Elliptic Cohomology

200 Pages·2002·15.459 MB·English

by  Charles B. Thomas (auth.)

Elliptic Cohomology
THE UNIVERSITY SERIES IN MATHEMATICS
Series Editors: Sylvain E. Cappell, New York University
Joseph J. Kohn, Princeton University
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Elliptic Cohomology
Charles B. Thomas
University of Cambridge
Cambridge, England
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Preface
Elliptic Cohomology was developed in the mid-1980s following the discovery
that a multiplicative genus defined on oriented cobordism, localized away from
the prime 2, has a surprising relation to Jacobi elliptic functions. After an
initial flurry of activity, interest waned in the new theory, partly because of
slowness in the appearance of applications but mostly because of the lack of
a satisfactory bundle-theoretic model. In one direction at least, the situation
is now more promising — moonshine phenomena of interest to finite group
theorists provide the means to give a usable geometric definition at least
for the classifying space of a finite group, and repaying the compliment,
elliptic cohomology appears to offer a good framework in which to formulate
moonshine for a preferred family of simple groups. As a stimulus to further
activity, it seems right to me to attempt to give a survey, however provisional, of
what is currently known. It has been plausibly claimed also that one variant
of the theory captures much of what is known about the stable homotopy
groups of spheres; it is my own hope that with more geometric input, elliptic
cohomology may resolve some of the open questions, which seem just beyond
the reach of K-theory.
This survey is an expanded version of lectures that I originally gave at
Ohio State University in the spring quarter, 1993. These have circulated for
some time in handwritten form, and I hope that the present version is both
clearer and more complete. Prerequisites are a good knowledge of algebraic
topology and the theory of finite groups. Some acquaintance with the theory
of modular forms will also be of assistance. In writing the book I have
increasingly realized that it is a survey in the sense that I have attempted
to give the flavor of what is known and that in many places I have referred
the reader to more detailed texts. In a subject that straddles several areas
of mathematics, this may be inevitable, but if the book encourages further
research along the lines I have indicated, I will be more than content.
V
vi Preface
As with my earlier book on the cohomology of finite groups, this book is
dedicated with love and gratitude to my wife, Maria.
Cambridge, September 1998.
Contents
Introduction ................................... 1
1. Elliptic Genera ............................... 7
1.1. Oriented Cobordism Ring . . . . . . . . . . . . . . . . . . . 7
1.2. Genera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3. Strong Multiplicativity . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4. Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2. Cohomology Theory Ell*(X) ......................... 23
2.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2. Complex-Oriented Cohomology Theories . . . . . . . . . . . . . 25
2.3. Baas–Sullivan Construction . . . . . . . . . . . . . . . . . . . . . 27
2.4. Construction of and Ell* (X) away from the Prime 2 . . 29
2.5. Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3. Work of M. Hopkins, N. Kuhn, and D. Ravenel ............ 35
3.1. Bundlesover the Classifying Space BG . . . . . . . . . . . . . . 35
3.2. General Character Theory . . . . . . . . . . . . . . . . . . . . . . . 36
3.3. Character Rings for h* (BG) . . . . . . . . . . . . . . . . . . . . . 38
3.4. Lubin–Tate Construction . . . . . . . . . . . . . . . . . . . . . . . . 39
3.5. Reduction of the Theorem to the Cyclic Case . . . . . . . . . . . 42
3.6. Proof of Proposition 3.3 . . . . . . . . . . . . . . . . . . . . . . . . 43
3.7. Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4. Mathieu Groups ................................. 49
4.1. Construction of Mathieu Groups . . . . . . . . . . . . . . . . . . 49
4.2. Conjugacy Classes and Modular Forms according to
G. Mason . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
vii
viii Contents
4.3. Eigenforms for Hecke Operators . . . . . . . . . . . . . . . . . . 54
4.4. Eight Elliptic Genera of Mathieu Type . . . . . . . . . . . . . . . 56
4.5. Thompson Series and Ramanujan Numbers . . . . . . . . . . . . 58
