Lecture Notes in Mathematics A collection of informal reports and seminars
Edited by A. Dold, Heidelberg and B. Eckmann, ZOrich
299
Proceedings of the Second Conference on Compact Transformation Groups University of Massachusetts, Amherst, 1971
Part II
ISBN 3-540-06078-2 Springer-Verlag Berlin' Heidelberg· New York ISBN 0-387-06078-2 Springer-Verlag New York· Heidelberg· Berlin
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COMMENTS BY THE EDITORS
The Second Conference on Compact Transformation Groups was held at the University of Massachusetts, Amherst from June 7 to June 18, 1971 under the sponsorship of the Advanced Science Education Program of the National Science Foundation. There were a total of 70 parti­ cipants at the conference.
As was the case at the first conference at Tulane University in 1967, the emphasis continued to be on differentiable transformation groups. In this connection there was a continued application of surgery typified by the lectures of Browder, Shaneson, and Yang (joint work with Montgomery). A new feature was the applications of the Atiyah-Singer Index Theorem to differentiable transformation groups typified by the lectures of Hinrichsen, Petrie, and Rothenberg. In connection with topological and algebraic methods significant innovations were made by Raymond (joint work with Conner) in the con­ struction of manifolds admitting no effective finite group action, by R. Lee in studying free actions of finite groups on spheres using ideas and methods derived from algebraic K-theory and by Su (joint work with W.Y. Hsiang) in applying the notion of geometric weight systems developed recently by W. Y. Hsiang. There were several lec­
tures on algebraic varieties by Michael Davis, Peter Orlik, and Philip Wagreich. Interest in this area arose from the application several years ago of Brieskorn varieties to the study of actions on
homotopy spheres. These Proceedings contain not only material presented at the con-
ference but also articles received by the editors up to the summer of 1972. We have divided the articles into two volumes; the first volume is devoted to smooth techniques while the second to non-smooth tech­ niques. While the proper aSSignment of a few papers was not obvious, the editors felt that this classification offered, in general, the
most natural division of the material.
H. T. Ku L. N. Mann
J. L. Sicks
J. C. Su
Amherst, Mass., July 1972
E. M. Bloomberg J. M. Boardman G. Bredon W. Browder H. Cohen F. Connolly Bruce Conrad W. D. Curtis Michael Davis Murray Eisenberg Dieter ErIe I. Fary L. A. Feldman Gary Ford V. Giambalvo David Gibbs R. Goldstein M. Goto E. Grove Stephen Halperin Gary Hamrick Douglas Hertz John Hinrichsen Edward Hood Norman Hurt Sl::iren I1lman Stephen Jones Louis Kauffman K. Kawakubo S. K. Kim Larry King S. Kinoshita Robert Koch H. T. Ku Mei Ku K. W. Kwun
Timothy Lance Connor Lazarov R. Lee
CONFERENCE PARTICIPANTS University of Virginia Johns Hopkins University Rutgers University Princeton University University of Massachusetts Notre Dame University Temple University Kansas State University Princeton University University of Massachusetts University of Dortmund University of California Stanislaus State College Radford College University of Connecticut University of Virginia SUNY at Albany University of Pennsylvania University of Rhode Island University of Toronto Institute for Advanced Study University of Massachusetts University of Virginia M.LT.
University of Massachusetts Princeton University University of Massachusetts University of Illinois at Chicago Osaka University University of Connecticut University of Massachusetts Florida State University Louisiana State University University of Massachusetts University of Massachusetts Michigan State University Princeton University Lehman College Yale University
L. Lininger R. Livesay S. L6pez de Medrano Larry Mann Deane Montgomery P. Orlik E. Ossa J. Pak Ted Petrie F. Raymond Richard Resch Robert Rinne M. Rothenberg Loretta J. Rubeo H. Samelson James Schafer V. P. Schneider Reinhard Schultz J. Shaneson Jon Sicks J. C. Su Richard Summerhill Mel Thornton Philip Wagreich Shirley Wakin H. C. Wang Kai Wang A. Wasserman Steven Weintraub J. E. West C. T. Yang
VI
SUNY at Albany Cornell University University of Mexico University of Massachusetts Institute for Advanced Study University of Wisconsin University of Bonn Wayne State University Rutgers University University of Michigan University of Connecticut Sandia Laboratory, Livermore, California University of Chicago University of Virginia Stanford University University of Maryland S. W. Louisiana University Purdue University Princeton University University of Massachusetts University of Massachusetts Institute for Advanced Study University of Nebraska University of Pennsylvania University of Massachusetts Cornell University University of Chicago University of Michigan Princeton University Cornell University University of Pennsylvania
J. Boardman G. Bred on
w. Browder M. Davis D. Erle I. Fary L. Feldman V. Giambalvo
R. Goldstein
S. Kinoshita
E. Ossa
T. Petrie
CONFERENCE LECTURE TITLES Cobordism of Involutions Revisited Strange Circle Actions on Products of Spheres, and Rational Homotopy Equivariant Differential Topology Actions on Exotic Stiefel Manifolds On Unitary and Symplectic Knot Manifolds Group Action and Betti Sheaf Reducing Bundles in Differentiable G-spaces Cobordism of Line Bundles with Restricted Characteristic Class Free Differentiable Circle Actions on 6-Manifolds Orbits of One-Parameter Groups Classical Group Actions on Manifolds with Vanishing First and Second Integral Pontrjagin Classes Operators Elliptic Relative to Group Actions Equivariant Singular Homology Cyclic Branched Covering Spaces and O(n)-Actions Invariants for Certain Semi-Free Sl-Actions Topological S1 and Z2n Actions on Spheres The Index of Manifolds with Toral Actions On Infinite Cyclic Covering Transformation Groups on Contractible Open 3-Manifolds Characteristic Invariants of Free Differentia­ ble Actions of Sl and S3 on Homotopy Spheres Transfer Homomorphisms of Whitehead Groups of Some Cyclic Groups Semi-Characteristic Classes The Topological Period of Periodic Groups Cobordism of Diffeomorphisms of (k-l)-Con­ nected 2k-Manifolds Degree of Symmetry of Compact Manifolds Introductory Remarks The Picard-Lefschetz Monodromy for Certain Singularities Actions of the Torus on 4-Manifolds Complex Bordism of Isometries and Monogenic Groups Applications of the Index Theorem to Smooth Actions on Compact Manifolds Applications of the Index Theorem to Smooth Actions on Compact Manifolds II
T. Petrie
F. Raymond
M. Rothenberg
R. Schultz
J. Shaneson
VIn
Applications of the Index Theorem to Smooth Actions on Compact Manifolds III Torus Actions on 4-Manifolds nl in Transformation Groups Closed Manifolds with no Action Except for Z2 G-Signature and Eauivariant Characteristic Classes I G-Signature and Equivariant Characteristic Classes II
Odd Primary Homotopy Theory and Applications to Transformation Groups Surgery on Four-Manifolds and Topological Transformation Groups Torus Actions on Homology Quaternionic Pro­ jective Spaces Equivariant Resolution of Singularities of Algebraic Surfaces Some Results on Free and Semi-Free Sl and S3 Actions on Homotopy Spheres Differentiable Pseudo-Free Circle Actions Differentiable Pseudo-Free Circle Actions II
INTRODUCTORY REMARKS The subject of transformation groups is in an active period and
it is good for all of us interested to meet and exchange ideas at first hand. A generation ago the fewer people then working in a field could manage to keep in touch by correspondence or occasional contacts at general meetings, but this is now more difficult, and specialized conferences of this kind perform an important service not easily achieved in any other way. Transformation groups is an area of topology which has connections with most of the other areas of topol­ ogy. In the past, progress in any part of topology has often led to progress in transformation groups. This is likely to continue and all of us must keep as well informed as we can about what others are doing at the same time as we are continuing with our own problems. Converse­ ly transformation groups has sometimes contributed to other areas, at the very least by suggesting questions and problems. It is a great pleasure to attend a conference on a very interesting subject under such convenient conditions and congenial surroundings as have been provided here.
