The Relation of Cobordism to K-Theories
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P. E. Conner · E. E. Floyd University of Virginia, Charlottesville
The Relation of Cobordism to K-Theories 1966
Springer-Verlag· Berlin· Heidelberg· New York
All rights, especially that of translation into foreign languages, reserved. It is also forbidden to reproduce this book, either whole or in part, by photomechanical means (photostat, microfilm and/or microcard) or by other procedure without written permission from Springer Verlag. (l:) by Springer-Verlag Berlin' Heidelberg 1966. Library of Congress Catalog Card Number 66-30143. Printed in Germany. Title No. 7348.
INTRODUCTION These lectures treat certain topics relating K-theory and cobordism. Since new connections are in the process of being discovered by various workers, we make no attempt to be definitive but simply cover a few of our favorite topics.
If there is any unified theme it is that we treat
various generalizations of the Todd genus. In Chapter I we treat the Thom isomorphism in K-theory.
The
families U, SU, Sp of unitary, special unitary, symplectic groups generate spectra MU, MSU, MSp of Thom spaces.
In the fashion of
G. W. Whitehead [26J, each spectrum generates a generalized cohomology theory and a generalized homology theory.
The cohomology theories are
a (.)
* denoted by.a * (.), £). * (.), and are called cobordism theories; U SU Sp U SU Sp the homology theories are denoted by {}. (.), 1l. (.), fl (.) and are called bordism theories.
n
*
*
*
The coefficient groups are, taking one case
a Un = .()..nu
as an example, given by un = 11 un (point), U U n related by..o = 11 u- • Moreover.o. is just n n bordism classes [MnJ of closed weakly almost
(point) and are
the bordism group of all complex manifolds Mn ,
similarly forll SU andll. SP. on the other hand there are the n n Grothendieck-Atiyah-Hirzebruch periodic cohomology theories K* (·),KO *
of K-theory.
The main point of Chapter I, then, is to
define natural transformations
-+
KO*(.)
: 0. *{.) -4
K*(.)
!"': I1
*(.)
SU
U
of cohomology theories.
Such transformations have been folk theorems
since the work of Atiyah-Hirzebruch [6], Dold [13], and others.
It
should be noted that on the coefficient groups, 1\
fV c
:Jr..L.
-2n U
-2n
K
lpt)
is identified up to sign with the Todd genus Td
=z
. .0. U
z.
2n In Chapter II we show among other things that the cobordism
theories determine the K-theories. ring
For example,
Z and makes Z into
generates a
c
It is
shown that K* (X,A)
as Z2-graded modules. KO * l·).
.0..* lX,A)@ .0 * U
U
Z
Similarly symplectic cobordism determines
The isomorphisms are generated by
t"c' F
respectively.
Various topics are treated along the way, in particular cobordism characteristic classes. There is the sphere 1\
framed bordism
J is nembedded
lPoint)
n The spectrum form
MUI 4.
=
whose homology groups are the fr
l·).
The coefficient group
are just the stable stems 7T
n+k in a natural way in MU, and one can thus
In Chapter III we s tudy the group
.0. u, fr = 7T
=
[-:
if Y
=X
if Y = -X otherwis
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