Internat. J. Math. & Math. Sci.
17
(1987) 17-33
Vol. i0 No.
DIFFEOMORPHISM GROUPS OF CONNECTED SUM OF A PRODUCT OF
SPHERES AND CLASSIFICATION OF MANIFOLDS
SAMUEL OMOLOYE AJALA
School of Mathematics
The Institute for Advanced Study
and
Princeton, New Jersey 08540
(Received April
ABSTRACT.
In
given where
i] and
Hp(M)
H
2]
Mathematics Department
University of Lagos
bkoka-Yaba
Lagos-Nigeria, West frica
21. 1986)
M
a classification of a manifold
n.p(M)
Z
of the type
(n,p,l) was
only non-trivial homology groups.
is the
In this
paper we give a complete classification of manifolds of the type (n,p, 2) and we
(n,p,r)
extend the result to manifolds of type
and
p=
3,5,6,7
where
r
is any positive integer
mod (8).
KEY WORDS AND PHRASES. Pseudo-diffeotopy classes of diffeomorphisms, diffeotopy.
1980 AMS SUBJECT CLASSIFICATION CODE. 57R55
0. INTRODUCTION.
In I] Edward C. Turner worked on a classification of a manifold M of the type
(n,p,r)
H (M)
p
where this means that M is simply connected smooth n-manifold and
r
the only non-trivial homology groups except for the top and
Z
(M)
i
bottn groups. He gave a classification of such manifolds for the case r= I and
Hn_
35,6,7 (mod 8). So Turner gave a classification of M of type (n,p,l) and
p= 3,5,67 (rood 8). In 2] Hajime Sato independently obtained similar results for
M of the type (n,p, i). The question which naturally follows is: Suppose r= 2 3,4
and so on, what is the classification of such M
i.e., what is the classification
of M of the type (n,p, 2), (n,p, 3) and so on? In this paper we will study
p=
manifolds for the type
eralize the result
and
(n,p, 2)
and give its complete classification and then gen-
M
to manifolds
p=
3,5,6,7 (mod 8).
In
I
of the type
(n,p,r) where r
is an integer
we prove the following
THEOREM i.i
Let M
be an n-dimensional oriented, closed, simply connected
manifold of the type (n,p, 2) with
q+l
p
p
to S X
U SP X
X D
p=
Dq+l#s p
Dq+l#s
3,56,7 (mod 8).
X D
q+l
where
Then
n= p+q+l,
M
is dlffeomorphlc
means connected
h
sum along the boundary as defined by Milnor and Karvaire
---> S p X sq# S p X S q is a diffeomorphism.
p
p
In 2 we compute the group
X sq#s X S q)
classes of diffeomorphisms of S p X sq# S P X S q p < q
0Diff(S
[3] and h Sp S q # S p x sq-->
of pseudo-diffeotopy
S.O. AJALA
I
2 X 2 unimodular matrices and H the subgroup
ab
the
=cd=O rood 2 and Z
such that
of GL(2, Z) consisting of matrices
4
c d
0 1
We will adopt the notation
subgroup of GL(2, E) of order 4 generated by
0
p
p
S q) and M+
the subgroup of M
consisting of
M
Diff(S
X
X
P, q
P, q
P, q
diffeomorphisms which induce identity map on all homology groups. We will then prove
Let
GL(2, Z)
denote the set of
6
-i
sqs
the following
THEOREM 2.1 (i)
p+q
If
even, then
is
Z
qO (M, q)
p+q
If
4
p
if
even, q
is
GL(2, Z) O GL(2, Z)
T0 (Mp, q)
(ii)
Z
4
H
H
p,q
if
GL(2, Z)I{
if
Z4
p
if
is even
p,q= 1,3,7
1,3,7
odd but
1,3,7,
p
1,3,7
q is odd but
is odd then
Z
H
if
GL(2, Z)
4
is even q
if
p
is odd but
is even and
I, 3, 7
i, 3, 7
q
We will further prove the following
EOREM 2.15
If
p
< q
and
is twice the order of the group
p=
q
(mod..8)
3,5,6,7
the order of the group
@..i
(SO(p+l)) O
va(M,q)
In 3 we apply the result in 2 to prove the following
Let M
THEOREM 3.7
such that
n= p+q+l
be an n-dimensional,
H
p
oriented manifold
and
n i0)
then if
smooth, closed,
zH
i=O,n
i=p,q+l
0
elsewhere
the number of differentiable manifolds up to dlffeo-
3,5,6,7 (mod 8)
morphism satisfying the above is equal to twice the order of the group
@p+q+l With induction hypothesis and technique used in i and
(SO(p+I))
Wq
2,
one can prove the following
{EOREM
(n,p,r)
3.8
If
M
is a
n=p+q+l
where
and
smooth, closed simply connected manifold of type
then the number of differentlable
3,5,6,7 (rood 8)
p=
manifolds up to diffeomorphism satisfying the above is equal to
r times the order of
7
q
SO(p+l)
e @p+q+l
I. MANIFOLDS OF TYPE (n, p, r)
Let M
DEFINITION:
be of type
(n,p, r)
H (M)
i
where
be a
closed, simply connected n-manlfold.