Deane Montgomery
TABLE OF CONTENT~
Conner. PtE. and Raymond. F.: Manifolds with Few Periodic Homeomorphisms ••••••••••••••••••••••••••••••••••••••••••••• 1
Koch. R.J. and Pall. G.: Centralizers of Rootless Integral Matrices •...•......................•...•......•....••.••••• 76
Yonner,~~~ond. F. and Weinberger. P.: Manifolds with No Periodic Maps •••••.•••••••••••.•••••••••.••••••••••••••• 81
QQnne~E. and Raymond. F.: Injective Operations of the Toral Groups II ••••••••••••.•••••••••••••••••• ~ ••.••••••••• 109
Conner, PtE. and Raymond, F.: Holomorphic Seifert Fibering ••••• 124
Carrell, J.B.: Ho1omorphically Injective Complex Toral Actions. 205
Conner, P.E. and Raymond. F.: Derived Actions •••••••••••••••••• 237
Farro I.: Group Action and Betti Sheaf ••••••••••••••••••••••••• 311
Kinoshita, S.: On Infinite Cyclic Actions on Contractible Open 3-Manifolds and Strong Irreducibility •••••••••••••••••••••• 323
TABLE OF CONTENTS
VOLUME I ----- Kawakubo, K.: Invariants for Semi-Free Sl Actions •••••••••••••• 1
Kawakubo. K.~ Topological Sl and Z2k Actions on Spheres •••••••• 14
Ku, H.T. and Ku, M.C.: Characteristic Invariants of Free Differentiable Actions of Sl and 83 on Homotopy Spheres 19
Montgomery. D. and Yang. C.T.: Differentiable Pseudo-Free Circle Actions on Homotopy Seven Spheres ••••••••••••••••••• 41
Schultz. R.: Semi-Free Circle Actions with Twisted Fixed Point Sets ••....•...•.•.........••....................•••.•.•••.• 102
Schultz, R.: Z2-Torus Actions on Homotopy Spheres •••••••••••••• 117
Wan~. K.: Free and Semi-Free Smooth Actions of Sl and S3 on Homotopy Spheres ••••••••••••••••••••••••••••••••••••••••••• 119
Boardman, J.M~: Cobordism of Involutions Revisited ••••••••••••• 131
tom Dieck, T.: Bemerkunger uber Aquivariante Euler-Klassen ••••• 152
tom Dieck, T.: Existence of Fixed Points 163
Giambalvo, V.: Cobordism of Line Bundles with Restricted Characteristic Class ••••••••••••••••••••••••••••••••••••••• 169
Hamrick, G. and Ossa, E.: Unitary Bordism of Monogenic Groups and Isometries ••••••••••.•••••••••••••••••••••••••••••••••• 172
Lazarov, C.: Quillen'S Theorem for MO· ••••••••••••••••••••••••• 183
Lee. C.N. and Wasserman. A.: Equivariant Characteristic Numbers •••••.••••••••••••••••.••••.•••••••••••••••••••••••• 191
LOpez de Medrano, S.: Cobordism of Diffeomorphisms of (k-l)- Connected 2k-Manifolds ••.•••••••••••••••.•••••••••••••••••• 217
Kawakubo, K. and Raymond, F.: The Index of Manifolds with Toral Actions and Geometric Interpretations of thery(m,(Sl,~» Invariant of Atiyah and Singer ••••••••••••••••••••••••••••• 228
Petrie, T.: Involutions on Homotopy Complex Projective Spaces and Related Topics ••••••••••••••••••••••••••••••••••••••••• 234
Orlik, P.: On the Homology of Weighted Homogeneous Manifolds 260
Orlik, P. and Wagreich, P.: Equivariant Resolution of Singu- larities with C· Actions ••••••••••••••••••••••••••••••••••• 270
Bredon, G.E.: Strange Circle Actions on Products of Odd Dimensional Spheres •••••••••••••••••••••••••••••••••••••••• 291
Davis, M.: Examples of Actions on Manifolds Almost Diffeomor- phic to Vn +l ,2 ••••••••••••••••••••••••••••••••••••••••••••• 300
XIV
Erle, D.: On Unitary and Symplectic Knot Manifolds 314
Goldstein, R.Z. and Lininger, L.: A Classification of 6- Manifolds with Free Sl Actions ••...••••.•••••••••••.••..••• 316
Grove, E.A.: SU(n) Actions on Manifolds with Vanishing First and Second Integral Pontrjagin Classes ••••••••••••••••••••••••• 324
Hsiang, W.Y.: On the Splitting Principle and the Geometric Weight System of Topological Transformation Groups,I ••••••• 334
Illman, S.: Equivariant Singular Homology and Cohomology for Actions of Compact Lie Groups •••••.•.•••••.•••••••••••••••• 403
Kauffman, L.: Cyclic Branched Covers and O(n)-Manifolds •••••••• 416
Ku, H.T., Mann, L.N., Sicks, J.L. and Su, J.C.: Degree of Symmetry of Closed Manifolds ••••••••••••••••••••••••••••••• 430
Kwun, K.W.: Transfer Homomorphisms of Whitehead Groups of Some Cyclic Groups, II •••••••••••••••••••••••••••••••••••••••••• 437
Shaneson, J.L.: Surgery on Four-Manifolds and Topological Transformations Groups ••.•••••••••••••••.•••••••••••••••••• 441
MANIFOLDS WITH FEW PERIODIC HOMEOMORPHISHS
* * by P. E. Conner and Frank Raymond
Louisiana State University and The University of Michigan
1. INTRODUCTION
In this paper we construct, in §R, a family of distinct compact
connected 4-manifolds V(k), k > 1, with the property that every finite
group must act trivially on V(k). The boundary of each of the V{k)
is the 3-sphere and hence the distinct open 4-manifolds
U{k) V(k) - 8{V{k» also possesses a total lack of non-trivial peri-
odic homeomorphisms.
The result is obtained by adjoining 4-dimensional cells along the
boundaries of each V (k) ,.,hich yield distinct closed aspherical mani-
folds, B (k), (that is, clol§ed manifolds which are K (TI ,1) 's). Nhat we
show is that each non-trivial periodic homeomorphism of B(k) has no
fixed points. Now, since any self homeomorphism h of V(k) may be
extended to B(k) by introducing a fixed point at the center of the
added 4-cell, h cannot have finite period unless it is the identity.
Let {nl, ... ,n~} be any set of distinct positive integers. Let .;-
D be the dihedral group given by 0 ~ Z ~ D ~ Z2 ~ 0, In n i n i n i
Section 7 we construct a closed aspherical (2~+1)-manifold
2HI M (nl, ... ,n~) for each {nl,n2, ... ,n~}. Let (G,~') denote an
effective action of a finite group G on~'. We show that G must be
a subgroup of De ... e D . In particular, (7.2), M3 (1) is a n l n~
non-orientable closed aspherical 3-manifo1d for which G must be
Z2 ~ D1 and must have exactly 2 circles of fixed points. This is the
closest we have been able to get to the trivial group for a closed
*supported in part by the National Science Foundation
-2- manifold.
Let (G,X) he an action of a oroup G on a path connected space
X and x s X a base point. If x is a fixed point then there is a
homomorphism 8 : G ->- ]I.ut TIl (x,x) given by
e (g) = g* : "1 (X,x) TIl(X,x). Let the outer automorphisms of
TIl(X,x), the automorphisms of TIl(X,x) modulo the inner automorphisms,
be denoted by Out(TIl(X,x». Even if x is not left fixed by G,
there still exists a homomorphism ~ : G ->- out(TI1{X,x», §3.
Basic for the estimation of the size of finite effective G in
the examples cited when X is a closed connected aspherical manifold
are the follOl.;ring;
is ~ monomorphism.
monomorphism.
A proof of 1.1. appears in [5; 6.2], and also here in A.ll. The second
theorem is an unpublished result of A. Borel. A proof of Borel's
theorem, suitable for our purpose, will be given in §3. In the APpen-
dix and §7 we extend 1.1. and 1.7.. as part of the Smith theory for
actions of p-groups on aspherical manifolds.
The claim of freeness for (G,B(~» is achieved by showinq that
Aut(IT1(B(k) ,x» has no elements of finite order other than the identity,
§ 8.
Out (1[1 (M»
and that the center of "l(M) is trivial, §6 and §7.
Let us explain how such a calculation is made. Take a group IT
and a homomorphism 1> : Z ->- Aut IT and form the semi-direct product
L = IT 01> Z. In Section 4 we develop a method for calculating Aut(L)
and Out(L) in terms of Aut(rr), Out(rr), knowledge of the cyclic group
-3- generated by ~ in Aut ~ and Out~, and the center K of~.
Under suitable assumptions we find L has trivial center and the se-
quence
o + H~(Z;K) + Out(L)
is exact where N denotes the normalizer of the group, (~) , generated
by ~ in Out (~). The sequence is split if ~ is abelian. Section
<I> 5 tells us that the contribution of the non-zero elements of HO(Z;K)
can only arise from automorphisms of infinite order. One now tries to
find a closed aspherieal manifold yn whose fundamental group is ~
and a homeomorphism h, with fixed point y 8 yn, so that
P = h* : ~l(yn,y) + ~l(yn,y). One may then construct a closed aspher­
ieal manifold Mn+l as a fiber bundle over the circle sl, fiber yn
and structure group the cyclic group generated by
L =: ~ 1 (Y) 01> Z.
in n out(~l(Y »
In Section 6 we look at ~ = zk and consider certain ~ E GL(k,Z).
Fundamental for our calculations is the matrix
¢n = (~-l i) e GL(2,Z), n ~ 1. We show that
is split exact and that the action of Z2 on
by -l. Hence, Out (L(n» = Out(Z Ell z 0 Z) = ~ n
Z is multiplication n
n n' Note that
PI = (~ i) and Out(L(l» Z2' The specific manifolds
2Hl ~1 (nl, ..• ,n t ), and others similar to them, are produced from Y
equal to the 2t-dimensional torus with P E GL(2t,Z) coming from
blocks of 2 x 2 matrices <l>n along the diagonal. The 4-manifolds
B(k) arise by taking for y3 principal circle bundles over the 2-
torus with Euler class 2k, and a suitable ~.
Our interest here has been in the action of finite groups G. If,
on the other hand, G is assumed to be a compact, connected Lie group
then there are a number of results known that guarantee that (G,M)
-4- must be trivial. For example, if M is closed and aspherical, r,
must be a torus Tk, [5], the Euler characteristic X(Mn ) = 0, [4], and
the rank of the center of Wl(Mn ) must be greater than or equal to k.
Obviously these criteria yield many examples where G must be trivial
to be effective. Other examples are any connected sum of closed orient-
ed 3-manifolds where one of the factors has fundamental group not
cyclic. Also, Atiyah and Hirzebruch have shown [l] that orientable
closed 4k-dimensional spin manifolds which admit non-trivial smooth A
circle actions have A genus 0. We do not use, however, any smoothness
assumptions throughout this paper.
We would like to express our appreciation to Professor Borel for
having shown us his result, (1.2.). Its use is fundamental for our
work.
We shall be concerned with a group extension
1 + N + G + F + 1.
We shall write N additively. We recall that G is the set N x F
with the group operation given by
wherein
(i) cp F Aut(N) is a function with cp (e) identity
(ii) f F x F + N is a function satisfying
(a) iii! (x) (cp (y) (g» == f (x,y) + (cp (xy) (g» - f (x,y)
(b) f(x,e) = f(e,x) = 0
(c) iii! (z) (f (x,y» + f (z ,xy) = f (z ,x) + f (zx,y) •
Our primary intention is the application of the followinq.
2.1. Lemma: If h N + L is ~ homomorphism then h ~ be ex-
-5- ter,deci to a homomorphism H G + L if and only if there is a function ----- T : F + L satisfying
T (x)
T (x)
h (g)
h(f(x,y)) . T(xy).
Proof: Suppose first that such an extension exists, H G + L.
Put T(x) = H(O,x) E L. Then T(x) . T(y) = H((O,x) . (O,y))
= H (f (x,y) ,xy) = H (f (x,y) ,e) . H «(0 ,xy)) = h (f (x,y)) • 'r (xv). In addition
T(x) . h(g) = H«O,x) . (g,e)) = H((l'j)(x) (g) ,x) = H(l'j)(x) (g) ,e)H«O,x))
h (l'j) (x) (g)) • T (x) •
Conversely, if such a function T exists then we put
H(g,x) = h(g) . T(x).
T(xy) h(gl)' h(fI)(x) (g2)) . T(x)
T (y) =
T(y) = h(gl) . T(x)
h (f (x,y))
h (g2) .
We shall apply this lemma when L is a group of homeomorphisms on a
space.
Suppose now (F,X) is a group of homeomorphisms on a pathwise
connected, locally pathwise connected space which is also semi-Iocally-
I-connected. Select a base point a s X and proceed to define a group
extension
I + ~l(x,a) + G + F + 1
as follows. For each x E F choose a path Px(t) in X issuing from
a with P x (1) = x·a. We assume Pe(t) - a.
First define It' F + Aut(~l (X,al l. If (J (tl is a loop based at
a, then l'j) (x) (0) is represented by
-6-
1
xo (3t-l)
Px (3-3t)
For any pair (x,y) we denote by f(x,y) £ ITl(x,a) the element
represented by
It is clear that f(x,e) f(e,x) O. Let us consider
f(z,x) + f(zx,y).
This is represented by the sum of paths
This must be compared with ~(z) (f{x,y» + f(z,xy), which is represented
by
A cursory inspection, however, proves that in 'rrl(x,a)
(c) ~ (z) (f (x,y» + f (z ,xy) = f (z ,x) + f (zx,y).
