Andrew Ranicki
- Published in print:
- 2002
- Published Online:
- September 2007
- ISBN:
- 9780198509240
- eISBN:
- 9780191708725
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509240.001.0001
- Subject:
- Mathematics, Geometry / Topology
This book is an introduction to surgery theory, the standard algebraic topology classification method for manifolds of dimension greater than 4. It is aimed at those who have already been on a basic ...
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This book is an introduction to surgery theory, the standard algebraic topology classification method for manifolds of dimension greater than 4. It is aimed at those who have already been on a basic topology course, and would now like to understand the topology of high-dimensional manifolds. This text contains entry-level accounts of the various prerequisites of both algebra and topology. Surgery theory expresses the manifold structure set in terms of the topological K-theory of vector bundles and the algebraic L-theory of quadratic forms. While concentrating on the basic mechanics of surgery, this book includes many worked examples, useful drawings for illustration of the algebra and references for further reading.Less
This book is an introduction to surgery theory, the standard algebraic topology classification method for manifolds of dimension greater than 4. It is aimed at those who have already been on a basic topology course, and would now like to understand the topology of high-dimensional manifolds. This text contains entry-level accounts of the various prerequisites of both algebra and topology. Surgery theory expresses the manifold structure set in terms of the topological K-theory of vector bundles and the algebraic L-theory of quadratic forms. While concentrating on the basic mechanics of surgery, this book includes many worked examples, useful drawings for illustration of the algebra and references for further reading.
Friedhelm Waldhausen, Bjørn Jahren, and John Rognes
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157757
- eISBN:
- 9781400846528
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157757.001.0001
- Subject:
- Mathematics, Geometry / Topology
Since its introduction by the author in the 1970s, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a ...
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Since its introduction by the author in the 1970s, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a full proof of the equivalence relating the two areas has not appeared until now. This book presents such a proof, essentially completing the author's program from more than thirty years ago. The main result is a stable parametrized h-cobordism theorem, derived from a homotopy equivalence between a space of PL h-cobordisms on a space X and the classifying space of a category of simple maps of spaces having X as deformation retract. The smooth and topological results then follow by smoothing and triangulation theory. The proof has two main parts. The essence of the first part is a “desingularization,” improving arbitrary finite simplicial sets to polyhedra. The second part compares polyhedra with PL manifolds by a thickening procedure. Many of the techniques and results developed should be useful in other connections.Less
Since its introduction by the author in the 1970s, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a full proof of the equivalence relating the two areas has not appeared until now. This book presents such a proof, essentially completing the author's program from more than thirty years ago. The main result is a stable parametrized h-cobordism theorem, derived from a homotopy equivalence between a space of PL h-cobordisms on a space X and the classifying space of a category of simple maps of spaces having X as deformation retract. The smooth and topological results then follow by smoothing and triangulation theory. The proof has two main parts. The essence of the first part is a “desingularization,” improving arbitrary finite simplicial sets to polyhedra. The second part compares polyhedra with PL manifolds by a thickening procedure. Many of the techniques and results developed should be useful in other connections.
Friedhelm Waldhausen, Bjørn Jahren, and John Rognes
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157757
- eISBN:
- 9781400846528
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157757.003.0001
- Subject:
- Mathematics, Geometry / Topology
This book presents a proof of the stable parametrized h-cobordism theorem, which deals with the existence of a natural homotopy equivalence for each compact CAT manifold. In this theorem, a stable ...
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This book presents a proof of the stable parametrized h-cobordism theorem, which deals with the existence of a natural homotopy equivalence for each compact CAT manifold. In this theorem, a stable CAT h-cobordism space is defined in terms of manifolds, whereas a CAT Whitehead space is defined in terms of algebraic K-theory. This is a stable range extension to parametrized families of the classical hand s-cobordism theorems first stated by A. E. Hatcher, but his proofs were incomplete. This book provides a full proof of this key result, which provides the link between the geometric topology of high-dimensional manifolds and their automorphisms, as well as the algebraic K-theory of spaces and structured ring spectra.Less
This book presents a proof of the stable parametrized h-cobordism theorem, which deals with the existence of a natural homotopy equivalence for each compact CAT manifold. In this theorem, a stable CAT h-cobordism space is defined in terms of manifolds, whereas a CAT Whitehead space is defined in terms of algebraic K-theory. This is a stable range extension to parametrized families of the classical hand s-cobordism theorems first stated by A. E. Hatcher, but his proofs were incomplete. This book provides a full proof of this key result, which provides the link between the geometric topology of high-dimensional manifolds and their automorphisms, as well as the algebraic K-theory of spaces and structured ring spectra.
