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String Topology and Cyclic Homology

Ralph L. Cohen Kathryn Hess Alexander A. Voronov

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Palabras clave – provistas por la editorial

Algebraic Topology; Manifolds and Cell Complexes (incl. Diff.Topology); Mathematical Methods in Physics

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Institución detectada Año de publicación Navegá Descargá Solicitá
No detectada 2006 SpringerLink

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Tipo de recurso:

libros

ISBN impreso

978-3-7643-2182-6

ISBN electrónico

978-3-7643-7388-7

Editor responsable

Springer Nature

País de edición

Reino Unido

Fecha de publicación

Información sobre derechos de publicación

© Birkhäuser Verlag 2006

Tabla de contenidos

Introduction

Ralph L. Cohen; Kathryn Hess; Alexander A. Voronov

The origin of the Moon 4.56 Gyr ago, its subsequent evolution, and the implications of both relative to the Earth remain subject to lively debate. Because the internal geochemistry and geophysics of the Moon does not appear consistent with an origin by the giant impact of a Mars-sized asteroid on the Earth, this hypothesis is challenged by one that proposes the capture of an independently formed planetesimal. The Moon—s internal structure also indicates that it and all the terrestrial planets initially had relatively cool, chondritic proto-cores prior to formation of metallic cores. Evidence exists that these proto-cores delayed formation of metallic cores for periods that correlate with the final mass of a planet.

The impact history of the inner solar system has been broadly outlined by the modern investigation of the Moon. First, soon after the formation of a coherent lunar crust and against an intense background of smaller cratering events, the Moon was subjected to extremely large impacts that formed basins up to 3200km in diameter. On Earth, the melt sheets from these continental-scale impacts may have been responsible for the formation of the first continental crust at ∼4.4 Gyr. Second, ∼50 impact basins >300 km in diameter formed between 4.5 and 3.8 Gyr probably by pulses of impactors produced during the migration and interaction of the giant planets within a structured solar disk of planetesimal rings. The last of these pulses at about 3.85 Gyr, producing the ∼14 mascon basins, resurfaced most of the Moon, and suggested an apparent “cataclysm” at that time. This period of 700 Myr may have been one of “punctuated cataclysm” as one or more giant planets encountered separate planetesimal rings and gaps during outward orbital migration. Finally, the implication of this violent impact history in the inner solar system prior to 3.8 Gyr relative to the surfaces of the hydrous terrestrial planets, that is, Earth, Mars and probably early Venus, is that clays were the dominant mineral species. These clays, as well as volcanic sulfides, may have provided the templates for the formation of complex organic precursors that made up the first living cells.

Part I - Notes on String Topology | Pp. 3-4

Intersection theory in loop spaces

Ralph L. Cohen; Kathryn Hess; Alexander A. Voronov

The origin of the Moon 4.56 Gyr ago, its subsequent evolution, and the implications of both relative to the Earth remain subject to lively debate. Because the internal geochemistry and geophysics of the Moon does not appear consistent with an origin by the giant impact of a Mars-sized asteroid on the Earth, this hypothesis is challenged by one that proposes the capture of an independently formed planetesimal. The Moon—s internal structure also indicates that it and all the terrestrial planets initially had relatively cool, chondritic proto-cores prior to formation of metallic cores. Evidence exists that these proto-cores delayed formation of metallic cores for periods that correlate with the final mass of a planet.

The impact history of the inner solar system has been broadly outlined by the modern investigation of the Moon. First, soon after the formation of a coherent lunar crust and against an intense background of smaller cratering events, the Moon was subjected to extremely large impacts that formed basins up to 3200km in diameter. On Earth, the melt sheets from these continental-scale impacts may have been responsible for the formation of the first continental crust at ∼4.4 Gyr. Second, ∼50 impact basins >300 km in diameter formed between 4.5 and 3.8 Gyr probably by pulses of impactors produced during the migration and interaction of the giant planets within a structured solar disk of planetesimal rings. The last of these pulses at about 3.85 Gyr, producing the ∼14 mascon basins, resurfaced most of the Moon, and suggested an apparent “cataclysm” at that time. This period of 700 Myr may have been one of “punctuated cataclysm” as one or more giant planets encountered separate planetesimal rings and gaps during outward orbital migration. Finally, the implication of this violent impact history in the inner solar system prior to 3.8 Gyr relative to the surfaces of the hydrous terrestrial planets, that is, Earth, Mars and probably early Venus, is that clays were the dominant mineral species. These clays, as well as volcanic sulfides, may have provided the templates for the formation of complex organic precursors that made up the first living cells.

