Andrew Ranicki | University of Edinburgh (original) (raw)

Papers by Andrew Ranicki

Above all, Wall was responsible for major advances in the topology of manifolds. Our aim in this ... 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 a... 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 ... 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-... 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 critic... 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 spec... 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. A... 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 inje... 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 fi... 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... 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 combinatoria... 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... 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 we... 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 i... 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 wi... 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].

Let M be a compact oriented and triangulated manifold of dimension 4k. If we orient each simplex,... more Let M be a compact oriented and triangulated manifold of dimension 4k. If we orient each simplex, we obtain geometric bases for the chains and cochains, which are thus identified The classical boundary and coboundary operators δ '. Cι -> C$_! , d: Ci -* C i+1 are transposes of one another. We shall describe below a symmetric transformation If M has no boundary, we then form the symmetric transformation G o> c c • where C = C 2k 0 C 2k+1 . It M has a boundary, we form the transformation \d 0<

Abstract The Wall finiteness obstruction is the principal application of the projective class gro... 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 ... 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 clas... 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 manifol... 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.

Above all, Wall was responsible for major advances in the topology of manifolds. Our aim in this ... 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 a... 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 ... 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-... 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 critic... 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 spec... 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. A... 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 inje... 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 fi... 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... 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 combinatoria... 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... 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 we... 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 i... 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 wi... 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].

Let M be a compact oriented and triangulated manifold of dimension 4k. If we orient each simplex,... more Let M be a compact oriented and triangulated manifold of dimension 4k. If we orient each simplex, we obtain geometric bases for the chains and cochains, which are thus identified The classical boundary and coboundary operators δ '. Cι -> C$_! , d: Ci -* C i+1 are transposes of one another. We shall describe below a symmetric transformation If M has no boundary, we then form the symmetric transformation G o> c c • where C = C 2k 0 C 2k+1 . It M has a boundary, we form the transformation \d 0<

Abstract The Wall finiteness obstruction is the principal application of the projective class gro... 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 ... 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 clas... 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 manifol... 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.