Friedhelm Waldhausen, Bjørn Jahren, and John Rognes
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157757
- eISBN:
- 9781400846528
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157757.003.0002
- Subject:
- Mathematics, Geometry / Topology
This chapter deals with the stable parametrized h-cobordism theorem. It begins with a discussion of the manifold part; here DIFF is written for the category of Csuperscript infinity smooth manifolds, ...
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This chapter deals with the stable parametrized h-cobordism theorem. It begins with a discussion of the manifold part; here DIFF is written for the category of Csuperscript infinity smooth manifolds, PL for the category of piecewise-linear manifolds, and TOP for the category of topological manifolds. CAT is generically written for any one of these geometric categories. Relevant terms such as stabilization map, simple map, pullback map, PL Serre fibrations, weak homotopy equivalence, PL Whitehead space, and cofibration are also defined. The chapter proceeds by describing the non-manifold part, the algebraic K-theory of spaces, and the relevance of simple maps to the study of PL homeomorphisms of manifolds.Less
This chapter deals with the stable parametrized h-cobordism theorem. It begins with a discussion of the manifold part; here DIFF is written for the category of Csuperscript infinity smooth manifolds, PL for the category of piecewise-linear manifolds, and TOP for the category of topological manifolds. CAT is generically written for any one of these geometric categories. Relevant terms such as stabilization map, simple map, pullback map, PL Serre fibrations, weak homotopy equivalence, PL Whitehead space, and cofibration are also defined. The chapter proceeds by describing the non-manifold part, the algebraic K-theory of spaces, and the relevance of simple maps to the study of PL homeomorphisms of manifolds.
Araceli Bonifant, Misha Lyubich, and Scott Sutherland
- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159294
- eISBN:
- 9781400851317
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159294.001.0001
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, ...
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John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, and is the only one to have received all three of the Leroy P. Steele prizes. In honor of his eightieth birthday, this book gathers together surveys and papers inspired by Milnor's work, from distinguished experts examining not only holomorphic dynamics in one and several variables, but also differential geometry, entropy theory, and combinatorial group theory. The book contains the last paper written by William Thurston, as well as a short paper by John Milnor himself. Introductory sections put the papers in mathematical and historical perspective, color figures are included, and an index facilitates browsing.Less
John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, and is the only one to have received all three of the Leroy P. Steele prizes. In honor of his eightieth birthday, this book gathers together surveys and papers inspired by Milnor's work, from distinguished experts examining not only holomorphic dynamics in one and several variables, but also differential geometry, entropy theory, and combinatorial group theory. The book contains the last paper written by William Thurston, as well as a short paper by John Milnor himself. Introductory sections put the papers in mathematical and historical perspective, color figures are included, and an index facilitates browsing.
T. N. Krishnamurti, H. S. Bedi, and V. M. Hardiker
- Published in print:
- 1998
- Published Online:
- November 2020
- ISBN:
- 9780195094732
- eISBN:
- 9780197560761
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195094732.003.0010
- Subject:
- Earth Sciences and Geography, Meteorology and Climatology
In this chapter we present some of the physical processes that are used in numerical weather prediction modeling. Grid-point models, based on finite differences, and ...
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In this chapter we present some of the physical processes that are used in numerical weather prediction modeling. Grid-point models, based on finite differences, and spectral models both generally treat the physical processes in the same manner. The vertical columns above the horizontal grid points (the transform grid for the spectral models) are the ones along which estimates of the effects of the physical processes are made. In this chapter we present a treatment of the planetary boundary layer, including a discussion on the surface similarity theory. Also covered is the cumulus parameterization problem in terms of the Kuo scheme and the Arakawa- Schubert sheme. Large-scale condensation and radiative transfer in clear and cloudy skies are the final topics reviewed. There are at least three types of fluxes that one deals with, namely momentum, sensible heat, and moisture. Furthermore, one needs to examine separately the land and ocean regions. In this section we present the socalled bulk aerodynamic methods as well as the similarity analysis approach for the estimation of the surface fluxes. The radiation code in a numerical weather prediction model is usually coupled to the calculation of the surface energy balance. This will be covered later in Section 8.5.6. This surface energy balance is usually carried out over land areas, where one balances the net radiation against the surface fluxes of heat and moisture for the determination of soil temperature. Over oceans, the sea-surface temperatures are prescribed where the surface energy balance is implicit. Thus it is quite apparent that what one does in the parameterization of the planetary boundary layer has to be integrated with the radiative parameterization in a consistent manner.