Part I - Notes on String Topology | Pp. 5-23

The cacti operad

Ralph L. Cohen; Kathryn Hess; Alexander A. Voronov

The origin of the Moon 4.56 Gyr ago, its subsequent evolution, and the implications of both relative to the Earth remain subject to lively debate. Because the internal geochemistry and geophysics of the Moon does not appear consistent with an origin by the giant impact of a Mars-sized asteroid on the Earth, this hypothesis is challenged by one that proposes the capture of an independently formed planetesimal. The Moon—s internal structure also indicates that it and all the terrestrial planets initially had relatively cool, chondritic proto-cores prior to formation of metallic cores. Evidence exists that these proto-cores delayed formation of metallic cores for periods that correlate with the final mass of a planet.

The impact history of the inner solar system has been broadly outlined by the modern investigation of the Moon. First, soon after the formation of a coherent lunar crust and against an intense background of smaller cratering events, the Moon was subjected to extremely large impacts that formed basins up to 3200km in diameter. On Earth, the melt sheets from these continental-scale impacts may have been responsible for the formation of the first continental crust at ∼4.4 Gyr. Second, ∼50 impact basins >300 km in diameter formed between 4.5 and 3.8 Gyr probably by pulses of impactors produced during the migration and interaction of the giant planets within a structured solar disk of planetesimal rings. The last of these pulses at about 3.85 Gyr, producing the ∼14 mascon basins, resurfaced most of the Moon, and suggested an apparent “cataclysm” at that time. This period of 700 Myr may have been one of “punctuated cataclysm” as one or more giant planets encountered separate planetesimal rings and gaps during outward orbital migration. Finally, the implication of this violent impact history in the inner solar system prior to 3.8 Gyr relative to the surfaces of the hydrous terrestrial planets, that is, Earth, Mars and probably early Venus, is that clays were the dominant mineral species. These clays, as well as volcanic sulfides, may have provided the templates for the formation of complex organic precursors that made up the first living cells.

Part I - Notes on String Topology | Pp. 25-45

Field theoretic properties of string topology

Ralph L. Cohen; Kathryn Hess; Alexander A. Voronov

The origin of the Moon 4.56 Gyr ago, its subsequent evolution, and the implications of both relative to the Earth remain subject to lively debate. Because the internal geochemistry and geophysics of the Moon does not appear consistent with an origin by the giant impact of a Mars-sized asteroid on the Earth, this hypothesis is challenged by one that proposes the capture of an independently formed planetesimal. The Moon—s internal structure also indicates that it and all the terrestrial planets initially had relatively cool, chondritic proto-cores prior to formation of metallic cores. Evidence exists that these proto-cores delayed formation of metallic cores for periods that correlate with the final mass of a planet.

The impact history of the inner solar system has been broadly outlined by the modern investigation of the Moon. First, soon after the formation of a coherent lunar crust and against an intense background of smaller cratering events, the Moon was subjected to extremely large impacts that formed basins up to 3200km in diameter. On Earth, the melt sheets from these continental-scale impacts may have been responsible for the formation of the first continental crust at ∼4.4 Gyr. Second, ∼50 impact basins >300 km in diameter formed between 4.5 and 3.8 Gyr probably by pulses of impactors produced during the migration and interaction of the giant planets within a structured solar disk of planetesimal rings. The last of these pulses at about 3.85 Gyr, producing the ∼14 mascon basins, resurfaced most of the Moon, and suggested an apparent “cataclysm” at that time. This period of 700 Myr may have been one of “punctuated cataclysm” as one or more giant planets encountered separate planetesimal rings and gaps during outward orbital migration. Finally, the implication of this violent impact history in the inner solar system prior to 3.8 Gyr relative to the surfaces of the hydrous terrestrial planets, that is, Earth, Mars and probably early Venus, is that clays were the dominant mineral species. These clays, as well as volcanic sulfides, may have provided the templates for the formation of complex organic precursors that made up the first living cells.