Now we must verify the relation
-7-
The left side is represented by
Px (3t) 0 < t < 1 "3
xP y (9t-3) 1 < t <
4 "3 9
< t < 2
< t < 1 "3
[
+ XPy (2-3t)
Px (3-3t) .
1 4 TIl (X,a) 4 G + F + 1.
2.3. Theorem: We mav geometrically realize this extension as a
* group of homeomorphisms of the universal covering space of X in such
* ~ ~ that (G,X) covers the action (F,X) ;.Jnder the map
n x v * * (G,X) + (F,X) where n(o,x) = x ~nd v : X 4 X is the
covering projection. The action (G,x*) is properly discontinuous if
(F,X) is properly discontinuous. Furthermore, there exists ~ canoni-
cal isomorphism of the isotropy groups, 'V * (h) Gb , b EX.
Now TIl (X,a) operates freely from the left as the group (>f ~ovz;ring
* transformations on X, the universal covering space. We wish to ex-
* tend this to an action of G on X * To say TII(x,a) acts on X is
to say that there is a homomorphism of TIl(x,a) into the group of all
homeomorphisms of x*. We wish to extend this homomorphism to all of
-8- G. To each IT £ ~l (X,a) we associate a homeomorphism
h (IT) * * * IT# : X ~ X as follows. A point b E X is represented by a
path pet) issuing from a. Let IT(t) be a loop at a representing
* IT. Then IT#(b) E X is the element represented by the path
To each
) IT (2t)
1 p(2t-l)
* X
{
If
We must consider first the composition (Tx 0 ~y) (b). This would be
represented by the path
xP (4t-2)
Xy~ (4t-3)
Compare this with (f(x,y)# 0 Txy) (b) which is qiven by
By inspection we then see
ToT x Y
We must also examine (Tx 0 IT#) (h), represented by
b
and «~(x) (a))# 0 Tx) (b) which is represented by
and again we have
Px (4t-2)
xp (4t-3)
According to our opening lemma we can extend a ~ a# to a homomorph­
* ism of G into the group of all homeomorphisms of X. Hence G acts
* * on X. If v : X ~ X is the projection map then
v ( (a , x) # (b) )
xv(b) by definition. This completes the first part of
We would now like to determine the isotropy group Gb c: G. To
each x E Fv(b)
a# 0 Tx(b) = b.
-1 Simply choose a with Tx(b) = (a )#(b). This
defines a function Fv(b) ~ Gb which is a 1-1 correspondence.
Suppose (al,x), (a 2 ,y) lie in Gb , then
b (a 1) # o T 0 (a 2) # 0 Ty(b) x
(a 1) ~ 0 ( ~(x) (a 2) ) # 0 Tx 0 Ty(b)
(a 1) # ° (~(x) (a 2 ))#o f(x,y)# ° T xy
* Hence (a l + ~(x) (a 2
) + f(x,y) ,xy E Gb so that at each b E X we
have a canonical isomorphism Fv(b) ~ Gb .
-10- Let us recall the definition of a properly discontinuous action
* (G,X). The discrete group G is said to operate properly discontin-
* uously on X if
(a) If b' I Gb, then there are neighborhoods U' b
and Ub with
and
the
(b)
(c)
such
We
* For each b s X , the isotropy group Gb is finite.
* At each b s X , there is a neighborhood Ub with GbUb
that if Ub n gUb <p, then g s Gb •
shall now show that G acts properly discontinuously on
action (F,X) is properly discontinuous.
Let b' ,e' G(b) • Then v (b') E' F (v (b)). Choose U v (b') and
* X
Uv(b) so that Uv(b') n F(Uv(b)) ¢. Since X is semi I-connected
we may also choose Uv(b) and Uv(b') so that
if
i* : ITI (U v (b)'v (b) -)- ITI (X,v (b)) and i*
are trivial. Hence, xUv(b)' and Uv(b') -1
IT 1 (U v (b' ) , v (b ')) ->-'1[ 1 (X, v (b' ) )
are evenly covered. In fact,
v (Uv(b») = ITl(X,a)ub , where Ub is a lift of Uv (b) to a neighbor-
-1 hood of b, and G(Ub ) = v (F(Uv(b))' Thus condition (a) is satis-
fied.
Let Uv(b) be a neighborhood of v(b) satisfyinq condition (c).
Let V be a neighborhood of v(b) which is evenly covered and choose
W v (b) = n x (V" U v (b) ) • x s F\) (b)
W\)(b) is evenly covered and satisfies
* condition (c) as a neighborhood of v (b). For b EX, the lift of
Wv(b) to b, Wb , has the desired property for (c). Finally we point
out that (G,x*) is a group of covering transformations if and only if
(F,X) is a group of covering transformations. This completes the
proof of the theorem.
aspherical space X then the following are equivalent.
-11-
* (iii) G acts on X as ~ group of covering transformations.
* Since X is contractible and finite dimensional any element of
G with prime order has a fixed point. If G has no elements of
finite order then * is trivial for all b ~ X .
2.5. The converse problem for a finite F might be phrased as
follows. Suppose we are given a group extension
1 + ~l (X,a) + G + F + 1
of TII(x,a) by a finite group. Can this extension be realized by the
foregoing geometric construction? This can be done as follows. Con­
* sider the product of X with itself, Y, as the set of all functions
* X F + X
recalling F is finite. We use the function~: F + Aut(TI1(x,a)) to
define an action (TIl (X,a) ,Y). To each (J ~ 'IT 1 (X,a) associate a homeo-
morphism on Y by
E (X) (z) " (ep (z ) (J ) # (X (z) ) .
It is readily seen (J + E is a homomorphism defining a left action of
'IT 1 (X,a) on Y as a group of covering transformations. We use f(x,y)
to define a homeomorphism T : Y + Y x by
(Tx(X))(Z) "f(z,x)#(x(zx)).
We wish to verify that this extends the homomorphism of 'ITI(x,a). Now
(TX 0 Ty)(X) (z) = (f (z ,x) # 0 f (zx,y) #) X (zxy). But ~ (z) (f (x,y)) +
f (z ,xy) = f (z ,x) + f (zx,y), hence (Tx 0 Ty) (X) (z) =
(~(z) f (x ,y)) # 0 f (z,xy) #) (X) (z) = (F (x,y) 0 Txy) (X) (z). Again
-12- (Tx 0 l:) (x) (Z) = f (z ,x) # 0 (Ql(zx) (a)) #x (zx) (f (z ,x) +
Ql(zx) (a))#x(zx) (f(z,x) +Ql (zx) (a) f(z,x) + f(z,x))#X(zx)
«Ql(z) (Ql(x)a))# 0 f(z,x)#)x(zx) = «Ql(z) (Ql(x))a)# 0 Tx)x(Z). Thus by
Lemma 2.1 the action of TIl(x,a) on Y extends to an action of G on
Y, and yields (F,Y/TI I
(X,a)). Application of the preceding construc­
tion to (F,Y/TIl(X,a)) yields G with its action on Y.
2.6. Corollary: If X is a finite dimensional aspherical space ---- and 1 -> TIl (X,a) -> G ->- F -> 1 is an extension by ~ finite ~ for
which G is torsionless then Y/G is an aspherical finite dimensional --- space with fundamental group G. In addition F acts freely on
Y/TI I (X,a).
This corollary may be regarded as a geometric formulation of a
theorem of J. P. Serre, see Swan [7J. The geometric dimension of a
group is finite if and only if the algebraic dimension is finite. The
corollary says that if G is a torsionless extenzion of a group TI
with finite algebraic (i.e., cohomological) dimension by a finite group,
then G has finite algebraic dimension.
3. CENTERLESS FUNDAHENTAL GROUPS
To any action (G,X) of a finite group on a pathwise connected
space there is canonically associated an abstract kernel (~,G,TIl(X));
that is, a homomorphism ~ : G + Out(TI) together with a group exten-
sion
1 + TI
which realizes this abstract kernel, (§2.). We shall set the notation
for this section. Choose any basepoint x € X and for each g E G
select a path pg(t) joining x to gx. The corresponding auto-
morphism g* on TIl (X,x) is the composition of the translation iso-
-13-
morphism TIl (X,x) ~ TIl (X,gx) with Pi(l-t) : TIl(X,gx) ~ TIl{X,x). This
g* is unique up to an inner-automorphism and yields the abstract
kernel '¥: G ->- Out (n). The extension cocycle f : G x G ->- n assigns
to (g,h) the element represented by the closed loop
{pg (3t) , 0 < t < 1 "3
gph (3t-l) , 1 < t <
pgh (3-3t) , 2 < t < 1. "3
The extension group L acts from left on the universal covering
* * space X and under the covering map p: X ->- X we have
p(llb) '" B (t)p(b), all * £ L, b £ X •
3.1. Lemma: If (G ,11) j sap-group acting on an aspherical man­
ifold with centerless fundamental group and if the associated abstract
kernel 'f: G ->- Out (TI) is trivial then (G ,~1) has 3!!: non-empty fixed
point ~ with mod p cohomology * isomorphic to H (M;Zp)'
Since n is centerless and 'jI is trivial it follows from Theorem Cl B
(8.8) (Homology, S. Maclane, p. 128) that 1 ->- n L ->- G ->- 1 is equi-
valent to 1 ->- TI ->- TI X G ->- G 1. In particular admits a splitting
homomorphism n : G ->- L whose image will be a p-group acting on a con-
tractible manifold. The image has a fixed point, hence by projection
so does the original (G,H). Thus we may choose the g* so that
g ->- g* is a homomorphism G Aut(TI). But 'Y is still trivial,
hence g* £ Inn(rr) and r is still centerless so each g* is con-
jugation by a unique Cl q £ n. Since g ->- Cl q is a ho~omorphism of a finite
group into a torsionless group we see g* = identity for all 9 £ G.
Accordingly, in the terminology of the appendix, (G;TI) = Hom(G;rr)
{OJ and rO = 7f, which completes the proof. (1).lternatively, we now
* have G x rr acting on M so that Tg 0 (a#b) = (g*(a»# 0 Tg(b)
a# 0 Tg(b). Theorem 6.1 of [5) now implies our result. That is,
* F(G,M) = E has the property that aIlE) = E, for all a £ rr1(M).
-14- Since E is path connected, veE) c Fa' where F a is the component
* of G(G,M) containing a = v(b), b is a basepoint of M in E and
v is the covering map. Finiteness of G and connectedness of E
implies that veE) = Fa = E/rr, cf. [5; Lemma 3.4]. Since E is acyclic
mod Zp' H* (Fa;Zp) % H* (rr;Zp) % H* (M;Zp)')
3.2. Theorem (A. Borel): If (F,M) is ~ finite group acting
effectively on ~ closed dspherical manifold M with centerless fund­
amental group ~ the associated abstract kernel T: F + Out(rr) is
~ monomorphism.
