Andrew Ranicki
University of Edinburgh, School of Mathematics, Faculty Member
Above all, Wall was responsible for major advances in the topology of manifolds. Our aim in this survey is to give an overview of how his work has advanced our understanding of classification methods. Wall's approaches to manifold theory... more
Above all, Wall was responsible for major advances in the topology of manifolds. Our aim in this survey is to give an overview of how his work has advanced our understanding of classification methods. Wall's approaches to manifold theory may conveniently be divided into three phases, according to the scheme:
This is the first treatment in book form of the applications of the lower K-and L-groups (which are the components of the Grothendieck groups of modules and quadratic forms over polynomial extension rings) to the topology of manifolds... more
This is the first treatment in book form of the applications of the lower K-and L-groups (which are the components of the Grothendieck groups of modules and quadratic forms over polynomial extension rings) to the topology of manifolds such as Euclidean spaces, via Whitehead torsion and the Wall finiteness and surgery obstructions. The author uses only elementary constructions and gives a full algebraic account of the groups involved; of particular note is an algebraic treatment of geometric transversality for maps to the circle.
Abstract: We prove that the Waldhausen nilpotent class group of an injective index 2 amalgamated free product is isomorphic to the Farrell-Bass nilpotent class group of a twisted polynomial extension. As an application, we show that the... more
Abstract: We prove that the Waldhausen nilpotent class group of an injective index 2 amalgamated free product is isomorphic to the Farrell-Bass nilpotent class group of a twisted polynomial extension. As an application, we show that the Farrell-Jones Conjecture in algebraic K-theory can be sharpened from the family of virtually cyclic subgroups to the family of finite-by-cyclic subgroups.
The Bass–Heller–Swan–Farrell–Hsiang–Siebenmann decomposition of the Whitehead group K 1 (A?[z, z-1]) of a twisted Laurent polynomial extension A?[z, z-1] of a ring A is generalized to a decomposition of the Whitehead group K 1 (A?((z)))... more
The Bass–Heller–Swan–Farrell–Hsiang–Siebenmann decomposition of the Whitehead group K 1 (A?[z, z-1]) of a twisted Laurent polynomial extension A?[z, z-1] of a ring A is generalized to a decomposition of the Whitehead group K 1 (A?((z))) of a twisted Novikov ring of power series A?((z))= A?[[z]][z-1]. The decomposition involves a summand W 1 (A,?) which is an Abelian quotient of the multiplicative group W (A,?) of Witt vectors 1+ a 1 z+ a 2 z 2+···? A?[[z]].
Abstract: 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... more
Abstract: 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 the Novikov inequalities.
Abstract: 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... more
Abstract: 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. Yamasaki.
Contemporary Mathematics Volume 279, 2001 ALGEBRAIC POINCARE COBORDISM ANDREW RANICKI ABSTRACT. An introduction to the cobordism theory of algebraic Poincare complexes and some its applications to manifolds, vector bundles and quadratic... more
Contemporary Mathematics Volume 279, 2001 ALGEBRAIC POINCARE COBORDISM ANDREW RANICKI ABSTRACT. An introduction to the cobordism theory of algebraic Poincare complexes and some its applications to manifolds, vector bundles and quadratic forms. Introduction This paper gives a reasonably leisurely account of the algebraic Poincare cobordism theory of Ranicki [16],[17] and the further development due to Weiss [20], along with some of the applications to manifolds, vector bundles and quadratic forms.
Abstract 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... more
Abstract 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.
Abstract 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... more
Abstract 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.
Abstract The geometric Hopf invariant of a stable map F is a stable\ mathbb Z/2-equivariant map h (F) such that the stable\ mathbb Z/2-equivariant homotopy class of h (F) is the primary obstruction to F being homotopic to an unstable map.... more
Abstract The geometric Hopf invariant of a stable map F is a stable\ mathbb Z/2-equivariant map h (F) such that the stable\ mathbb 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\ looparrowright N^ n in terms of the double point set of f.
The Hauptvermutung is the conjecture that any two triangulations of a polyhedron are combinatorially equivalent. This conjecture was formulated at the turn of the century, and until its resolution was a central problem of topology.... more
The Hauptvermutung is the conjecture that any two triangulations of a polyhedron are combinatorially equivalent. This conjecture was formulated at the turn of the century, and until its resolution was a central problem of topology. Initially, it was verified for low-dimensional polyhedra, and it might have been expected that further development of high-dimensional topology would lead to a verification in all dimensions.