Part I - Notes on String Topology | Pp. 47-65

A Morse theoretic viewpoint

Ralph L. Cohen; Kathryn Hess; Alexander A. Voronov

The origin of the Moon 4.56 Gyr ago, its subsequent evolution, and the implications of both relative to the Earth remain subject to lively debate. Because the internal geochemistry and geophysics of the Moon does not appear consistent with an origin by the giant impact of a Mars-sized asteroid on the Earth, this hypothesis is challenged by one that proposes the capture of an independently formed planetesimal. The Moon—s internal structure also indicates that it and all the terrestrial planets initially had relatively cool, chondritic proto-cores prior to formation of metallic cores. Evidence exists that these proto-cores delayed formation of metallic cores for periods that correlate with the final mass of a planet.

The impact history of the inner solar system has been broadly outlined by the modern investigation of the Moon. First, soon after the formation of a coherent lunar crust and against an intense background of smaller cratering events, the Moon was subjected to extremely large impacts that formed basins up to 3200km in diameter. On Earth, the melt sheets from these continental-scale impacts may have been responsible for the formation of the first continental crust at ∼4.4 Gyr. Second, ∼50 impact basins >300 km in diameter formed between 4.5 and 3.8 Gyr probably by pulses of impactors produced during the migration and interaction of the giant planets within a structured solar disk of planetesimal rings. The last of these pulses at about 3.85 Gyr, producing the ∼14 mascon basins, resurfaced most of the Moon, and suggested an apparent “cataclysm” at that time. This period of 700 Myr may have been one of “punctuated cataclysm” as one or more giant planets encountered separate planetesimal rings and gaps during outward orbital migration. Finally, the implication of this violent impact history in the inner solar system prior to 3.8 Gyr relative to the surfaces of the hydrous terrestrial planets, that is, Earth, Mars and probably early Venus, is that clays were the dominant mineral species. These clays, as well as volcanic sulfides, may have provided the templates for the formation of complex organic precursors that made up the first living cells.

Part I - Notes on String Topology | Pp. 67-77

Brane topology

Ralph L. Cohen; Kathryn Hess; Alexander A. Voronov

The origin of the Moon 4.56 Gyr ago, its subsequent evolution, and the implications of both relative to the Earth remain subject to lively debate. Because the internal geochemistry and geophysics of the Moon does not appear consistent with an origin by the giant impact of a Mars-sized asteroid on the Earth, this hypothesis is challenged by one that proposes the capture of an independently formed planetesimal. The Moon—s internal structure also indicates that it and all the terrestrial planets initially had relatively cool, chondritic proto-cores prior to formation of metallic cores. Evidence exists that these proto-cores delayed formation of metallic cores for periods that correlate with the final mass of a planet.

The impact history of the inner solar system has been broadly outlined by the modern investigation of the Moon. First, soon after the formation of a coherent lunar crust and against an intense background of smaller cratering events, the Moon was subjected to extremely large impacts that formed basins up to 3200km in diameter. On Earth, the melt sheets from these continental-scale impacts may have been responsible for the formation of the first continental crust at ∼4.4 Gyr. Second, ∼50 impact basins >300 km in diameter formed between 4.5 and 3.8 Gyr probably by pulses of impactors produced during the migration and interaction of the giant planets within a structured solar disk of planetesimal rings. The last of these pulses at about 3.85 Gyr, producing the ∼14 mascon basins, resurfaced most of the Moon, and suggested an apparent “cataclysm” at that time. This period of 700 Myr may have been one of “punctuated cataclysm” as one or more giant planets encountered separate planetesimal rings and gaps during outward orbital migration. Finally, the implication of this violent impact history in the inner solar system prior to 3.8 Gyr relative to the surfaces of the hydrous terrestrial planets, that is, Earth, Mars and probably early Venus, is that clays were the dominant mineral species. These clays, as well as volcanic sulfides, may have provided the templates for the formation of complex organic precursors that made up the first living cells.