4.6. Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5. Cohomology of Certain Simple Groups ................. 61
5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2. Remaining Mathieu Groups . . . . . . . . . . . . . . . . . . . . . . . 71
5.3. Groups and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.4. Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6. Ell*(BG) — Algebraic Approach .................... 79
6.1. Mackey and Green Functors . . . . . . . . . . . . . . . . . . . . . . 79
6.2. Generalized Group Cohomology . . . . . . . . . . . . . . . . . . . 82
6.3. Morava K -Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.4. Rank Two p-Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.5. Groups of Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.6. Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7. Completion Theorems ........................... 103
7.1. Equivariant Coefficient Ring . . . . . . . . . . . . . . . . . . 104
7.2. Example: Metacyclic p-Groups . . . . . . . . . . . . . . . . . . . . 112
7.3. Equivariant Elliptic Cohomology of G-Complexes . . . . . . . . 112
7.4. Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
8. Elliptic Objects ............................... 119
8.1. Quantum Field Theories . . . . . . . . . . . . . . . . . . . . . . . . 119
8.2. Free Loop Space LBG . . . . . . . . . . . . . . . . . . . . . . . . . 123
8.3. EllipticSystem for . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
8.4. Bundles over Arbitrary Loop Spaces LX . . . . . . . . . . . . . . 131
8.5. Constructing Moonshine-like Virasoro-Equivariant Vector
Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
8.6. Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
9. Variants of Elliptic Cohomology ..................... 143
9.1. Elliptic Genera of Level N . . . . . . . . . . . . . . . . . . . . . . 144
9.2. Projective Plane Functors . . . . . . . . . . . . . . . . . . . . . . . 147
9.3. Atiyah Invariant and the Ochanine Genus . . . . . . . . . . . . . 149
9.4. Kernel and Images of the Ochanine Genus . . . . . . . . . . . . . 150
9.5. Localization of at the Prime 2 . . . . . . . . . . . . . . . . . 153
9.6. Introduction to the Spectrum . . . . . . . . . . . . . . . . . . . . 156
9.7. Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .157
Contents ix
10. K3-Cohomology .............................. 159
10.1. Toward a Homology Theory . . . . . . . . . . . . . . . . . 159
10.2. Abelian Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
10.3. K3-Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
10.4. Siegel ModularForms and Open Questions . . . . . . . . . . . . 173
10.5. Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
A. Brown–Peterson Cohomology ...................... 179
B. Cayley Projective Plane ....................... 183
C. Index of ........................187
References .................................... 191
Index ....................................... 197
Introduction
Elliptic cohomology combines ideas and results from algebraic topology, arith-
metic, representations of finite groups and theoretical physics in the shape of
topological field theories. From the algebraic point of view, it is a quotient of spin
cobordism, and as such it forms part of a chain:
with each link corresponding to a 1 -dimensional commutative formalgroup law. In
the case of ellipticcohomology this was written down by Euler in the eighteenth
century, and the validity of Eilenberg–Steenrod axioms for the corresponding
cohomology theory follows from properties of addition on a class of elliptic
curves in characteristic p. Universality of the formal group law for cobordism
implies that there is a corresponding ring homomorphism or genus taking values
in the coefficients of the theory concerned.
In 1988 G. Segal gave a talk in the Bourbaki Seminar [101] in which he
summarized what was known at the time under the headings:
(cid:127) Ell*(X) is a cohomology theory [69, 70].
(cid:127) The structural genus is rigid with respect to compact, connected group
actions [25].
(cid:127) The completed localized ring is determined by elliptic characters
[56].
Segal also explained that a genus related to the universal elliptic genus should
be defined as the index of a Dirac operator in infinite dimensions and suggested
that a geometric model for Ell*(X) could be constructed by using ideas from
conformal field theory.
We devote Chaps. 1–3 to a survey of this material. The first two headings are
related, since we can use rigidity to prove the exactness of Ell* (X). As Segal
1