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... more
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 thatWe combine this with Weiss' universal chain bundle theory to produce almost complete calculations of UNil*(Z; Z, Z) and the Wall surgery obstruction groups L*(Z [D∞]) of the infinite dihedral group D∞= Z2* Z2.
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... more
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,?) with c 1| p2 (M)=[?]| p2 (M)= 0.
The ends of a topological space are the directions in which it becomes noncompact by tending to infinity. The tame ends of manifolds are particularly interesting, both for their own sake, and for their use in the classification of... more
The ends of a topological space are the directions in which it becomes noncompact by tending to infinity. The tame ends of manifolds are particularly interesting, both for their own sake, and for their use in the classification of high-dimensional compact manifolds. The book is devoted to the related theory and practice of ends, dealing with manifolds and CW complexes in topology and chain complexes in algebra. The first part develops a homotopy model of the behavior at infinity of a noncompact space.
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... more
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. Math. 195 (2005) 205–258], Connolly and Davis [Geom. Topol. 8 (2004) 1043–1078, e-print http://arXiv. org/abs/math/0306054].
Abstract The Wall finiteness obstruction is the principal application of the projective class group K0 (Λ) to topology, with Λ= Z [G] the group ring of the fundamental group G. The finiteness obstruction is an element of the reduced... more
Abstract The Wall finiteness obstruction is the principal application of the projective class group K0 (Λ) to topology, with Λ= Z [G] the group ring of the fundamental group G. The finiteness obstruction is an element of the reduced projective class group K0 (Λ)= coker (K0 (Z)−−→ K0 (Λ)).
The quadratic L-groups Ln (A)(n≥ 0) of Wall [10] are defined for any ring with involution A, and are 4-periodic. An n-dimensional normal map (f, b): M−−→ X determines its quadratic signature σ∗(f, b)∈ Ln (Z [π]) for any oriented covering... more
The quadratic L-groups Ln (A)(n≥ 0) of Wall [10] are defined for any ring with involution A, and are 4-periodic. An n-dimensional normal map (f, b): M−−→ X determines its quadratic signature σ∗(f, b)∈ Ln (Z [π]) for any oriented covering X with group of covering translations π. If X is the universal cover, σ∗(f, b) is the surgery obstruction, and σ∗(f, b)= 0 if (and for n≥ 5 only if)(f, b) is normally bordant to a homotopy equivalence.
This book presents the definitive account of the applications of this algebra to the surgery classification of topological manifolds. The central result is the identification of a manifold structure in the homotopy type of a Poincaré... more
This book presents the definitive account of the applications of this algebra to the surgery classification of topological manifolds. The central result is the identification of a manifold structure in the homotopy type of a Poincaré duality space with a local quadratic structure in the chain homotopy type of the universal cover.
ABsTRACT. This is a new edition of the classic 1970 book on the surgery theory of compact manifolds, which is the standard book on the subiect. The original text has been supplemented by notes on subsequent developments, and the... more
ABsTRACT. This is a new edition of the classic 1970 book on the surgery theory of compact manifolds, which is the standard book on the subiect. The original text has been supplemented by notes on subsequent developments, and the references have been updated. The book should appeal to any mathematician interested in the algebraic and geometric topology of manifolds.
V is a direct sum of T-invariant subspaces Vj= ker (T-2jI)" J, one for each eigenvalue 2j, such that T-2jI: Vk~ Vk is nilpotent forj= k and an automorphism forj 4= k. We obtain in this paper an explicit formula for the projection pj (T)=... more
V is a direct sum of T-invariant subspaces Vj= ker (T-2jI)" J, one for each eigenvalue 2j, such that T-2jI: Vk~ Vk is nilpotent forj= k and an automorphism forj 4= k. We obtain in this paper an explicit formula for the projection pj (T)= pj (T) Z: V~ V onto the subspace V~, as a polynomial in T.
