Smooth S1 Manifolds
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557 Wolf Iberkleid Ted Petrie
Smooth S 1 Manifolds
Springer-Verlag Berlin. Heidelberg. New York 1976
Authors
Wolf Iberkleid C E N T R O de INV. del IPN Apdo. Postal 1 4 7 4 0 Mexico, 14 DF/Mexico
Ted Petrie Rutgers University Department of Mathematics N e w B r u n s w i c k N. J./USA
Library of Congress Cataloging in Publication Data
roerkleid, Wolf, 1946Smooth S 1 manifolds. (Lecture notes in mathematics ; 557) Bibliogr~,ph=¢: p. Includes index. i. Differential topolo~TJ. 2. ~v~nifolds (Mathematics) 3. Characteristic classes. 4. Topological transform&tion groups. I. Petrie~ Ted, 1939joint author, llo Title. III~ Series: Lecture ~.otes in mathematics (Berlin) ; 557. QA3oTP8 no. 557 [QA613.6] 510'.8s [514'o7] 76-50065
AMS Subject Classifications (1970): 57D20, 57D65, 57E25, 5 5 B 2 5
ISBN 3-540-08002-3 ISBN 0-387-08002-3
Springer-Verlag Berlin- Heidelberg" New York Springer-Verlag New York • Heidelberg • Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin. Heidelberg 1976 Printed in Germany. Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.
Smooth
S1
Manifolds
by Wolf Iberkleid and Ted Petrie Introduction. Part i, Part iT, Consequences of non-singularity,
Recent developments. PART I.
The Algebraic Tools
1
25
1. Preliminary remarks on G spaces 2. Structure of smooth G manifolds
26 in terms of handles
46
3. Multiplicative properties of h~( )
54
4. Fixed point free actions
61
5. The universal coefficient theorem
74
6. Poincar~ duality
83
References
PART II.
A setting for smooth
99
SI
actions
lOl
I. Introduction and notation
102
2. Specifics about S 1
105
3. The relation between completion and localization
109
4. Remarks on spin c structures
ll2
5- The induction homomorphism
ll6
6. Differential Structure and the representations TXp
ll9
7. The topology of some real algebraic varieties
122
8. Real algebraic actions on P(C 4n)
125
9. The case of finite isotropy groups
130
Io. Induction
146
References
160
Symbol index
161
INTRODUCTION Part
I We single
complex
representation
(localization
1
at
manifold
underlying
G.
Here The
of
in more
Z.
the
prime
ideals
of
R ( S I)
prime
ideals
of
R(S
R = R ( S I) primes of
P.
R(SI),
prime
denote
P
1
to all
the ideals
R = R( S I) ~ Q. Z
of
the R ( S l)
if
is a spin
F
P.
denotes
and to study
compact
integers
Z.
generated denotes
is d e n o t e d of
the
C
interest Lie groups without
of the b i l i n e a r
PZ
consists in
P
that w h e n
we do so.
P n PZ = ~"
minus
smooth closed
manifolds
that
the l o c a l
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