Part I - Notes on String Topology | Pp. 79-87

Preliminaries

Ralph L. Cohen; Kathryn Hess; Alexander A. Voronov

The origin of the Moon 4.56 Gyr ago, its subsequent evolution, and the implications of both relative to the Earth remain subject to lively debate. Because the internal geochemistry and geophysics of the Moon does not appear consistent with an origin by the giant impact of a Mars-sized asteroid on the Earth, this hypothesis is challenged by one that proposes the capture of an independently formed planetesimal. The Moon—s internal structure also indicates that it and all the terrestrial planets initially had relatively cool, chondritic proto-cores prior to formation of metallic cores. Evidence exists that these proto-cores delayed formation of metallic cores for periods that correlate with the final mass of a planet.

The impact history of the inner solar system has been broadly outlined by the modern investigation of the Moon. First, soon after the formation of a coherent lunar crust and against an intense background of smaller cratering events, the Moon was subjected to extremely large impacts that formed basins up to 3200km in diameter. On Earth, the melt sheets from these continental-scale impacts may have been responsible for the formation of the first continental crust at ∼4.4 Gyr. Second, ∼50 impact basins >300 km in diameter formed between 4.5 and 3.8 Gyr probably by pulses of impactors produced during the migration and interaction of the giant planets within a structured solar disk of planetesimal rings. The last of these pulses at about 3.85 Gyr, producing the ∼14 mascon basins, resurfaced most of the Moon, and suggested an apparent “cataclysm” at that time. This period of 700 Myr may have been one of “punctuated cataclysm” as one or more giant planets encountered separate planetesimal rings and gaps during outward orbital migration. Finally, the implication of this violent impact history in the inner solar system prior to 3.8 Gyr relative to the surfaces of the hydrous terrestrial planets, that is, Earth, Mars and probably early Venus, is that clays were the dominant mineral species. These clays, as well as volcanic sulfides, may have provided the templates for the formation of complex organic precursors that made up the first living cells.

Part II - An Algebraic Model for Mod 2 Topological Cyclic Homology | Pp. 103-113

Free loop spaces

Ralph L. Cohen; Kathryn Hess; Alexander A. Voronov

The origin of the Moon 4.56 Gyr ago, its subsequent evolution, and the implications of both relative to the Earth remain subject to lively debate. Because the internal geochemistry and geophysics of the Moon does not appear consistent with an origin by the giant impact of a Mars-sized asteroid on the Earth, this hypothesis is challenged by one that proposes the capture of an independently formed planetesimal. The Moon—s internal structure also indicates that it and all the terrestrial planets initially had relatively cool, chondritic proto-cores prior to formation of metallic cores. Evidence exists that these proto-cores delayed formation of metallic cores for periods that correlate with the final mass of a planet.

The impact history of the inner solar system has been broadly outlined by the modern investigation of the Moon. First, soon after the formation of a coherent lunar crust and against an intense background of smaller cratering events, the Moon was subjected to extremely large impacts that formed basins up to 3200km in diameter. On Earth, the melt sheets from these continental-scale impacts may have been responsible for the formation of the first continental crust at ∼4.4 Gyr. Second, ∼50 impact basins >300 km in diameter formed between 4.5 and 3.8 Gyr probably by pulses of impactors produced during the migration and interaction of the giant planets within a structured solar disk of planetesimal rings. The last of these pulses at about 3.85 Gyr, producing the ∼14 mascon basins, resurfaced most of the Moon, and suggested an apparent “cataclysm” at that time. This period of 700 Myr may have been one of “punctuated cataclysm” as one or more giant planets encountered separate planetesimal rings and gaps during outward orbital migration. Finally, the implication of this violent impact history in the inner solar system prior to 3.8 Gyr relative to the surfaces of the hydrous terrestrial planets, that is, Earth, Mars and probably early Venus, is that clays were the dominant mineral species. These clays, as well as volcanic sulfides, may have provided the templates for the formation of complex organic precursors that made up the first living cells.