D∞= Z2∗ Z2∼= ZZ 2 where the generator of the cyclic groupZ2 acts onZby− idon the right-hand side. A discrete group G is said to be over D∞ if it admits a surjective homomorphism G→ D∞. Such a homomorphism induces an amalgamated free... more
D∞= Z2∗ Z2∼= ZZ 2 where the generator of the cyclic groupZ2 acts onZby− idon the right-hand side. A discrete group G is said to be over D∞ if it admits a surjective homomorphism G→ D∞. Such a homomorphism induces an amalgamated free product decomposition G∼= G1∗ H G2, and at the same time an HNN-structure G∼= H α Z for some automorphism α∈ Aut (H) where G⊂ G is a subgroup of index 2. Groups over D∞ show up naturally in the study of virtually cyclic groups. Now let R be a ring.
The stable classification of quadratic forms over a field is given by the Witt group. Topology, via surgery theory, has embedded the Witt groups in a general theory of forms over any ring with involution. In this paper we use... more
The stable classification of quadratic forms over a field is given by the Witt group. Topology, via surgery theory, has embedded the Witt groups in a general theory of forms over any ring with involution. In this paper we use geometrically inspired methods to make computations. General results on the L-theory of a Laurent polynomial extension are used to study the Witt groups of genus 0 function fields.
A semi-invariant in surgery is an invariant of a quadratic Poincaré complex which is defined in terms of a null-cobordism. We define five such gadgets: the semicharacteristic, the semitorsion, the cross semitorsion, the torsion... more
A semi-invariant in surgery is an invariant of a quadratic Poincaré 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 obstructions, especially in the odd-dimensional case.
There have been dramatic advances in algebraic K-theory recently, especially in the computation and understanding of negative K-groups and of nilpotent phenomena in algebraic K-theory. Parallel advances have used remarkably different... more
There have been dramatic advances in algebraic K-theory recently, especially in the computation and understanding of negative K-groups and of nilpotent phenomena in algebraic K-theory. Parallel advances have used remarkably different methods. Quite complete computations for the algebraic K-theory of commutative algebras over fields have been obtained using algebraic geometric techniques. On the other hand, the Farrell-Jones conjecture implies results on the K-theory for arbitrary rings.
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... more
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 structure. Thus, if X is a finitely dominated CW complex and Y is a round finite CW complex, the product X× Y is a CW complex with a round finite homotopy structure.
Reidemeister torsion (R-torsion for short) is an algebraic topology invariant which takes values in the multiplicative group of the units of a commutative ring. Although R-torsion has been largely subsumed in the more general theory of... more
Reidemeister torsion (R-torsion for short) is an algebraic topology invariant which takes values in the multiplicative group of the units of a commutative ring. Although R-torsion has been largely subsumed in the more general theory of Whitehead torsion, it has retained a life of its own. There are applications to the structure theory of compact polyhedra and manifolds, especially in the odd dimensions (eg 3), knots and links, dynamical systems, analytic torsion, Seiberg-Witten theory,
The topological applications of algebraic K- and L-theory involve a chain level procedure which assembles an R [hi-module chain complex from a local system of R-module chain complexes over a space X, with R a commutative ring and n the... more
The topological applications of algebraic K- and L-theory involve a chain level procedure which assembles an R [hi-module chain complex from a local system of R-module chain complexes over a space X, with R a commutative ring and n the group of covering translations of a regular covering I~ of X. In this paper we investigate the assembly of chain complexes from the categorical point of view, replacing X by a A-set.
The cobordism groups of quadratic Poincaré complexes in an additive category with involution A are identified with the Wall L-groups of quadratic forms and formations in A, generalizing earlier work for modules over a ring with involution... more
The cobordism groups of quadratic Poincaré complexes in an additive category with involution A are identified with the Wall L-groups of quadratic forms and formations in A, generalizing earlier work for modules over a ring with involution by avoiding kernels and cokernels.
Introduction. For surgery on codimension 1 submanifolds with nontrivial normal bundle the theory of Wall [13, Section 12C] has obstruction groups LN*(tt'—» 77), with 77 a group and it'a subgroup of index 2, such that there is defined an... more
Introduction. For surgery on codimension 1 submanifolds with nontrivial normal bundle the theory of Wall [13, Section 12C] has obstruction groups LN*(tt'—» 77), with 77 a group and it'a subgroup of index 2, such that there is defined an exact sequence involving the ordinary L-groups of rings with involution
An algebraic theory of surgery on chain complexes with an abstract Poincar6 duality should be a'simple and satisfactory algebraic version of the whole setup'to quote 5 17G of the book of Wall [25] on the surgery of compact manifolds. The... more
An algebraic theory of surgery on chain complexes with an abstract Poincar6 duality should be a'simple and satisfactory algebraic version of the whole setup'to quote 5 17G of the book of Wall [25] on the surgery of compact manifolds. The theory of Mishchenko [l01 describes the symmetric part of the surgery obstruction, and so determines it modulo 8-torsion. The theory presented here obtains the quadratic structure as well, capturing all of the surgery obstruction.
