Homology of Classical Groups Over Finite Fields and Their Associated Infinite Loop Spaces
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674 Zbigniew Fiedorowicz Stewart Priddy
Homology of Classical Groups Over Finite Fields and Their Associated Infinite Loop Spaces
Springer-Verlag Berlin Heidelberg New York 1978
Authors Zbigniew Fiedorowicz Department of Mathematics University of Michigan Ann Arbor, Michigan 48104/USA Stewart Priddy Department of Mathematics Northwestern University Evanston, Illinois 60201lUSA
AMS Subject Classifications (1970): 18HlO, 20G40, 55D35, 55F40
ISBN 3-540-08932-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-08932-2 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 1978 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
Table of Contents
Introduction............ ..•.•... .. .. .. ..•. .. ....•.. ..•.. .. .. .
1
Chapter I
Infinite Loop Spaces Associated with ImJ..........
9
1.
Introduction...........................................
9
2.
The real image of
spaces............... ..........••
10
3.
The mod-2 homology of JO(q)...... ..... ...............•.
17
4.
The symplectic and complex image of
20
J
J
spaces... .....•
The odd primary homology of JO( q) •.........•.....•...•. The odd primary homology of JSp(q) and JU(±q)" ........ The integral cohomology of JO (q) •...........•........•.
8.
The integral cohomology of JSp(q) and JU(±q) .•.......••
Chapter II
45
permutative categories of Classical Groups over Finite Fields....................................
49
1.
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . • . . . . . . • . . . . . • • • •
49
2.
permutative categories and their associated infinite loop spaces............................................
49
3.
The orthogonal groups and their subgroups..............
67
4.
Quadratic forms and extraordinary orthogonal groups....
82
5.
Spinor groups over finite fields.......................
97
6.
The general linear, symplectic and unitary groups......
110
7.
Orthogonal groups over finite fields of characteri stic 2.......................................
8.
117
Permutative functors associated with the classical groups.. . . . • . . . . . .• . . .. . • . . . . . . . . . . . .••• • . . • .• . .. . • . . ••
145
IV
Chapter III
K-Theory of Finite Fields and the ImJ spaces....
166
1.
Introduction. . . . . . . . . . . . . . . . . . . • . . . • . • . . . • . . . • . . . . . . . . •
166
2.
The Brauer lift........................................
166
3.
The main results.......................................
176
4.
Mod
186
5.
General pattern
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