Suppose this is false, then for some prime p there is a Sylow p-
subgroup, G, of the kernel of 0/. Applying the lemma we see that G
has a fixed point set whose mod p cohomology is isomorphic to that of
M, a closed manifold. If M is orient able then G acts trivially on
M, contradicting effectiveness.
Suppose M is not orientable, then the action of G, G in the
kernel of 0/, can certainly be lifted to the oriented double covering
Md of M. Since G acts trivially on rr l (M) it also acts trivially
d on rrl(M) and once again effectiveness on M is contradicted.
3.3. Remarks: Actually in proof of 3.1 we only use the facts
that M is a finite dimensional space whose universal covering is
acyclic mod Zp' and whose fundamental group is torsionless and center­
less. In particular if M is'a finite dimensional K(rr,l) with
centerless rr, then for any finite group G to act freely on M, the
group G must be embedded in the group of homotopy classes of self
homotopy equivalences of M. And, more generally, if G is a finite p-
group acting without fixed points, then Out(rr) must contain p-tor-
sion.
Corollary 6.2 of [5] asserted that if (G,Mn ) is an effective
action of a finite group on a closed aspherical generalized manifold
-15-
over Z with a fixed point, then the homomorphism G 7 Aut TI was a
monomorphism. No assumption on being centerless is made there.
The point of 3.1. is that the centerless assumption implies the exist-
ence of a fixed point set for each p-subgroup and consequently even
yields an embedding of the p-Sylow subgroup of G into Aut TI.
Another interesting interpretation is that if (Zp,Mn ) is an
effective free action on a closed aspherical manifold for which the
generator induces a homeomorphism homotopic to the identity, then TI
must have a center. In particular, if n = 3, M is orientable, and
TI is "sufficiently large" then M is a Seifert manifold modulo the
Poincare conjecture.
There are many examples of closed aspherical manifolds for which
center TI = 1. Moreover, A. Borel has exhibited examples from the the-
ory of symmetric spaces and algebraic groups for which Out TI is fi-
nite, [3].
4. AUTOMORPHISMS OF' A SE~U-DIRECT PRODUCT
In this section we shall consider a group automorphism ~ : TI 7 TI
for which I-~* : HI (n,Q) ~ Hl(n,Q). We shall form the semi-direct
product L TI 0 Z with multiplication given by (a,n) (S ,m) =
(a~n(B) ,n+m) and we shall determine the outer automorphism group,
Out(L) and the center of L.
4.1. Lemma: The ~ of the natural homomorphism
L ~ (L/[L,L]) x 7,0
is the subgroup TI L L.
Proof: The short exact sequence
1 ~ TI ~ L ~ Z ~ 0
is split by x(n) (e,n). We shall consider Q as a trivial L-module
-16- and sh~ll use the Lyndon spectral sequence to show that
There is then the spectral sequence {E~,t,dr} =) H*(L;Q) with
In particular,
x* : HI (Z;Q) ~ Hl(L;Q) and the composite homomorphism
is trivial. Since any element (a,n) E L may he written (a,O) • X(n)
the lemma follows.
This lemma proves that n C L is a characteristic subgroup; that
is, n is invariant under each automorphism of L. Thus every auto-
morphism of L induces an automorphism of Z • ~Ie denote by
Aut+(L) C Aut(L) the subgroup, which has index at most 2, of those
automorphisms which induce the identity on Z.
If a E Aut+(L), we may write
a (a, 0) c(a), c E Aut(n)
a (e,n) (~(n) ,n), ~ : Z + n a crossed-homomorphism.
The second assertion follows since (~(n+m) ,n+m) = (~(n) ,n) • (~(m) ,m)
(~(n)ln~(m),n+m). Since (a,n) = (a,O) (e,n) we have o(a,n) =
(c(a)~(n) ,n). Now in addition we have
o«e,n) (a ,0») = (J (In (a) ,n) = (c!J>n (a) ~(n) ,n).
But (J (e,n) • (J (et,O) = (J «e,n) (a,O» implies (~(nHnc(a) ,n)
(Cln(a)~(n),n). Thus we obtain the funcamental identity
-17-
To utilize this identity we now prove
4.2. Lemma: If c E Aut(rr) and if 8 E rr is an element for
which
~(l) = 8 and
Proof: Since Z is a free group there is a unique homomorphism
h Z ->- L with hill = (0,1). If we write h(n) = (~(n) ,n) then
~ Z ->- IT is the required crossed-homomorphism.
Observe that e = ~(O) = ~(-l+l) = ~(-l)1>-l(o) so that ~(-l)
1>-1(6- 1 ). Now ¢-1{8- l ) . (1)-l{c{a))) = 1>-1(8- l c{H- l (a»08-1 )
¢-1{8- l 61>c{¢-1{a» . 8- 1 ) = c(¢-l(a» • ¢-1(6- l ). The required iden-
tity may now be established by double induction since it was just ver-
ified for n = -1. Assume that the identity has been established for
some n, Inl > O. If n > 0 we write ~ (n+l)
~(l)¢~(n) . 1>n+l(c(a» q,(l)1>(q,(n)~n(c(a») =
rp(l)¢(c¢n(a) • rp(n» c(¢n+l(a» q,(l)¢(rp{n» c{¢n+l{a». q,(n+l).
If n < 0, we write q,(n-l) . (¢n-l{c{a») = q,{_l)~-l{~{n)~n(c(a»)
~(-l)¢-l{c{¢n{a»C!){n» = c{¢n-l{a»~{_l)1>-l{q,(n» = c{~n-l{a»C!)(n-l).
Thus the identity is established for all integers.
For each pair (c,8), where c E Aut(rr) and 8'1>(c(a»
(c¢(a»·6 we may define a EAut+{L) by
a (a,n) = (c(a)~(n) ,n).
-18- ~"e must show that 0 is multinlicative. Thus, 0 (cd,n{S) ,n+m) =
(c{S)CIl(m),m) = o(o;,n) o(S,m). The reader may show that hath the
kernel and cokernel of 0 are trivial.
Composition is the group operation in Aut+{L) and this corres-
ponds to composition in Aut(rr). However, 01{02(e,l» = 0 1 (0 2 ,1) =
(c l (02)ol,l). Denotinq (1(0) E Inn(rr), conjuqation by 0, we introduce
the group G of all pairs (c,o) for which 11 (o) 0 <I> 0 C
c 0 E Aut(rr). The qroup operation in G is
4.3.~: Aut+(L) ~ G.
Clearly Inn(L) C Aut+(L) so let us describe the correspondinq
subgroup of G. For (o;,n) E L we have
-n -1 (a,n) (e,l) {1> to; ) ,-n)
Thus to the inner-automorphism determined bv (o;,n) there corresponds
(ll{a) 0 <!,n,o;<!>{o;-l».
Let us denote hy Kerr the center. \'7e also denote by
C{1» C Out{rr) the centralizer in the outer automornhism qroup of rr
and by (1)) C Out{") the subgroup qenerated hy 1>. Finally, since
Inn{L) C Aut+(L) we have Out+(L) C Out{L) as a subgroup of index at
most 2.
4.4. Theorem: If the order of 1> in Out{rr) is equal to its
order in Aut{rr) then there is ~ short exact sequence
+ o ..;. HI) (7,;K) ..;. nut (L) ..;. C(1))/{4>) ..;. 1.
Proof: Surely there is an epimorphism G + C{¢) and
(\1{0;) 0 ¢n,o;4>(a- l » ..;. <l>n E Out{rr). 'T'hus we receive an epimorphism
-19- !Jut+(L) -)- C (p) I (<}). A pair (c, C) represents an element in the kernel
of this epimorphism if and only if c = 11 (a) 0 q,n for some a E 1T ,
n E: Z. But then lJ (6) 0 til 0 lJ (a) 0 tIl n = lJ (6) 0 lJ (<1J (a) ) 0 <1J n+l =
lJ (a) n+l
o <1J .
Thus Jl (a) ~(a<1J(a-l» and so we may write = B . o:t» (a-I) for some
B EO K and (e,a) = fi.d,B) (jJ(a) 0 <!I n , O:<!J(a- l ». Thus (e,a) =
fi.d, S) E: !Jut + (L) for some B E K. The orohlem DC"'" is to determine
when (id,B) E: G is an inner-automorohism. If (id,S) =
(jJ(a) 0 <!In,a<!J(a- l )) then ,n = I E: !Jut(1T). Since the order of
<!J E: Out(1T) is equal to the order of 'E: Aut(1T) by hypothesis we
have
Jl (a) id E: Aut(1T).
Thus a E: K and a'(a- l ) = B. On K, a + O:<1J(a- l ) is an endomorphism
with cokernel HO(Z;K). Hence we obtain
4.5. Corollary: If, under the hypothesis of the theorem, the
is abelian then the short exact sequence
+ o -)- HO(Z;1T) -)- Out (L) -)- C(<!J)/(<!J) -)- 1
is split.
Proof: Since 1T is abelian, K = 1T and Out(1T) = Aut(1T) and we
may define Crt) -)- r, hy c -)- (e,e), but then ,n -)- (,n,e), which is
just the inner-automorphism of L given bv (e,n). Thus the splitting
homomorphism C(tJ»/(tJ» -)- Out+(L) is induced.
Now let us discuss the full group Out(L). By analogy an element
a EO Out-(L) may be written a(a,n) = (c(o:)o/(n) ,-n) where o/(n+m) =
0/ (n) t»-no/ (m) and
-20-
Thus with <5 = '¥ (1) we find IJ (0 ) 0 <1'-1 0 c = c 0 <I> so that in
Out(TI) we receive <1>-1 = c 0 P 0 -1 Thus if N (<I»!: Out(TI) is the c
subgroup of all elements for which c 0 cD 0 -1 c = <I>±l then
(<1» !: C (<I» !: N('!» • N(ll) is the normalizer of (¢ ) in Out (TI) .
4.6. Theorem: If ¢ has order > 2 under th~ hypothesis of the
theorem 4.4. we again have a short exact sequence
o + HO(Z;K) + Out(L) + N(<l»/(<I» + 1,
which splits if is abelian. --- ----- On the other hand, if <!> = <1>-1 in Aut(,r) then we may set up a
1-1 correspondence hetween nut-(L) and Out+(L). Namely, if
da,n) = (c(a)~(n),n) E Aut+(L), then al(a,n)
(c(a)~(n) ,-n) E Aut-(L) since <1>-1 = <1>. This induces the correSDon­
dence between Out + (L) and Out (L) (One may easily illustrate this
case by the fundamental group L of the Klein bottle. Here L is the
non-trivial semi-direct product of 7, by Z and Out+(L) ~ Z2' while
Out (L) ~ Z2 and the full automorphism group Out(L) ~ Z2 + Z2')
In the remaining part of this section we shall determine when the
semi-direct product, under the conditions at hand, possesses a center.