A simple (resp. finite) n-dimensional Poincaré complex X (n≥ 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 νX admits a CAT... more
A simple (resp. finite) n-dimensional Poincaré complex X (n≥ 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 νX admits a CAT reduction for which the corresponding normal map (f, b): M→ X from a compact CAT manifold M has Wall surgery obstruction σs∗(f, b)= 0∈ Ls n (π1 (X))(resp. σh∗(f, b)= 0∈ Lh n (π1 (X))). The surgery obstruction groups Ls∗(π)(resp.
Bringing together many results previously scattered throughout the research literature into a single framework, this work concentrates on the application of the author's algebraic theory of surgery to provide a unified treatment of the... more
Bringing together many results previously scattered throughout the research literature into a single framework, this work concentrates on the application of the author's algebraic theory of surgery to provide a unified treatment of the invariants of codimension 2 embeddings, generalizing the Alexander polynomials and Seifert forms of classical knot theory.
Introduction The theory of algebraic surgery on chain complexes with an abstract Poincare duality developed in Part I (Ranicki [22]) is applied here to the study of geometric surgery on manifolds.
Page 1. Geometric Topology Localization, Periodicity, and Galois Symmetry (The 1970 MIT notes) by Dennis Sullivan Edited by Andrew Ranicki February 2, 2005 Page 2.
IN order to distinguish between the combinatorial properties of finite simplicial complexes and the topology of compact polyhedra and compact manifolds it is necessary to consider infinite simplicial complexes, non-compact polyhedra, open... more
IN order to distinguish between the combinatorial properties of finite simplicial complexes and the topology of compact polyhedra and compact manifolds it is necessary to consider infinite simplicial complexes, non-compact polyhedra, open manifolds, and algebraic K-and L-theory.
The Novikov Conjecture has to do with the question of the relationship of the characteristic classes of manifolds to the underlying bordism and homotopy theory. For smooth manifolds, the characteristic classes are by definition the... more
The Novikov Conjecture has to do with the question of the relationship of the characteristic classes of manifolds to the underlying bordism and homotopy theory. For smooth manifolds, the characteristic classes are by definition the characteristic classes of the tangent (or normal) bundle, so basic to this question is another more fundamental one: how much of a vector bundle is determined by its underlying spherical fibration?
The finiteness obstruction [X] eK0 (Z [7ci {X)']) of Wall [13],[14] is an algebraic K-theory invariant of a finitely dominated CW complex X such that\ _X~\= 0 if and only if X is homotopy equivalent to a finite CW complex. We develop here... more
The finiteness obstruction [X] eK0 (Z [7ci {X)']) of Wall [13],[14] is an algebraic K-theory invariant of a finitely dominated CW complex X such that\ _X~\= 0 if and only if X is homotopy equivalent to a finite 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->\ _X~\ from topology to algebra.
The papers in this volume were typeset by the editors using the TEX typesetting program and the AMS-TEX, LATEX, LAMS-TEX, AMS-LATEX, and XY-pic macro packages. Style files were prepared by the editors using templates prepared by Cambridge... more
The papers in this volume were typeset by the editors using the TEX typesetting program and the AMS-TEX, LATEX, LAMS-TEX, AMS-LATEX, and XY-pic macro packages. Style files were prepared by the editors using templates prepared by Cambridge University Press. Some of the figures were prepared in PostScript. TEX and AMS-TEX are trademarks of the American Mathematical Society; PostScript is a registered trademark of Adobe Systems Incorporated.
Signatures of quadratic forms play a central role in the classification theory of manifolds. The Hirzebruch theorem expresses the signature σ (N)∈ Z of a 4k-dimensional manifold N4k in terms of the L-genus L (N)∈ H4∗(N; Q). The 'higher... more
Signatures of quadratic forms play a central role in the classification theory of manifolds. The Hirzebruch theorem expresses the signature σ (N)∈ Z of a 4k-dimensional manifold N4k in terms of the L-genus L (N)∈ H4∗(N; Q). The 'higher signatures' of a manifold M with fundamental group π1 (M)= π are the signatures of the submanifolds N4k⊂ M which are determined by the cohomology H∗(Bπ; Q).
