the algebraic $L$-spectrum $\LL_{\bullet}(\A)$. Previously such groups had only been defined for simplicial complexes $X$...

Without Abstract

The structure set $\ST^{TOP}(M)$ of an $n$-dimensional topological manifold $M$ for $n \geqslant 5$ has a homotopy invariant functorial abelian group structure, by the algebraic version of the Browder-Novikov-Sullivan-Wall surgery theory. An element $(N,f) \in \ST^{TOP}(M)$ is an equivalence class of $n$-dimensional manifolds $N$ with a homotopy eq...

The difference between the quadratic L-groups L*(A) and the symmetric L-groups L*(A) of a ring with involution A is detected by generalized Arf invariants. The special case A=Z[x] gives a complete set of invariants for the Cappell UNil-groups UNil*(Z;Z,Z) for the infinite dihedral group D∞=Z2*Z2, extending the results of Connolly and Ranicki [Adv....

The meeting was devoted to the Kirby-Siebenmann structure theory for high-dimensional topological manifolds and the related disproof of the Hauptvermutung. We found nothing fundamentally wrong with the original work of Kirby and Siebenmann, which is solidly grounded in the literature. Their determination of T OP/P L depends on Kirby's paper on the...

The classification of high-dimensional μ-component boundary links motivates decomposition theorems for the algebraic K-groups of the group ring A[F μ ] and the noncommutative Cohn localization Σ -1 A[F μ ], for any μ≥1 and an arbitrary ring A, with F μ the free group on μ generators and Σ the set of matrices over A[F μ ] which become invertible ove...

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 in Korzeniewski [11]. 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 f...

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-->B be the localisation with respect to a set sigma of maps between finitely generated projective A-modules. Suppose that Tor_n^A(B,B) vanishes for all n...

An explicit (-1)^n-quadratic form over Z[Z^{2n}] representing the surgery problem E_8 x T^{2n} is obtained, for use in the Bryant-Ferry-Mio-Weinberger construction of 2n-dimensional exotic homology manifolds.

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...

The Waldhausen construction of Mayer-Vietoris splittings of chain complexes over an injective generalized free product of group rings is extended to a combinatorial construction of Seifert-van Kampen splittings of CW complexes with fundamental group an injective generalized free product.

Cappell's codimension 1 splitting obstruction surgery group UNiln is a direct summand of the Wall surgery obstruction group of an amalgamated free product. For any ring with involution R we use the quadratic Poincaré cobordism formulation of the L-groups to prove thatLn(R[x])=Ln(R)⊕UNiln(R;R,R).We combine this with Weiss’ universal chain bundle the...

Traditionally, localization is defined in the context of commutative algebra. However, ever since the work of Ore it has also been possible to localize noncommutative rings. High-dimensional knot theory requires the noncommutative localization matrix inversion method of Cohn [53], [54]. The algebraic K- and L-theory invariants of codimension 2 embe...

The Blanchfield and Seifert forms of knot theory have algebraic analogues over arbitrary rings with involution. The covering Blanchfield form of a Seifert form is an algebraic analogue of the expression of the infinite cyclic cover of a knot complement as the infinite union of copies of a cobordism between two copies of a Seifert surface. The inver...

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...

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 relatio...

We provide a proof of the controlled surgery sequence, including stability, in the special case that the local fundamental groups are trivial. Stability is a key ingredient in the construction of exotic homology manifolds by Bryant, Ferry, Mio and Weinberger, but no proof has been available. The development given here is based on work of M. Yamasak...

The algebraic theory of surgery gives a necessary and sufficient chain level condition for a space with n-dimensional Poincaré duality to be homotopy equivalent to an n-dimensional topological manifold. A relative version gives a necessary and sufficient chain level condition for a simple homotopy equivalence of n-dimensional topological manifolds...

Traditional Morse theory deals with real valued functions f : M R and ordinary homology H (M ). The critical points of a Morse function f generate the Morse-Smale (f) over Z, using the gradient flow to define the differentials. The isomorphism H (M) imposes homological restrictions on real valued Morse functions. There is also a universal coefficie...

The algebraic theory of surgery on chain complexes C with Poincaré duality H * (C)≅H n-* (C) describes geometric surgeries on the chain level. The algebraic effect of a geometric surgery on an n-dimensional manifold M is an algebraic surgery on the n-dimensional symmetric Poincaré complex (C,ϕ) over ℤ[π 1 (M)] with the homology of the universal cov...