is said to
if
H
if
.r
if
0
elsewhere
i=0,n
i p, q+l
n= p+q+l
We recall from Milnor and Kervaire
DEFINITION:
M
Let M
I
and M 2
be
3]
(p+q+l)-manifolds with boundary and H
p+q+l
DIFFEOMORPHISM GROUPS OF CONNECTED SUM
19
be half-disc, i.e.,
Hp+q+l
pqLet D
H
be the subset of
(H
i
so that
i
-I
,xp+q+ll
ix= Xl, X2,...
pql--
p+q+I,Dp+q)
for which
>
by identifying
We can choose embeddings
0
I
O?
1,2
(Ml-i I(0))_+ (M2-i 2 (0))
We then form the sum
i2((l-t)u)
with
il(tU
x
i, x I _>
(M,SM)
reverses orientation
2. i I
Ixl --<
<
0
for
<
t
[I-6sqNH
i u
is called the connected sum along the boundary and will be denoted by
REMARK: (i) Notice that the boundary of
M
(2) M I
2
M
M
1
M1
is
2
MIV M2
has the homotopy type of
M
I
This sum
M
2
1t 2
the union with a single
point in connon.
THEOREM i.I
h
S
p
M
If
3,5,6,7 (mod 8)
p
X S
q
# Sp
S
X
q
--->
S
p
X S
q
which induce identity on homology such that
Sp
D q+l
(n, p, 2)
is a smooth manifold of type
where
#
Sp X
Dq+Ius p
[M, %1’ 2
q+l
D
#
#
p
S
S
embeddings
p
<
q
Let
Sp
0.i
and
M
S
q
is diffeomorphic to
p
X D
q+l
(n,p, 2) and
Z
the generators of the first and second sun,hands of H (M)
H
PROOF:
p+q+l
n
then there exists a diffeomorphism
be a manifold of type
kl’ 2
We can choose
P
--> M
so as to represent the homology class
Let
two homotopic embeddings are isotopic
characteristic class of the embedded sphere
S
p
i
since
6
represent
k.1 i
p_iSO(q+l))
I, 2
p= 3,5,6,7 (rood
normal bundle of the embedded sphere is trivial It follows that
q+l
p
embedding 0i S X D
--> M such that its homology class is
0i
Since
be the
8), the
extends to an
we can
k.l Then
q+l
Sp D
to
q+l
sPxDq+IsPxD --> M
q+
q+l
boundary’of sPxD SPxD
form a connected sum along the boundary of the two embedded copies of
q+l
q+l
p
p
S X D
We then have an embedding i
get S X D
i,[ S p] k I + k 2 6 Hp(M)
q
S pXS q#S pXS
and since S pX
such that
is
sPxDq+Iv sPxDq+l
Notice that the
Dq+l#spXD q+l
then it is easy to see that
p
p
H (S XD q+l#s XD
i
q+l)
It is also easy to see that
H (M-Int(S p X D
i
Now since
SPX D q+l
q+l
# S p X D q+l))
[
has the homotopy type of
-H
H
L"
is a trivial disc bundle over
Z
Sp
for
i=0
for
i=p
for
i= 0
for
i=p
then it has cross sections;
q+l
q+1
p
S XD
onto
# Sp X D
hence, there exists orientation reversing diffeomorphism of
itself.