We shall eventually prove
4.7. Theorem: If the center of TI is finitely generated then
L has a trivial center if and only if
(i) <I> leaves no central element in TI, other than the identity,
(ii) <!> has infinite order in Out TI.
-21-
Proof: Suppose first that L is centerless. If there were a
central element a E K, a f e and t(a) = a, then (a,O) E L is a cen-
tra1 element because
(Sa ,n) (as ,n) .
to its order in Aut~. If leaves no central element of IT oth----- er than the identity fixed, then the center of L is the suhgroup of
elements (e,n) with ¢n = I E nut n.
Proof: Suppose (a,n) is central in L then (a,n) (e,l)
(a,n+l) = (e,l) (a,n) = (¢(a) ,n), hence a = <p(~<l. On the other hand,
(a , n) (B, ()) = (a q, n ( B) , n) = (B, rl) (a, n) ( B a , n ) so th a t ~,n ( B) =
a-lea and q,n = I E Out(n). But q, is assumed to have the same order
in Aut(n), so <p n I,B = a3a- l which implies a E K and this to-
gether with <P(a) = a means that a = e. This completes the proof.
In particular, if this mutual order is k, 0 < k < 00, then the center
of L is the infinite cyclic subgroup of elements of the form (e,kn)
and if has infinite order, then L has trivial center.
Let us now suppose that in Out(n), <P has finite order k,
o < k < 00. There is then, for each integer n, aYE IT, uniquely de­
termined modulo k, such that q,nk(B) = y-ISY.
4 . 9. Lemma: y q, (y -1) E K.
q,nk(¢(S)), which yields the
identity <l>(y-l)<l>(S)<l>(y) = y-lq,(S)Y. Since ¢ is an automorphism it
follows that y<p(y-l) is central.
W 1 by K But t hen y¢(y-l) is replaced e may rep ace y ya, a E. _
by ya¢(a-ly-l) = y<l>(y-l)a~(a-l). That is, the homology class
-22- -1
cQ,(y1>(y )) E HO(Z;K) is well defined. We define X Z -+ H(J (Z;K)
by
4.10~: The function X Z -+ H(J(Z;K) is ~ homomorphism.
Proof: Let Y E '1[. qik(y) = y-lyy y.
Hence, by induction, ,kn(S) =' y-nSyn, for each integer n, and of
course, yn,(y-n) lies in K also. We may write yn,(y-n) =
yn-ly¢(y-l),(yl-n) = y¢(y-l)yn-l,(yl-n), n > 0 and yn¢(y-n)
yn+ly-ll(y)I(y-l-n) = y-l'(y)yn+l,(y-l-n), n < n. So bv induction we
find that
in HO(Z;K). Thus we have defined a homorphism
Now under the assumption that I leaves no element of K fixed
we know that the endomorphism of K given by cr -+ crl(cr- l ) must be a
monomorphism. If K is assumed finitely generated then HO(Z;K), the
cokernel of K -+ K by cr -+ crl(cr- l ), must be a finite group. Hence
there is an integer n, 0 < n < 00, with cQ,(ynl(y-n)) OEHO(Z;K).
Thus, yn'(y-n) = crl(cr- l ), for some cr E K. Replace yn by
p = yncr-l Then ¢ (p) = p and Ink (S) =' y -nSyn = CI. -lp -lSp CI.
We now claim that (p,nk) E L is a central element, for
(6 ,m) (p ,nk)
(p ,nk) (S ,m)
(Slm(p),m+nk) = (Sp,m+nk),
(p,nk(B),nk+m) = (pp-lsp,nk+m)
(Sp ,m+nk) •
-1 p SP.
Since nk = 0, this is a non-trivial central element. \'lJe found this
central element by assuming has finite order in Out TI. Thus if L
is centerless, I has infinite order in Out TI. We have already shown
the converse and so this completes the proof of the The~rem 4.7.
-23-
4.11 Corolla£[: If the center of L is trivial then Out L is
finite if and only if C(¢l/(¢) is finite.
Proof: If the center of L is trivial then a + aw(a- l ) is a
monomorphism for all a E K. But then is a finite group.
Finally nut+(L) is a normal subgroup of Out(L) with index at most
2.
There is an analogue of Theorem 4.4 when the order of in
Out(rr) is not equal to its order in Aut(rr). It is a corollary of
above together with the proof of ~heorem 4.4.
4.12. Corollary: If we take as in the beginning of this sec-
tion, then there is ~ exact sequence
x Z + HO(Z;K) + Out+(L) + C(w)/(w) + 1.
The imAge of X is trivial if order w in Out(rr) is equal to its
order in Aut(1').
4.13. Remark: lAuch of the analysis that we used here in this sec-
tion only depends upon workinq with a semi-direct product where the
normal subgroup l' is a characteristic subgroup of L. The quotient
group need not necessarily be Z for obtaining a representation of
1I.ut + (Ll as in 4.3. For example, if L = zk x rr where l' is center-
less, then
-24-
is also a split exact sequence, k > O. As an immediate application
consider a closed orientable 2-manifold M of genus greater than 1.
Let L = Z x nl(M,x). It is known that the isotopy classes of homeo­
morphismsof M x Sl is iso~orphic to Out nl(M x Sl), and hence iso­
morphic to Out(L).
5.1. Lemma: Let ¢: Z ~ Aut(K) be ~ homomorphism, where K is
an abelian group. Then Hl(Z;K) is canonically isomorphic to
HO (Z ;K) .
Proof: Ioet Hom¢ (z ;K) be the abelian group of crossed-homomor-
phisms. NQw K 'V
Hom<D(Z;K) since to each y 2 K there corresponds a
unique crossed-homomorphism If' : Z ->- K with tl)(n+n:) = /()(n) ¢n (Ij)(m)) and
~(l) = y, (see 4.2). Furthermore, by uniqueness, If' is a principal
crossed-homomorphism if and only if -1 y = a1' «('1 ). This correspondence
establishes the isomorphism.
In the remainder of this section n is a torsionless group whose
center, K, is finitely generated. Furthermore, ¢ : n ->-" is an auto-
morphism for which
(iii) ¢ leaves no central element of ~ fixed other than the
identity.
It follows now that L = n a is a torsionless group with trivial
center and there is a short exact sequence
Because we assumed K is finitely generated and that the endomorphism
of K given by y ~ y¢(y-l) is a monomorphism, it follows that
-25- H1(Z;K) is a finite group.
Now L is centerless and the embedding Hl(Z;K) + Out+(L) may
be regarded as an abstract kernel and hence there is a unique group
extension 1 + L + G + Hl(Z;K) + 0 which realizes this abstract kernel.
5.2. Theorem: If the quotient ~/K is also torsionless, the
group extension which realizes Hl(Z;K) + Out+(L) is torsion free. - -- We must examine first the embedding K + Aut+(L). To any y s K
there is a unique crossed-homomorphism ~ : Z + K with
!pel) = y
The corresponding automorphism 'l' s Aut+ (L) y
'l'y (a,n) = (a'¥l(n) ,n).
is
5.3. Lemma: There is a k > 0 and a g
(~(g) 0 o/y)k = I s Aut+(L) if and only if q
(s,m) s L for which
(8,01, Sk s K and
Suppose first that such a k > 0 and g s L exist. Obviously
That is, for some a = ( a , n), ~ (a) = o/k. y
But II (a)
corresponds to the pair (~(a) 0 <lln, a<ll(a- l » while o/y corresponds
to (I,yl. Since has infinite order in Out(rr), n = 0 and a £ K.
Further, a<ll(a- l ) = yk.
k Now (~(g) 0 o/yl )J (g) 0 ~ (0/ (g»... 0 II ( o/k -1 (g) 0 'l'k
y y - y
ever,
e. Thus k y
-26- I s Aut+(L).
Let us denote by S C Aut+(L) the subgroup generated by Inn(L}
and the image of K .... Aut+(L). Then S is the counter-image of the
subgroup Hl(Z;K) with respect to the quotient homomorphism
Aut+(L} .... Out+(L}.
5.4. Lemma: If ~/K is torsionless, then S is ~ torsionless
subgroup of Aut+(L).
Proof: Any element of S can be written v(g) 0 o/y for some
g s L, Y E K, (since o/y 0 v(g) = >l('fy(g» 0 'fy)' If >leg) 0 'fy has
order k, then g = (B,O) and B-k¢(Sk) yk. Our previous proof
shows Bk E K, hence E K since ~/K is torsionless. NOw, hO'"ever,
using the fact that ~ is torsionless, 8- l ¢(B) = y. This implies
is the inner-automorphism corresponding to (a-l,O) and hence
v (g) o 'f y
Now we can prove the theorem stated. If in
\!
'l' Y
there is an h s G of order k we see that r (h) s Hl(Z;K) has order
exactly k also since L is torsionless. Thus the automorphism of
L given by conjugation with h defines an element of S whose order
is exactly k and under Aut+(L)"" Out+(L) this automorphism corres-
ponds to 11 (h) .
With this result we may construct a torsionless finite extension
of L. If h has finite algebraic dimension, then G will also have
the same finite dimension by the result of Serre (see 2.6 for a slight-
ly weaker statement).
A convenient source of examples to illustrate our general con-
structions is
( nn-l matrix
GL(2,Z). For each integer n > 0 we introduce the \ i ), which has determinant -1, in GL (2, Z) •
6.1.~: There is 122 ~ M £ GL(2,Z) for which
Proof: We write
(n~l i )(~
(~ ~) (n~l i)
( bn+a (n-l) dn+c (n-l)
d+b(n-l») d+bn
a+b ) c+d .
From c + an = -dn - en + c we have a + c = -d, while -a - c + bn - b
= -a - b yields c = bn. Now, however, d + bn = -c - d = -bn - d
implies that d = -bn = -c and, therefore, a = O. But then
c + a(n-l) = -bn - a(n-l) implies c = 0, b = 0 and d = 0 also.
Now, let GL(2,Z) PGL(2,Z) denote the quotient by the sub-
group of order 2 which -I generates. We have shown that if
'1 (M) commutes with (n-l ,., n i) then ~1 commutes w;.th ( n~l i) in
GL(2,Z) •
6.2. Lemma: There does not exist a matrix M £ GL(2,Z) for which
1)-1 1 .
From the pair of equations
-28-
d+b (n-l) 1 d+bn ')
( bn-a dn-c
an + c = dn - c
we find that c = (d-b)n. However, bn + d = (l-n)d + c = d - nd + nd
bn shows that b o since n > O. From the relation b(n-l) + d =
b(l-n) + a we then obtain a = d. However, c = nd so that finally
an + c dn - c also shows an = -c = -nd or a -d. Hence the
O-matrix is the only possibility for M.
6.3. Lemma: The centralizer of
subgroup of matrices of the form
a(a+2b-bn) = +1 + b 2 .