L~(~× Z)= Ln (~) _ The hamiltonian formalism of [4] allowed a unified approach to the three L-theories, and a purely algebraic description of these decompositions. This was done in parts I. and II. of this paper ([5]), which will be... more
L~(~× Z)= Ln (~) _ The hamiltonian formalism of [4] allowed a unified approach to the three L-theories, and a purely algebraic description of these decompositions. This was done in parts I. and II. of this paper ([5]), which will be denoted I., II.. In I. there were defined abelian groups l Un (A){fg projective I Vn (A)'using quadratic forms on~ fg free Wn (A)[based
For n> 4 and i-connected (f, b) with 2i≤ n there is a one-one correspondence between geometric surgeries on (f, b) killing elements x∈ Ki (M) and algebraic surgeries on (C, ψ) killing x∈ Hi (C). The Wall surgery obstruction of an... more
For n> 4 and i-connected (f, b) with 2i≤ n there is a one-one correspondence between geometric surgeries on (f, b) killing elements x∈ Ki (M) and algebraic surgeries on (C, ψ) killing x∈ Hi (C). The Wall surgery obstruction of an n-dimensional normal map (f, b): M→ X σ∗(f, b)∈ Ln (Z [π1 (X)]) was originally defined by first making (f, b)[n/2]-connected by geometric surgery below the middle dimension, using forms for even n and automorphisms of forms for odd n.
This paper introduces the notion of an intrinsic transversality structure on a Poincare duality space X". Such a space has an intrinsic transversality structure if the embedding of X" into its regular neighborhood Wn+ k in Euclidean space... more
This paper introduces the notion of an intrinsic transversality structure on a Poincare duality space X". Such a space has an intrinsic transversality structure if the embedding of X" into its regular neighborhood Wn+ k in Euclidean space can be made" Poincare transverse" to a triangulation of Wn+ k. This notion relates to earlier work concerning transversality structures on spherical fibrations, which are known to be essentially equivalent to topological bundle reductions.
PK:^ 0 (Ж [р])> K0 {2Z [tìì with я= тг,(E), p= тт,(В). This is the transfer map (1.4) of the preceding paper, Munkholm and Pedersen [4], to which we refer for terminology and background material.
The Wall finiteness obstruction of a finitely dominated CW complex X is an element [X] eÄT0 (Z [7t1 (A')]) of the reduced projective class group such that [X~\= 0 if and only if X is homotopy equivalent to a finite CW complex. The... more
The Wall finiteness obstruction of a finitely dominated CW complex X is an element [X] eÄT0 (Z [7t1 (A')]) of the reduced projective class group such that [X~\= 0 if and only if X is homotopy equivalent to a finite CW complex. The finiteness obstruction of a pointed homotopy idempotent p^ p2:
Abstract. The Bryant-Ferry-Mio-Weinberger surgery exact sequence for compact ANR homology manifolds of dimension ≥ 6 is used to obtain transversality, splitting and bordism results for homology manifolds, generalizing previous work of... more
Abstract. The Bryant-Ferry-Mio-Weinberger surgery exact sequence for compact ANR homology manifolds of dimension ≥ 6 is used to obtain transversality, splitting and bordism results for homology manifolds, generalizing previous work of Johnston.
Abstract: For an $ n $-dimensional normal map $ f:{M^ n}\ to {N^ n} $ with finite fundamental group ${\ pi _1}(N)=\ pi $ and PL $1$ torsion kernel $ Z [\ pi] $-modules ${K_ {\ ast}}(M) $ the surgery obstruction ${\ sigma _ {\ ast}}(f)\ in... more
Abstract: For an $ n $-dimensional normal map $ f:{M^ n}\ to {N^ n} $ with finite fundamental group ${\ pi _1}(N)=\ pi $ and PL $1$ torsion kernel $ Z [\ pi] $-modules ${K_ {\ ast}}(M) $ the surgery obstruction ${\ sigma _ {\ ast}}(f)\ in L_n^ h (Z [\ pi]) $ is expressed in terms of the projective classes $[{K_ {\ ast}}(M)]\ in {\ tilde K_0}(Z [\ pi]) $, assuming ${K_i}(M)= 0$ if $ n= 2i $.