Table Of ContentDifferential Algebras in Topology
Research Notes in Mathematics
Volume 3
Differential Algebras in Topology
David Anick
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, Massachusetts
Boca Raton London New York
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ISBN 13: 978-1-56881-001-0 (hbk)
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Library of Congress Cataloging-in-Publication Da1a
Anick, David Jay.
Differential algebras in topology / David Anick.
p. cm.
Includes bibliographic references.
ISBN 1-56881-001-6
1. Algebraic topology. 2. Complexes. I. Title.
QA612.A76 1993
514' .2-dc20 93-12604
CIP
To Flora and George
Who always encouraged me
In my love of math
Tay or & Francis
&
Taylor Francis Group
http://taylorandfrancis.com
AUTHOR’S FOREWORD
For the gifts and the call of God are irrevocable.
Romans 11:29
I offer the ideas in this book as my parting gift to the mathematical community. Re
sponding to what I think is an irrevocable call, I am leaving to become a medical doctor.
As I write, my medical training is set to commence in less than a week. I want to share
my reflections on the process of creating the book, and to acknowledge the assistance of
others.
The ideas in this monograph have been brewing inside me on an intuitive level for
several years. I didn’t want them simply to be lost when I stopped doing mathematics.
Motivated by the desire to leave with a minimum of loose ends, I undertook the project of
writing it up.
When I started, I had no clue that what I envisioned as a sixty- to eighty-page paper
would blossom into a three hundred page book! The turning point came at the end of
the fall semester of 1990, when I finally grasped that the classical techniques I had been
counting upon — slight generalizations of those of Cohen-Moore-Neisendorfer — might
be inadequate for the task. Gradually my focus shifted. Instead of seeing the problem (of
constructing £>*) as principally one of clever combinatorics and well-coordinated equations,
I began to think in terms of larger questions and in terms of the development of wholly
new methods. The result is what you see before you.
Every chapter of the book was written or substantially rewritten during the four-month
period of May through August of 1991. This haste shows, I know, in the book’s many rough
edges — chapters of greatly unequal length, an inconsistent labeling system for results and
formulas, large quantities of notation, minimal provision of background, and so on. For
these weaknesses I ask the reader’s forbearance and forgiveness. Nearly all of the results,
including most of the concepts and definitions, were unknown to me just five months ago.
I had to concentrate on working out the details, and the larger picture correspondingly
suffered.
There is nothing like an absolute deadline to motivate someone. During this summer
viii DIFFERENTIAL ALGEBRAS IN TOPOLOGY
of writing I have been constantly aware that it was “now or never” for developing these
ideas. I recalled frequently the image of Evariste Galois during the final hours of his life,
frantically writing down as much as he could of his mathematical ideas so that the ideas
would not die too. I have never before worked so intensely, nor with so single a focus.
In spite of the exhaustion and the sacrifices, the process has been satisfying. For a
few months I have given myself over entirely to the pursuit of new knowledge, immersing
myself in a totally abstract yet utterly beautiful portion of All That Is. I will never regret
this experience, and I am grateful for the opportunity to have had it.
It has been said that in a field of wildflowers, an entire book could be written about
any one of them. As I have dwelled in the world of the TVs, I have simultaneously been
aware that even this fascinating complex world represents but one flower in the vast field
of mathematical truth. Still, I hope that my readers will agree the story of the ZVs is
a story that should be told, and that others will be able to appreciate the beauty and
intricacy of this particular flower.
There are many who helped to make this book a reality. Financial and institutional
support came from the Massachusetts Institute of Technology, the University of Rochester,
the Sloan Foundation, and the National Science Foundation. Technical typing was done
by Joan Robinson, Margaret Beucler, Robert Becker, and Judy Romvos.
For appreciation I wish to single out Brayton Gray and Steven Halperin, without whose
staunch encouragement I might well have quit the entire project at any number of points.
Those with whom my mathematical discussions directly influenced the content of the book
include Fred Cohen, Brayton Gray, Mike Hopkins, Haynes Miller, Joe Neisendorfer, Bill
Richter, and Paul Selick. To these individuals, as well as to the many others who helped
me to feel valued and successful during my eleven years as a professional mathematician,
I extend my profound gratitude.
TABLE OF CONTENTS
Abstract ............................................................................................................................ xi
Preface ............................................................................................................................ xiii
Summary of Global Notations ................................................................................. xix
1. Introduction and Mathematical Overview ........................................................ 1
The Cohen-Moore-Neisendorfer (CMN) exponent theorem for spheres. Review of the
CMN program for the Moore space. CMN diagrams. Generalizing the CMN program:
what new techniques are needed and why? Chapter by chapter overview of the highlights
of the book.
PART Is PROPERTIES OF SPECIFIC DGL’s
2. The DGL’s Lk and I* ............................................................................................ 15
Hopf, Lie, and differential graded algebras. Hopf algebra kernels. Altered binomial
coefficients. The special dgL’s Lk and L , and the homology of their enveloping algebras.
♦-Degree.
3. The DGL Mk ........................................................................................................... 27
Construction of Mk and its relations with L* and Lk. Adams-Hilton models for D*
and G*.
4. The Bockstein Spectral Sequence for UMk ................................................... 35
A polynomial trick. The special cycles {c5}. Two ways of looking at UMk. The
homology of UMk.
5. The DGL Nkl ......................................................................................................... 49
Hopf commutators. DGL’s with linear differentials. Definition and properties of Nkl.
The folding homomorphism V.
PART II: ALGEBRAIC AND TOPOLOGICAL PREPARATIONS
6. Three Topological Tools ....................................................................................... 67
Review of odd primary homotopy theory: exact sequences, Bocksteins, Samelson prod
ucts, Hurewicz homomorphism. Adams-Hilton models: definitions, dga homotopy, exten
sion problem. Homotopy class extension theorem and corollaries. Cyclic cotensor product
of three dgH’s. Thin product of three spaces.