(n~l i) in GL(2,Z)
~+2b-bn) with
Proof: Immediate by direct calculation.
6.4. Lemma: 1.f M = (~ ~) c GL (2 ,2) is .§ ma.trix for
is the
which
ad > 0, be > 0 then exactly Qn~ Q.f ~.9~ i9~ matrices M, -.H, -1 M ,
Proof: If M E SL(2,Z) then
M- l
-M -1
_M- l
Since ad > 0, bc > 0 all the possibilities are covered.
which
MK = + (n-l - n
If k > 1 there is no matrix M E GL(2,~) for
Proof: Suppose that such a matrix did exist. Since M commutes
wi th MK by Lemma 3 we can write M = (~n ~+2b-bn) with
a(a+2b-bn) = +1 + b 2n. We must eliminate some special cases when
n = 1. These are b = + 1, a 0; yieldinq the matrices,
and b + 1, a +1, yieldinq,
+1) (0 -0 =:':. 1 1 \ -1 1 ) .
" Mk -_ (° 1
Ubvlously these matrlces are not solutions of I') 1 with K > 1.
In general, thus we can assume a (a+2b-bn) > 0 and apply Lemma 4
to the equations
(_I)K n ~ ) (_M-l)k ( -1
(-I) K 1 l~n)
-30- We see immediately that no one of the matrices 1-'1, M- l , -M, _M- l
CI'I!1
k > 1. possibly have all terms positive since 'n-l 1°)
W~Uld sti 11 lie in the centralizer of (n 1 ' By observing(n M
n
- l
the case Mk = - 1) is shown not to occur by a similar argument.
6.6. Lemma: There is no element of finite order, other than -I, -(n=l-l~
in the centralizer of n 1/'
Proof: Such an element would have the form ("ban b \ a+2b-bn J. If b = 0 we have a 2
= 1 corresponding to +1. When n = 1 we have
exhibited all such matrices with a(a+b) = 0 and they are not of
finite order. If a(a+2b-bn) > 0, then by Lemma 4 we see that the
matrix cannot have finite order.
6.7. Theorem: In GL(2,Z) ( nn- l
the centralizer of
Proof: Let C L PGL(2,Z) be the centralizer of
11 ) is the
According to Lemmas 6.1 and 6.6 there are no elements of finite order
in C. Consider then the intersection C n PSL(2,Z). This is a tor-
sionless subgroup of the free product Z2 * Z3 and hence is itself free.
n C onn-l 11)2 But every element in C n PSL(2,Z) commutes with and
hence C n PSL(2,Z) must be infinite cyclic. We have now
o + en PSL(2,Z) + c + Z2 + 0 and since C is torsionless it also
follows that C L PGL (2, Z)
We assert that n (n:l
some k > 1 there would be
Mk (n-l ~) . = + Finally, - n
is also an infinite cyclic group.
~) must generate C, for if not then for
a matrix M € GL(2,Z) with
by Lemma 6.1, n-l(C) is the centralizer
e-l 1) n 1
(n:l ~), = (-n+2 -1 ) Putting 1> I - c!> o ' and
n n -n
-31- det(I-~n) = - n. Consequently, HO(7.,Zez) = cokernel of the monomorphism
-1 a ~ a~n(a ) has order n, and is isomorphic to Zn' The group
L(n) = (~) 0 ~ Z is ~entp.Tless by ~4 n
6.8. Corollary: There is ~ short exact sequence
In fact, Out(L(n» ~ Zn 0 Z2 where Z2 acts on Zn by ->- -\.
If n = 1 then det(I-~) -1 and hence Out(L(l» ~ Z2' This
represents the smallest group of outer automorphisms we have been able
to construct.
integer b 'I o.
(~ ~) in
C('I') in GL(2,Z) is the group generated by (-~ and SL(2,Z) and is isomorphic to ~Z2'
Proof: We first observe that and hence
Suppose that ~). We then
(: ~) = (~ ~) C- l -~ ) or , Z E: Z. Z
Since we may write
We see that C('I') is generated (-1
by ° -~ ) and(~ ~) , for each
C(~) ~ Z&Z2'
Proof: There is an isomorphism C (':I) il;
Z + Z2 given by
(-~ -~) -+ (0,1) € Z(E) Z2'
The group 'l' is (-1 generated by -b ~) = (-1 Je o - b ~) , or in terms of
zGlz 2 by (b, 1), The quotient group is Z2b'
6,11. ~: odd and -- -- b even,
Proof: We must calculate the cokernel of the monomorphism
I - 'l' : ~Z -+ ~z given by the matrix (~ ~), If we are given
( X) € ~Z its image is (2X ) The cokernel of this homomorphism y' bx+2y ,
is ZjBZ2 if b is even, but Z4 if b is odd,
Let us form Lb = (~Z) 0 'I' Z, b
6,12, Theorem: ~ is ~ centerless ~ and there exist split
exact seguences:
1 + Z4 + Out (Lb ) -+ Z2b -+ 1, b odd,
Proof: 'I' is of infinite order in GL(2,Z) and det(I-'l') 4
and 'l' leaves no element of ~Z fixed except the identity,
6,13. Lemma: There is an exact sequence
-33-
(/ -1 O)(X Y) -b -1 z w
We obtain
Hence,
These matrices do not lie in C(~) and hence there is an epimorphism
of N(~)/(~) onto Z2 with kernel C(~)/(~). It is easily seen that
this epimorphism splits.
6.14. Corollary: Out(~) = (Z~Z2) 0 N(~)/(~), b even, and
Z4 0 N(¥)/(~), b odd is ~ split extension of out+(~) by Z2'
The smallest example occurs when b = 1. Out (L l ) has order 16
and is non-abelian,
The next example will not yield a centerless group. Let
( -1 0) /0 1) _ ( 0 -1) to obtain a matrix of order 6. o -1 ~-l -1 - 0 1
6.15. Lemma: The centralizer C(T) C GL(2,Z) is the subgroup
generated by T.
~: From the computation
-34-
is the subgroup of matrices of the form ( -b a-b
with a 2 + b 2 - ab = +1. The condition on the determinant can be equi­
valently put as a 2 + b 2 + (a-b)2 = +2. Thus only +2 is possible and
there are six solutions, corresponding to the elements in the subgroup
generated by T.
Let TI = (Z+Z) °T Z. Note that (~ ~)- (~ -i)= (-i ~) has
determinant +1 so Hl(Z;Z+Z) = O. Since C(,)/(T) = {I} we conclude
that out+(TI) is trivial. Since has period 6, the center K is
the subgroup of all elements {(O,O,6j)}. Now we observe that
( 1 1)(0 -1) /1 0) (1 1)/1 1) ° -1 1 1 = tl -1 = \-1 0 lo -1
and that (~ Furthermore,
6.16. Proposition: Out(TI) ~ Z2 and the generator of Out(TI) is
given by an automorphism of period 2 in Aut(TI). Explicitly,
1> (p,q,r) (p+q, -q, -r).
Now we form L = TI 0 Z with respect to the automorphism ~. On
the center K C TI, ¢(O,O,6j) = (O,O,-6j), thus Hl(Z;K):: Z2 and the
center of L is the infinite cyclic subgroup of elements {(e,2k)}.
Furthermore I - ~* : Hl(~;Q) ~ Hl(~;Q). To see this we have only to
recall that the homomorphism Z + TI given by r + (O,O,r) induces
Hl(Z;Q) :: HI (TI;Q) and on this image subgroup ~ is r + -r.
Now Out(TI) ~ Z2 so C(~)/(~) = {a} since ~ generates Out(TI).
Thus we have
-35-
We need only show
splits. Since ~ has order 2 in Aut(n), however, the element of
order 2 in Aut - (L) is (a ,n) + (a, -n).
In the next part of this section we shall examine a matrix in
SL(3,Z) that arises from our earlier considerations.
Let,
which yields
-h
-36- This implies b = d, h = -g, g + h h hence g = h = 0, -c
and c + f = -f hence c = f = 0, and a + b = e. We get
b
~ ). i
a+b
° Since det = ::1, we must have i (a2+ab_b 2 ) = +1. Consequently
and 2 + ab b 2 +1. This is exactly what we have studied a - for the matrix (~ il, where we found the set of matrices ( ~
f
earlier
a~b ) in GL(2,Z) generated by (~ ~) and C~ -~) . Thus we have as
generators of C (r)
°
Next we claim that HO(Z;Z~Zfr~) = Z2 and that leaves only
(0,0,0) E Z$Z&Z fixed. This follows immediately from det(I-r) = 2.
As before N(r) = c(r). Let H = (Z~Z) or z.
6.18, Theorem: The group H is centerless Out (H) is iso-
Proof: We have already found the split exact sequence:
Since Aut Z2 = 1, we obtain the desired result,
We shall now give a procedure, similar to that just treated, for
finding interesting matrices E GL(£,Z) for any positive Q, The
matrices which we will describe first lie in GL(2£,Z),
-37-
"'n -- (nn- l 11_) xn ' n > O. Let us rename ~ y Let y denote an arbi-
trary 2 x 2 matrix with integer entries and determinant not necessarily
different from O.
Proof:
-1 y·x m .
b (m-I)+d) bm+d
We have b= (c-a)+(d-b), a= (n-l) (c-a)+n(d-b) , hence,
c = a+(c-a) = n«c-a)+(d-b» = nb. Therefore, a(n-l)+c = a(m-l)+c and
a = 0 since n t m. Also as c = nb = rob, c = b = 0, hence d = O.
consider now,
b) (n-I1) (-11 ,(a d n 1 = m -(m-1V c
( a(n-l)+bn
c(n-l}+dn
a+h,\
c+d/
-b+d'\
rob- {m-l) d J
We have c = (a+b)n and m(b-d) c = -(a+b)m. Hence, (a+b)n
-m(a+b) , and since n and m > 0, a+b 0, Therefore c = 0 , b
and d = b, dn = rna -rob and as n = -m, b d = 0, hence a =
Finally, we show x = -1 has only trivial solutions. m'Y y·x n
O.
-a
bm+d -c+dn
am-dn c = dn+bm
hence, 0 = b(n+m) or b 0, d a, am -c
a = d = c = O.
Let be distinct positive integers. Form the matrix
~(nl, ... ,ni) 8 GL(2£,Z} by considering blocks of (2 x 2)-matrices
x n l 0
0
n9,
We wish to compute the centralizer C(~l and C(~}/(<I» as well as
the normalizer. Let c be an arbitrary matrix in GL(2£,Z). We
wish to determine the solutions to c<l> = <l>c. Write c in (2 x 2)-
blocks and denote the (i , j ) th b lock by coo. l.J
(Cil» (i, j) c .. <1> ••
1J J,J
(i) c .. x x c. , i, 1J n. n. 1.