The noncommutative (Cohn) localization S^{-1}R of a ring R is defined for any collection S of morphisms of f.g. projective left R-modules. We exhibit S^{-1}R as the endomorphism ring of R in an appropriate triangulated category. We use this expression to prove that if S^{-1}R is "stably flat over R" (meaning that Tor^R_i(S^{-1}R,S^{-1}R)=0 for i>0)...

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 $c_{1}|_{\pi_{2}(M)}=[\omega]|_{\pi_{2}(M)}=0$. The rigidi...

This paper was originally planned when the only known fact

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...

. Ideas from the theory of topological stability of smooth maps are transported into the controlled topological category. For example, the controlled topological equivalence of maps is discussed. These notions are related to the classication of manifold approximate brations and manifold stratied approximate brations. In turn, these maps form a bund...

The Bryant-Ferry-Mio-Weinberger surgery exact sequence for compact A N R ANR homology manifolds of dimension ≥ 6 \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 ≥ 7 \geq 7 : if f :...

Introduction Surgery theory investigates the homotopy types of manifolds, using a combination of algebra and topology. It is the aim of these notes to provide an introduction to the more algebraic aspects of the theory, without losing sight of the geometric motivation. 0.1 Historical background Aclosedm-dimensional topological manifold M has Poinca...

We use noncommutative localization to construct a chain complex which counts the critical points of a circle-valued Morse function on a manifold, generalizing the Novikov complex. As a consequence we obtain new topological lower bounds on the minimum number of critical points of a circle-valued Morse function within a homotopy class, generalizing t...

this paper shows that the two components of M

this paper is to give a reasonably leisurely account of the algebraic Poincare cobordism theory of Ranicki [11], [12] and the further development due to Weiss [14], along with some of the applications to manifolds and vector bundles. Algebraic Poincare cobordism is modelled on the bordism groups#

The Bryant-Ferry-Mio-Weinberger surgery exact sequence for compact AN R homology manifolds of dimension ≥ G 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 ≥ 7: if f : M -≥ P is a map fro...

A manifold is a Poincare duality space without singularities.

The Novikov complex of a circle-valued Morse function is constructed algebraically from the Morse-Smale complex of the restriction to a fundamental domain of the real-valued Morse function on the pullback infinite cyclic cover.

Introduction. A triangulation (K; f) of a topological space X is a simplicial complex K together with a homeomorphism f : jK jGammaGamma!X , with jKj the polyhedron of K. A topological space X is triangulable if it admits a triangulation (K; f ). The topology of a triangulable space X is determined by the combinatorial topology of the simplicial co...

this paper the bordism of automorphisms of highdimensional manifolds is considered from the point of view of the localization exact sequence in algebraic L-theory.

The Hirzebruch theorem expresses the signature oe(N) 2 Z of a 4k-dimensional manifold N 4k in terms of the L-genus L(N) 2 H 4 (N ; Q ). The signature theorem plays a central role in the classification of simply-connected manifolds. The `higher signatures' of a manifold M with fundamental group 1 (M) = are the signatures of submanifolds N 4k ae M wh...

We use noncommutative localization to construct a chain complex which counts the critical points of a circle-valued Morse function on a manifold, generalizing the Novikov complex. As a consequence we obtain new topological lower bounds on the minimum number of critical points of a circle-valued Morse function within a homotopy class, generalizing t...

The controlled finiteness obstruction and torsion are defined using controlled algebra, giving a more algebraic proof of the topological invariance of torsion and the homotopy finiteness of compact ANRs.

this paper we apply the algebraic surgery transfer of Luck and Ranicki [18] for a fibration F B with the fibre F a d-dimensional Poincarecomplex to a further investigation of the behaviour of the Wall surgery obstruction and the Mishchenko symmetric signature in fibrations. These invariants are generalizations of the equivariant signature, and the...

this paper the Novikov rings of formal power series will be used to obtain a homological characterization of finite domination for an infinite cyclic cover of a finite CW complex. In Ranicki [13] this characterization will be applied to the study of fibre bundles over S , fibred knots, the bordism of di#eomorphisms and open book decompositions. In...

this paper we only deal with the cases A = Z and A = Z[Z 2 ], so assume that A is commutative

this paper the bordism of automorphisms of highdimensional manifolds is considered from the point of view of the localization exact sequence in algebraic L-theory

ensional lens spaces by means of the based simplicial chain complex of the universal cover. In the same year Franz [3] (a student of Reidemeister) defined R-torsion in general, and used it to obtain the combinatorial classification of the high-dimensional lens spaces. R-torsion is a combinatorial invariant : finite simplicial complexes with isomorp...