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In this chapter we present some of the physical processes that are used in numerical weather prediction modeling. Grid-point models, based on finite differences, and spectral models both generally treat the physical processes in the same manner. The vertical columns above the horizontal grid points (the transform grid for the spectral models) are the ones along which estimates of the effects of the physical processes are made. In this chapter we present a treatment of the planetary boundary layer, including a discussion on the surface similarity theory. Also covered is the cumulus parameterization problem in terms of the Kuo scheme and the Arakawa- Schubert sheme. Large-scale condensation and radiative transfer in clear and cloudy skies are the final topics reviewed. There are at least three types of fluxes that one deals with, namely momentum, sensible heat, and moisture. Furthermore, one needs to examine separately the land and ocean regions. In this section we present the socalled bulk aerodynamic methods as well as the similarity analysis approach for the estimation of the surface fluxes. The radiation code in a numerical weather prediction model is usually coupled to the calculation of the surface energy balance. This will be covered later in Section 8.5.6. This surface energy balance is usually carried out over land areas, where one balances the net radiation against the surface fluxes of heat and moisture for the determination of soil temperature. Over oceans, the sea-surface temperatures are prescribed where the surface energy balance is implicit. Thus it is quite apparent that what one does in the parameterization of the planetary boundary layer has to be integrated with the radiative parameterization in a consistent manner.
J. C. Kaimal and J. J. Finnigan
- Published in print:
- 1994
- Published Online:
- November 2020
- ISBN:
- 9780195062397
- eISBN:
- 9780197560167
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195062397.003.0006
- Subject:
- Earth Sciences and Geography, Atmospheric Sciences
Any land surface that receives regular rainfall is almost certain to be covered by vegetation. Most of the inhabitable regions of the globe fall into this category. ...
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Any land surface that receives regular rainfall is almost certain to be covered by vegetation. Most of the inhabitable regions of the globe fall into this category. Often the vegetation is tall enough to call into question the assumption, implicit in the discussion of the first two chapters, that the roughness elements on the ground surface are much lower than any observation height of interest to us. In fact, if we venture to make measurements too close to tall vegetation, we discover significant departures from many of the scaling laws and formulas that seem to work in the surface layer above the canopy. To take one example, momentum is absorbed from the wind not just at the ground surface but through the whole depth of the canopy as aerodynamic drag on the plants. Consequently, although we still observe a logarithmic velocity profile well above the canopy, its apparent origin has moved to a level z = d near the top of the plants. The precise position of this “displacement height,” d, depends on the way the drag force is distributed through the foliage and this in turn depends on the structure of the mean wind and turbulence within the canopy. Our interest in the nature of within-canopy turbulence, however, is not motivated solely by its influence on the surface layer above. The understanding of turbulent transfer within foliage canopies provides the intellectual underpinning for the physical aspects of agricultural meteorology. As such it has a history almost as venerable as investigations of the surface layer itself. The landmark study of Weather in Wheat by Penman and Long (1960) was the first of a series of seminal papers to establish the quantitative link between the turbulent fluxes in a canopy and the physiological sources and sinks of heat, water vapor, and carbon dioxide (CO2). Prominent and influential among these early publications were those by Uchijima (1962), Denmead (1964), Brown and Covey (1966), and Lemon and Wright (1969). Whereas these authors were motivated by curiosity about plant physiology and the transfer of water and other scalars through the soil-plant-air continuum, other workers forged the link between the classical surface layer studies detailed in Chapter 1 and the structure of within-canopy turbulence.
Less
Any land surface that receives regular rainfall is almost certain to be covered by vegetation. Most of the inhabitable regions of the globe fall into this category. Often the vegetation is tall enough to call into question the assumption, implicit in the discussion of the first two chapters, that the roughness elements on the ground surface are much lower than any observation height of interest to us. In fact, if we venture to make measurements too close to tall vegetation, we discover significant departures from many of the scaling laws and formulas that seem to work in the surface layer above the canopy. To take one example, momentum is absorbed from the wind not just at the ground surface but through the whole depth of the canopy as aerodynamic drag on the plants. Consequently, although we still observe a logarithmic velocity profile well above the canopy, its apparent origin has moved to a level z = d near the top of the plants. The precise position of this “displacement height,” d, depends on the way the drag force is distributed through the foliage and this in turn depends on the structure of the mean wind and turbulence within the canopy. Our interest in the nature of within-canopy turbulence, however, is not motivated solely by its influence on the surface layer above. The understanding of turbulent transfer within foliage canopies provides the intellectual underpinning for the physical aspects of agricultural meteorology. As such it has a history almost as venerable as investigations of the surface layer itself. The landmark study of Weather in Wheat by Penman and Long (1960) was the first of a series of seminal papers to establish the quantitative link between the turbulent fluxes in a canopy and the physiological sources and sinks of heat, water vapor, and carbon dioxide (CO2). Prominent and influential among these early publications were those by Uchijima (1962), Denmead (1964), Brown and Covey (1966), and Lemon and Wright (1969). Whereas these authors were motivated by curiosity about plant physiology and the transfer of water and other scalars through the soil-plant-air continuum, other workers forged the link between the classical surface layer studies detailed in Chapter 1 and the structure of within-canopy turbulence.