J l. J
(ii) c .. x xc .. , i j. 11 n i n i 11
c .. x 1J nj
-39-
By Lemma 6.19, we have seen that there are no solutions for (i) other
than the trivial ( ~ ~) solution. Thus, each diagonal c.. block 11
must have determinant + 1. But, then, we have found the solutions
for (ii) in 6.7. They are the matrices generated by x and n. 1 (-1 0) o -1 . Let us denote by -Ii' the diagonal matrix in GL(2£,Z)
whose entries are all 1 except for (i,i)-th block which is(~l
and by X the diagonal block matrix whose (i,i)-th block is n i and whose diagonal blocks are (~ ~) • Then C (11 (n l , ••. , n £)) is
x n. 1
generated by X ,X , ••. ,X ,-I , ... ,-1£. This group is obviously n l n 2 n£ n
£ (Z Gl Z2) GJ ••• ($ (Z C:B Z2) = (Z ffi Z2) The infinite cyclic subgroup
generated by is the diagonal subgroup of Z£. Thus
o .) -1 '
We now wish to show that the normalizer, N(11), is the same as the
centralizer. We must find all solutions c ~ GL(2£,Z) of the equation
-1 ~ c.
J
(q,-lc) .. -1 -1 -1 Ik~ikCkj cP .. c. x c.
1j n. 1j' 1,J 11 1
Again by Lemma 6.19 we see that each c ij =(~ N(11) = C(o).
0) o • Hence,
We now wish to determine the cokernel of a + (I-~) (a), where
a £ z2£. The matrix (I-1J) is a block
th blOCk( 2-ni -1) and with determinant -n i 0
splits up along the blocks and we obtain
diagonal matrix with the (i,i)­
£ t (-1) TIi=ln i . The cokernel
Z <3:> Z (f). •• ® Z . Th~ n l n 2 n£
action of 1 x 1 x ••• x 1 x Z2 xl ••• x 1, with Z2 in the i-th posi-
tion on Z ill ... ff> zED .•• ® Z is trivial on all factors except n l n i n£
the i-th factor where it sends \ + -A. The action of the free part
is trivial. The extension so defined is
-40- where the D are dihedral groups:
n i fine the semi-direct product
and have shown
o ..,. Z n.
group L is centerless.
Since ~ € GL(2~,Z) we may find an element of GL(2£+1,Z) by
adding the (1 x I)-matrix block (-l)to ~ as we did earlier in 6.18.
Also, if we wish our element in GL(£,Z) to have positive determina~t,
we may add to or replace the last (2 x 2) matrix block by
(=i ~l)' (;1 ~l) or (~ ~l). We shall now examine how these
changes slightly alter the computation above.
In c~ = 1>c where 1> E: GL(2~+1,Z}, '" = ¢(x ,x , ... ,x ,-I) we n l n 2 n£
obtain in addition to the equations (i) and (ii) of the previous
discussion additional equations from the bottom row and the last
X __ (n-ll') column. Using the fact that n n 1 ' one easily checks that c
has the previous form for the 2£ x 2£ submatrix and + 1 in the
(2£+1,2£+1)-st entry with o's othenvise in the last row and last c01-
umn. That is,
C(1)(x ,x , ... ,x ,-l))=C(1)(x ,x , •.• x »@uI ~ n l n 2 n t n l n 2 n£
o
= (Z(±)Z2)£ + Z2' We also claim that it is easy to check that
N(1)) = C(1)) and so
6.21. Corollary: L is centerless, and
We now wish to add one of the (2 x 2) matrix blocks
(a l
(a 3
) ( ~1 ~l')
to the matrix <I>(n 1 , ... ,n 2 ) to get <I>(n l ,n 2 , ... ,n 2 ,a j
) EO GL(2(2+1),Z).
In order to compute Out L we need to have the analogue of Lemma 6.19
for the matrices a j above. In fact, by direct computation,
y·a. J
y·x m
have only the trivial(~ ~) solution for y a (2 " 2)-matrix with
integer entries.
6.22. Corollary: Each of the groups
'" terless. Out L(n l ,n2 , ... ,n 2 ,a j ) Out L(n l , ..• ,n 2 ) x Aj , where
Al = Z 4 0 (Z 2 x z), A2 Z 6 ~ A3 = (Z 2 ® Z 2) x GL (2, Z) •
To show that Out L = Out+L one need only check that
y·x m
y·a. J
is cen-
C 'o 1\-1 (t
y'a2 1 1) .y has solution 0 However, the trivial solutions
to the first equation guarantee that the first column of the matrix c
-42-
has only 0 entries. Hence, there is no solution c in GL(2(~+1),Z)
of the equation c¢ = ¢-lc. We re-emphasize that we have introduced
these last complications so that one can, if one wishes, find desir-
able ¢ E GL(~,Z} with positive determinant.
7. TOPOLOGICAL EXAMPLES
7.1. Let Y be a topological space and ¢: (Y,yO) + (Y,yo) be a
homeomorphism. On Rl x Y we introduce an action of Z as a group
of covering transformations by n(r,y) = (r-n, ¢n(y». Then
X(¢} = R X z Y is the quotient, and a point in X{¢} is written
«r,y)}. There is the map v : X(¢) given by v«r,y)} =
exp(2TIir}. This is a fibre map with fibre Y and structure group Z.
Hence, X(¢} is ~ closed aspherical manifold if Y ~. The cross-sec­
tion X : sl X(¢) is given by x(exp(2TIir}) = «r,yO})' which is
well defined since is fixed. For the preferred base point of
X(¢) we use From the map given
by Y + «O,y» we then obtain a short exact sequence
1 1 + TIl(Y,yO) + TIl(X,xO) + TIl(S ) + 0 which is split by X*. Thus we
may canonically identify TIl(X(¢} ,xO} with the semi-direct product
K C TIl(Y,yo) be the center. Let N(¢*) C out(TIl(Y,yO}} be the norm­
alizer of the cyclic subgroup, (¢*), generated by
9* £ out(TIl(Y,yO)}' If ¢* satisfies the hypothesis of 4.6 we have
the short exact sequence:
when TIl(X(¢),xO) is centerless and 4.8 will compute its center
otherwise.
Let us apply this to the algebraic examples constructed in §6.
-43- Let ¢
n T2 ~ T2 be the automorphism of the 2-dimensional torus
given by
and let
pherical 3-manifold has centerless fundamental group isomorphic to
L(n) of 6.1 - 6.8.
7.2. Corollary: ~ manifold M (1), (¢* = (~ f), admits an in­
volution, but ~ periodic maps of larger period. Fl~rthermore I every
involution on M{l) ~ exactly two disjoint circles as ~ point
set.
For the first part of the lemma we apply Borel's Theorem (§3) and
the fact that Out(L(l» ~ Z2' For the second part we recall that
since L(l) is centerless there is an extension
1 + L(l) + G ~ Out(L(l» ~ 0
which is unique to within a Baer equivalence. But then
G ~ L(l) 0 Z2 since Out(L(l» is generated by the image of an auto­
morphism in Aut(L(l» which has period 2. By the realization pro-
cedure (§2) then any involution on M(l) is covered by some action
of L(l) 0 Z2 on the contractible unlversal covering space of M(l).
Finally to see that the fixed point set consists of exactly 2
circles one may apply the results of the Appendix.
7.3. Corollary: For M(n), all the elements of Out L(n) ~ Zn 0 Z2
may be realized 2Y periodic homeomorphisms.
Proof: -1 -1
For all n > 0, the involution «r,zl,z2» + «r,zl ,2 2 »
-44-
is well defined and has fixed points. Simply observe that
n-l n I-n -1 -n -1. . ((r-l,2 1 22 ,21 2 2 » -> ((r-l, 21 22 ,2 1 22 » lS equlvalent to the
above. Similarly, let be a primitive n-th root of unity, n > 1.
Define a free action of Zn by 2 ( (r , 21 ,2 2 » -> (( r , 21 A, 22 A ». One
may use these two actions to define the others. Observe that all these
manifolds are boundaries in the non-oriented sense.
7.4. Corollary: Let n > 2 and T be a non-trivial periodic
homeomorphism on N(n). Then the period k divides 2n. If k is
odd then the ~ generated £y T acts freely.
Proof: For n > 2, Out L(n) ~ Z 0 Z2 and the action of Z2 on n
Z is by n ..,. -A, by 6.8. Suppose T is a periodic homeomorphism of
period k. Then Zk C Z 0 Z2· n If Y E ?ok and Y = (A,l) then
Y"Y = (A+l.k (A), 1+1) (A - A ,0) = (0,0) . Hence Y is of order 2. Thus
Zk ~ Zn' if k is odd. We have seen from §5 that no element in the
image of Z n
Out L(n) can be represented by a periodic automorphism
in Aut L(n). Hence, the group generated by T must be free.
Using the action of Z2 with fixed points mentioned above we see
that M(n)/Z2 is the non-oriented 2-sphere bundle over the circle
because each torus fibre is invariant and the quotient space of the in-
volution is the 2-sphere.
Perhaps any smooth involution is equivalent to the one we have ex-
hibited. However, one can certainly find an infinite number of non-
conjugate topological involutions by simply choosing an invariant 3
ball around a point in the fixed Get and altering the linear involution
inside the ball and keeping it unchanged on the boundary. One does
this so that the fixed point set is no longer tamely embedded inside
this ball. This procedure was first described by D. Montgomery and
R. H. Bing.
-45-
7.5. The groups L = (z z) 0 z of 6.9 - 6.14 may be geometrl-b 'jib cally realized as the fundamental groups of the total space of 5 1 _
bundles over the Klein bottle with structure group 0(2) .
We fix an integer b of 0 and introduce an automorphism on T2
by
At the identity in T2 the automorphism 'JI* induced on the funda-
mental is given by the matrix ( -1 -n GL (2 ,Z) . We introduce group -b
E
T2 obtained
as before by identifying for each integer k
(_Ilk (-llkbk (_l)k (r, z l' z 2) '" (r-k , z 1 ' z 1 z 2 1 •
We can also fiber M(b) over the Klein bottle, K, with fibre 8 1 • We
regard K as obtained from T2 by the identification
A point in
well defined. Now
exp(rrir) ,zl >. We must show p is
however (-l)kexp(rrir) exp(rri(r-k)) and k (_l)k
< (-1) exp(rrirl,zl > < exp(rrir) ,zl> E K. Thus P is well defined.
-46-
k «r+k'Yl'h» = «r,¢ (h'h»)
rrir
rrir l
rrir l
given by
maps Sl homeomorphically onto the fibre over
< exp rrir, zl >. The structure group of the bundle H"" K is 0(2).
There is a bundle K ..,. Sl with fibre sl and structure group
given by > -+ The diagram
commutes. In fact H(b) is a Seifert manifold of type (O,n,II) and
it has a double covering a principal circle bundle over the torus with
Euler class 2b.
