Ebook: Elliptic Cohomology

Elliptic cohomology is an extremely beautiful theory with both geometric and arithmetic aspects. The former is explained by the fact that the theory is a quotient of oriented cobordism localised away from 2, the latter by the fact that the coefficients coincide with a ring of modular forms. The aim of the book is to construct this cohomology theory, and evaluate it on classifying spaces BG of finite groups G. This class of spaces is important, since (using ideas borrowed from `Monstrous Moonshine') it is possible to give a bundle-theoretic definition of EU-(BG). Concluding chapters also discuss variants, generalisations and potential applications.

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Elliptic cohomology is an extremely beautiful theory with both geometric and arithmetic aspects. The former is explained by the fact that the theory is a quotient of oriented cobordism localised away from 2, the latter by the fact that the coefficients coincide with a ring of modular forms. The aim of the book is to construct this cohomology theory, and evaluate it on classifying spaces BG of finite groups G. This class of spaces is important, since (using ideas borrowed from `Monstrous Moonshine') it is possible to give a bundle-theoretic definition of EU-(BG). Concluding chapters also discuss variants, generalisations and potential applications.

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Elliptic cohomology is an extremely beautiful theory with both geometric and arithmetic aspects. The former is explained by the fact that the theory is a quotient of oriented cobordism localised away from 2, the latter by the fact that the coefficients coincide with a ring of modular forms. The aim of the book is to construct this cohomology theory, and evaluate it on classifying spaces BG of finite groups G. This class of spaces is important, since (using ideas borrowed from `Monstrous Moonshine') it is possible to give a bundle-theoretic definition of EU-(BG). Concluding chapters also discuss variants, generalisations and potential applications.
Content:
Front Matter....Pages i-ix
Introduction....Pages 1-5
Elliptic Genera....Pages 7-21
Cohomology Theory Ell * (X) ....Pages 23-33
Work of M. Hopkins, N. Kuhn, and D. Ravenel....Pages 35-47
Mathieu Groups....Pages 49-60
Cohomology of Certain Simple Groups....Pages 61-77
Ell * (BG) — Algebraic Approach....Pages 79-101
Completion Theorems....Pages 103-117
Elliptic Objects....Pages 119-142
Variants of Elliptic Cohomology....Pages 143-158
K3-Cohomology....Pages 159-178
Back Matter....Pages 179-199

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Elliptic cohomology is an extremely beautiful theory with both geometric and arithmetic aspects. The former is explained by the fact that the theory is a quotient of oriented cobordism localised away from 2, the latter by the fact that the coefficients coincide with a ring of modular forms. The aim of the book is to construct this cohomology theory, and evaluate it on classifying spaces BG of finite groups G. This class of spaces is important, since (using ideas borrowed from `Monstrous Moonshine') it is possible to give a bundle-theoretic definition of EU-(BG). Concluding chapters also discuss variants, generalisations and potential applications.
Content:
Front Matter....Pages i-ix
Introduction....Pages 1-5
Elliptic Genera....Pages 7-21
Cohomology Theory Ell * (X) ....Pages 23-33
Work of M. Hopkins, N. Kuhn, and D. Ravenel....Pages 35-47
Mathieu Groups....Pages 49-60
Cohomology of Certain Simple Groups....Pages 61-77
Ell * (BG) — Algebraic Approach....Pages 79-101
Completion Theorems....Pages 103-117
Elliptic Objects....Pages 119-142
Variants of Elliptic Cohomology....Pages 143-158
K3-Cohomology....Pages 159-178
Back Matter....Pages 179-199
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