Part II - An Algebraic Model for Mod 2 Topological Cyclic Homology | Pp. 115-137

Homotopy orbit spaces

Ralph L. Cohen; Kathryn Hess; Alexander A. Voronov

The origin of the Moon 4.56 Gyr ago, its subsequent evolution, and the implications of both relative to the Earth remain subject to lively debate. Because the internal geochemistry and geophysics of the Moon does not appear consistent with an origin by the giant impact of a Mars-sized asteroid on the Earth, this hypothesis is challenged by one that proposes the capture of an independently formed planetesimal. The Moon—s internal structure also indicates that it and all the terrestrial planets initially had relatively cool, chondritic proto-cores prior to formation of metallic cores. Evidence exists that these proto-cores delayed formation of metallic cores for periods that correlate with the final mass of a planet.

The impact history of the inner solar system has been broadly outlined by the modern investigation of the Moon. First, soon after the formation of a coherent lunar crust and against an intense background of smaller cratering events, the Moon was subjected to extremely large impacts that formed basins up to 3200km in diameter. On Earth, the melt sheets from these continental-scale impacts may have been responsible for the formation of the first continental crust at ∼4.4 Gyr. Second, ∼50 impact basins >300 km in diameter formed between 4.5 and 3.8 Gyr probably by pulses of impactors produced during the migration and interaction of the giant planets within a structured solar disk of planetesimal rings. The last of these pulses at about 3.85 Gyr, producing the ∼14 mascon basins, resurfaced most of the Moon, and suggested an apparent “cataclysm” at that time. This period of 700 Myr may have been one of “punctuated cataclysm” as one or more giant planets encountered separate planetesimal rings and gaps during outward orbital migration. Finally, the implication of this violent impact history in the inner solar system prior to 3.8 Gyr relative to the surfaces of the hydrous terrestrial planets, that is, Earth, Mars and probably early Venus, is that clays were the dominant mineral species. These clays, as well as volcanic sulfides, may have provided the templates for the formation of complex organic precursors that made up the first living cells.

Part II - An Algebraic Model for Mod 2 Topological Cyclic Homology | Pp. 139-151

A model for mod 2 topological cyclic homology

Ralph L. Cohen; Kathryn Hess; Alexander A. Voronov

The origin of the Moon 4.56 Gyr ago, its subsequent evolution, and the implications of both relative to the Earth remain subject to lively debate. Because the internal geochemistry and geophysics of the Moon does not appear consistent with an origin by the giant impact of a Mars-sized asteroid on the Earth, this hypothesis is challenged by one that proposes the capture of an independently formed planetesimal. The Moon—s internal structure also indicates that it and all the terrestrial planets initially had relatively cool, chondritic proto-cores prior to formation of metallic cores. Evidence exists that these proto-cores delayed formation of metallic cores for periods that correlate with the final mass of a planet.

The impact history of the inner solar system has been broadly outlined by the modern investigation of the Moon. First, soon after the formation of a coherent lunar crust and against an intense background of smaller cratering events, the Moon was subjected to extremely large impacts that formed basins up to 3200km in diameter. On Earth, the melt sheets from these continental-scale impacts may have been responsible for the formation of the first continental crust at ∼4.4 Gyr. Second, ∼50 impact basins >300 km in diameter formed between 4.5 and 3.8 Gyr probably by pulses of impactors produced during the migration and interaction of the giant planets within a structured solar disk of planetesimal rings. The last of these pulses at about 3.85 Gyr, producing the ∼14 mascon basins, resurfaced most of the Moon, and suggested an apparent “cataclysm” at that time. This period of 700 Myr may have been one of “punctuated cataclysm” as one or more giant planets encountered separate planetesimal rings and gaps during outward orbital migration. Finally, the implication of this violent impact history in the inner solar system prior to 3.8 Gyr relative to the surfaces of the hydrous terrestrial planets, that is, Earth, Mars and probably early Venus, is that clays were the dominant mineral species. These clays, as well as volcanic sulfides, may have provided the templates for the formation of complex organic precursors that made up the first living cells.

Part II - An Algebraic Model for Mod 2 Topological Cyclic Homology | Pp. 153-159