We have calculated out('l(H(b») in 6.9 - 6.14, and shown that
rrl(M(b» is centerless. This gives us another class of manifolds
which allows only a few finite groups to act. Since these manifolds
are orientable out(ITl(M(b») is isomorphic to the group of isotopy
classes of homeomorphisms of M(b) onto itself. We further mention
that the groups Z4' b odd and Z2+Z2 can be realized as free actions
on M(b).
7.6. The group IT of 6.15 can be realized as the fundamental
group of an orientable closed flat 3-manifold as follows. On T2 let
-47- -1 be the automorphism T(zl,z2) = (z2 ,zlz2)' which has period 6.
Let A = exp 2r.i/6 and on 8 1 x T2 introduce the identification
-1 -1 1 2 (t,zl'Z2) ~ (tA ,z2 ,z l z2)' This yields Y = (8 x T )/Z6'
which is a closed aspherical 3-manifold, acted on by 8 1 , whose funda-
mental group is n. Denoting by <t,zl,z2> a point in Y we intro­
duce a diffeomorphism of period 2
which corresponds to the automorphism on 71. Again on
-1 -1 make the identification (t l ,< t 2 ,zl,z2> ) ~ (-t l ,< t2 ,zlz2,z2 > )
to obtain the closed aspherical 4-manifold X ~ (8 1 x Y)/Z2' whose
fundamental group is L (of 6.17) and which is acted on by 8 1
effectively. The group Out(L) was Z2~Z2' Out(71) was Z2' which
once again must be the group of isotopy classes of homeomorphisms as
we
well as the group of homotopy classes of self homotopy equivalences of
Y.
7.7. We may realize H of 6.18 as the fundamental group of the
orientable closed aspherical 4-manifold
The group Z generated by acts upon by
-1 (z2,z l Z2,z3 l. Any finite group which acts effectively on X(r) must
be a subgroup of Z2 $ Z2 $ Z2'
7.8. Let ~(nl,n2'" .,n~l E GL(2t,zl. There is defined a homeo­
morphism with fixed point on T2i by
n1-l n l n~-l <l>(n 1 ,n2 , .. ·,n t ) (zll'Z12,,,,,z~1'ZR,2)=(zll z12,zllz12 ... ,z~1 z£2'
n~ znz£2)'
-48- Define
We have seen from 6.20, that TIl (M2t+l) is centerless and that
Out TI l
l n
Hence, if (G,M2t + l ) is ~ finite group acting effectively, then
GC:D @ .•• $D n l n
t We also may realize the entire sum of dihedral
groups D I±l ••• ffJ D as an action on by the description n l
n t
acts on the 2 i, and (2i+1) -st
coordinates and trivially on the other coordinates as follows:
is a generator of Zn, and l
written multiplicatively. One can check that this action is compatible
with the identifications
(r,zll,···,zil,zi2,···, Zt 2) ~
n i n~
or GL(2 (HI) ,Z),
Z
similar to the previous case. This closed aspherical manifold of dimen-
sion 2(£+1) or 2t+l has centerless fundamental group and outer
automorphism group as in 6.21 or 6.22.
Let us call the action of on M2Hl des-
cribed in 7.8, the "stan?~rd action" of D on M. Because TIl(M) L
is centerless, A.4 and A.IO of the Appendix and knowledge of this
action will determine the cohomology of the fixed ,point sets for any
action (G,M) of a finite group G on M. We shall obtain generaliza-
-49- tions of 7.2 and 7.4.
7.10. Theorem: The effective action has ~ non-empty fix-
ed point set only if G is ~ subgroup of
G has odd order, then G acts freely.
In if
Let 2Hl V = V (n l ,n2 , ... ,n£) 2Hl M (n l , ... ,n~) minus the interior
of a tame (2£+1)-ball.
7.11. Corollary: If (G,V) denotes an effective action of a
finite group then G is isomorphic to ~ subgroup of
has non-empty fixed point set.
~ (2) and
Fairly simple proofs can be given if we assume that {nl,n2, •.. ,n~}
are all odd. We first treat this case. Let (G,M) be an effective
action. By the geometric realization procedure of §2 we may construct
an abstract kernel ~: G + out(TIl(M,x» = Out(L). The homomorphism
~ is an embedding of G into D. Let
be the natural projection. For each g E G, we may represent
~ (g) by r. 0 1.
k (A ,0) i ' where is a generator of Z n , k an integer
0 k and E Consider ~(g2) k 1. k < < n i , 6 Z2' r. 0 = (\ ,o)i'(\ ,o)i = - 1.
(\k\6k,6 2 ). = (\k(l+6),1) .. Thus, 2 is either identically 1 E G or 1. 1.
g
else 2 9.-g E HO(Z;Z ), and so the group generated by g2 will act freely
by §5. Consequently, the elements of G which fix anything must have
order 2. We have not used the fact that the n. 1.
are odd, yet.
Suppose H is a 2-group with non-empty fixed point set. Let
k k' g,h E Hand r i 0 'l'(g) ::: (\ ,-l)i and r i 0 ~(h) = (A ,-l)i' for some
k-k' 2 2 i. Then r i 0 'l'(gh) (\ ,l)i' Since (gh) ::: 1, r i 0 'l'(gh)
(\2(k-k') ,l)i = (l,l}i' But this means that 2(k-k') = 0 if n i is
odd. Thus, if r i 0 ~(g) ::: (\k,-l)i' \k is independent of g. Fur­
thermore, it is impossible that r i 0 'l'(g) have the form
-50- (\k,l), k I 0, for some i. For then
which implies that is even. It is clear now that
The corollary follows directly from the theorem since any effec-
tive action (G,V) extends to M by just extending the action over
the interior of the deleted ball. A fixed point is introduced at the
center of the added ball. Since the extension (G,M) is effective,
9-G must be, by the theorem, isomorphic to a subgroup of (Z2) • No sub-
group of G may act freely on V for otherwise this would introduce
an action of (Z2)jwith exactly one fixed point on the closed manifold
which is impossible.
To obtain these results in the generality stated we have to apply
some of the techniques and results of the Appendix. We intend to
compare an arbitrary action (G,M) with the standard action (D,M) •
The complications arise because we must take care of the base points.
The theorem is a consequence of the following considerations. Let
(G,M) and (G,M) , be two actions of a finite group G on an aspher-
ical manifold with centerless fundamental group. Choose a Ldse ~oint
x EM and construct abstract kernels ~, and ~' from the geometric
realization procedure of §2.
7.12. Theorem: If ~ = ~', then for any p-subgroup H of G
there is a one-one correspondence between the components of the fixed
point sets and their cohomology groups, coefficients in zp' of one
action with those of the other action.
The proof is fairly complicated. We shall impose the parts of the
hypothesis as needed. Let (G,M) be an action of a finite group on a
path connected space M. We shall assume M nice enough to admit
covering space theory. Choose a base point x E M and by means of the
geometric realization procedure of §2 define the homomorphism
~ : G + out(~l(M,X)). Suppose that h : (M,y) + (M,x) is a homeo­
morphism isotopic to the identity, then (G,M)+ defined by g(m) =
-51- -1 hgh (m) is equivariantly homeomorphic to (G,M). One can, by using
the path from y to x given by the isotopy, check that the homo-
morphism ",+ , '1'+ : G ->- Out "I (M,x) given by the geometric realization
procedure applied to (G,M)+, is equal to '1'.
Let E and E' to be induced crossed-product extensions of
'1"1 (M,x) defined by 'I' and '1". The groups E and E' operate on
* the universal covering M of M and cover the actions (G,M) and
(G,M)' .
7.13. Lemma: If "l(M,x) has trivial center then the extensions
E and E' are congruent. Moreover, if M is also a finite dimen-
sional K(~,l) and H eGis a p-subgroup, then the fixed point set
F = F(H,M) is non-empty, if and only if, F' = F((H,M) ') is non-empty.
Finally, if M is also a manifold then there is a bijection between
the components of the respective fixed point sets and an isomorphism
of their cohomology groups.
~roof: Since "l(M,x) is assumed to have trivial center,
'I' = ~', [6; p. 128] implies the extensions E and E' are congruent.
Therefore the finite subgroups of E and E' are isomorphic.
Let y be another point in M and choose a path p xy from
x to y. We may use the geometric realization procedure at y to
define an extension E Y
of by G on M*. One may choose
at will the particular representatives of paths from y to the various
gy. However, we only need make a convenient choice and so we choose
trivial paths whenever gy = y. Consequently, if H is a subgroup of
G which leaves y fixed then E Y
contains the semi-direct
"l(M,y) 0 H, (A.S). We want now to modify the extension E defined at
x, Ex' slightly. The paths from x to the various gx that were used
to define Ex can also be altered at will. One will get a new exten-
sion but it will be congruent to the old one since ", (M,x) is center-
-52- less. One chooses the new paths so that Pxy induces an isomorphism
of E Y
up to automorphisms of nl(M,x).
Suppose H c: G leaves y fixed. Then
E x and on agree
contains the semi-
direct product TIl(M,y) 0 Hand H leaves the base point over y
fixed. Also, since '\,
Ex Ey ' Ex contains a subgroup isomorphic to H
which projects to H under Ex + G. Thus, some point over y is left
fixed by this subgroup. We have shown that if y E M and H c: Gy '
then there is ~ subgroup isomorphic to nl(M,x) 0 H c: Ex' and ~ point
in M* over y which is left fixed by the embedding of H in Ex.
If we now assume that M is a finite dimensional K(TI,l) then
Smith theory may be applied and we find that for any finite p-subgroup
H of Ex' the fixed point set F* = F(H,M*) is not empty. Also from ,then H c: Ex and F(H,M*) -I r/J.
above, if H c: G and F - F{H,Mj t ¢,V We now wish to apply the results
of the Appendix. Let H c: G be a p-subgroup so that F = F(H,M) I ¢
and let M also be a manifold. We may assume without lOGS of gener-
ality that the base point x is in F by altering M by an isotopy
if necessary. Consider the other action (G,M) '. Since E' is
naturally isomorphic to Ex' E' and Ex contain isomorphic finite
suh;rroups. We wish to show F' = F ( (H ,M) ') is not empty. Clearly
there is a finite subgroup H' in E' which is isomorphic to Hand
projects to H in E' + G. Hence, F(H' ,M*) f ¢ and therefore,
F' f ¢. We may also perform another isotopy of M and get an action
equivalent to (G,M)' but which now has x E F'. We will denote it
also by (G ,M) , •
Now we have two actions (H,M) and (H,M) , with 1jf = 1jf' and
x E F n F'. By A.4 and A.IO of the Appendix we may conclude that the
components of the fixed point sets are in one-to-one correspondence
and their cohomology groups are isomorphic. This completes the proof
of Lemma 7.13 and Theorem 7.12.
-53-
We can apply 7.12 in the following situ