A manifold is a Poincar'e duality space without singularities. McCrory obtained a homological criterion of a global nature for deciding if a polyhedral Poincar'e duality space is a homology manifold, i.e. if the singularities are homologically inessential. A homeomorphism of manifolds is a degree 1 map without double points. In this paper combinato...

The Novikov Conjecture is the single most important unsolved problem in the topology of high-dimensional non-simply connected manifolds. These two volumes are the outgrowth of a conference held at the Mathematisches Forschungsinstitut Oberwolfach (Germany) in September 1993, on the subject of 'Novikov Conjectures, Index Theorems and Rigidity'. They...

The Novikov Conjecture is the single most important unsolved problem in the topology of high-dimensional non-simply connected manifolds. These two volumes are the outgrowth of a conference held at the Mathematisches Forschungsinstitut Oberwolfach (Germany) in September, 1993, on the subject of `Novikov Conjectures, Index Theorems and Rigidity'. The...

The controlled finiteness obstruction and torsion are defined using controlled algebra, giving a more algebraic proof of the topological invariance of torsion and the homotopy finiteness of compact ANRs.

Without Abstract

! Ln 1 p (A) 1 u ! Ln 1 h (A)

The cobordism groups of quadratic Poincar complexes in an additive category with involution \mathbbA\mathbb{A} are identified with the Wall L-groups of quadratic forms and formations in \mathbbA\mathbb{A} , generalizing earlier work for modules over a ring with involution by avoiding kernels and cokernels.

triangulation of W,. This notion relates to earlier work concerning transversality structures onspherical fibrations, which are known to be essentiall y equivalent to topological bundle reductions. Thus, for n > 5, a Poincar e duality space X'with a transversality structure onitsSpivak normal fibration (i. e., with an "extrinsic" transversality str...

The algebraic K-theory product K 0(A) K 1 B K 1(A B) for rings A, B is given a chain complex interpretation, using the absolute torsion invariant introduced in Part I. Given a finitely dominated A-module chain complex C and a round finite B-module chain complex D, it is shown that the A B-module chain complex C D has a round finite chain homotopy s...

A semi-invariant in surgery is an invariant of a quadratic Poincar complex which is defined in terms of a null-cobordism. We define five such gadgets: the semicharacteristic, the semitorsion, the cross semitorsion, the torsion semicharacteristic, and the cross torsion semicharacteristic. We describe applications to the evaluation of surgery obstruc...

For an n-dimensional normal map f: Mn → Nn with finite fundamental group π1(N) = π and \mathrm{PL} 1 torsion kernel Z[ π ]-modules K*(M) the surgery obstruction σ*(f) ∈ Lh n(Z[ π ]) is expressed in terms of the projective classes [ K*(M) ] ∈ K̃0 (Z[ π ]), assuming Ki(M) = 0 if n = 2 i. This expression is used to evaluate in certain cases the surger...

For an dimensional normal map (formula preesened) Nn with finite fundamental group (formula preesened) and PL 1 torsion kernel (formula preesened)-modules (formula preesened) the surgery obstruction (formula preesened) is expressed in terms of the projective classes (formula preesened), assuming (formula preesened). This expression is used to evalu...

algebra. An n-dimensional algebraic Poincare complex over a ring A with an involution -: A -+ A; a r+ a is an A-module chain complex G with an n-dimensional Poincare duality H*(G) = Hn_*(G). We shall use n-dimensional algebraic Poincare complexes to define two sequences of covariant functors Ln {Ln}: (rings with involution) -+ (abelian groups) (n E...

Without Abstract

A group G with an epimorphism,G ! D1 onto the infinite di- hedral group D1 = Z2 � Z2 = Z ⋊ Z2 inherits an amalgamated,free product structure G = G1 �F G2 with F an index 2 subgroup of G1 and G2. Also, there is an index 2 subgroup,¯ G � G with an HNN structure ¯ G = F ⋊� Z. For such a G we obtain an isomorphism of reduced Nil-groups f Ni(R[F];R[G1 F...

We prove a squeezing/stability theorem for delta-epsilon controlled L-groups when the control map is a polyhedral stratified system of fibrations on a finite polyhedron. A relation with boundedly-controlled L-groups is also discussed.

Incluye bibliografía