Thus there exists an orientation reversing embedding
j
S pxD
q+l#S pD q+l
---> M-Int(SpD q+l#S pD q+l)
S.O. AJALA
20
j,[S p] kl+ k 2 and in fact this embedding is a homotopy equivalence.
q+l
is diffeomorphic to
[4, Thin. 4.1] that sPxDq+IsPxD
such that
follows by
M-Int(S p
S
p
h
xD
q+l
D
@S
q+l
p
p
S XD
q
h
M
it follows that
is diffeomorphic to
for an orientation preserving diffeomorphism
b
S p S qS p XS q
--->
From the embeddlngs in the proof, it is clear
induce identity on homology.
h
that
Consequently,
q+l U S p X Dq+l # S p X D q+l
XD
xsqs p XS
Sp
q+l)
It
pxS q#Sp XS q)
2. THE GROUP
q
p
p
S
S q) and M+
For convenience, we adopt the notation M
Diff(S X S
P, q
P, q
q
q
p
p
the subset of M
S
S
consisting of diffeomorphisms of S x S
which induce
q
0Diff(S
P,
identity on all homology groups.
Let M
DEFINITION:
be an oriented smooth manifold.
Let
M
orientation preserving diffeomorphisms of
f,g 6 Dill(M)
(f(x),0)
H(x,O)
H(x,l)
and
class of diffeomorphisms of M
"’0(Mp, q)
p < q
for
If
Mp, q
Sp
f
then
of
x 6 M
for all
and
MXI
g
are
such
The pseudo-diffeotopy
We wish to compute
0(DiffM)
is denoted by
f, 6
p
f, :H,(S pxs q#s XS q)
of homology groups of
(g(x),l)
f
H
said to be pseudo-diffeotopic if there exists a dlffeomorphism
that
is the group of
Diff(M)
induces an automorphism
---> H,(S pxS q#spxS q)
x S q # Sp x S q
Since pseudo-diffeotopic diffeomorphisms
induce equal automorphism on homology then we have a well-defined homomorphism
0(Mp, q) --> Auto(H,(Sp X S q # Sp X S q)
where
of
Auto(H,(S
p
XS
q
# S p X S q)
denotes the group of dimension preserving automorphlsms
,(S pxs q#s pxs q)
THEOREM 2.1 (i)
If
p+q
is even then
p,q are even
if p,q are 1,3,7
if p,q are odd but
1,3,7
if pffi 1,3,7 and q is odd but
if
Z4 -4
$
q))=..O
(--(Mp,
GL(2, Z)
H
$
GL(2, Z)
+
H
GL(2, E)
H
The following propositions give the
PROPOSITION 2.1
If
0p,q)
PROOF:
Since
p+q
p+q
is
proo.f of Theorem 2.1.
even,
p
is
even, then
Z4 Z4
is even and
have
H i(SpXS q$S pS q)
Z
0
Generators of
H0(sPxs q#spxS q)
and
is even then
p
-
if
i=O
if
i
p
q
must also be even.
H
We
p+q
or
q
e I s ewhere
Hp+q(SpXS q#spxS q)
are mapped to the same
(sPs q#SpXS q) Z Z If f 6 Mp, q we
P
p
p
p
the automorphism f. :Hp(S XS qs XS q) ---> Hp(S xS qS PXS q)
Then (f)p
f under
in dimension p
f. :Z Z---> Z $
generators but
+ 1,3,7
-
shall denote by
(f)p
induced by the image
is the induced
DIFFEOMORPHISM GROUPS OF CONNECTED SUM
automorphism.
p
q#s p
H (S xS
P
e
e
l
i
2
If
el, e 2
S q)
if
and
e
are the generators of the first and second summand of
denotes the intersection then
e
2
21
Let
-i
I
e
E GL(2, Z)
a3 a4
e
lo
e
0
I
(f)p
if
2o
e
0
2
el, e 2
takes
to
el e{ (ale I + a2e2).(alel + a2e2)
alale l’e I + ala2e l-e 2 + a ale 2-e I + a2a2e2-e
ala2el’e 2 + a2ale2-e I ala 2-ala2 0
Similarly
e
2
0
2
+
a2e2) (a3e i + a4e2)
ala3elo I + ala4el e 2 + a2a3e2o e I + a2a4e2 e 2
ala4 a2a 3 I since GL(2, Z) is unimodular.
(a3e I + a4e2)’(ale I + a2e 2) a3alel e I + a3a2el e 2 + a4ale2o
+ a4a2e 2 e 2 a3a2-a4a I -i
’.
el e2
but
e
(ale I
e
e’el
hence for
(f)
even
p
0 1
-I0
Similarly for
of
Z
(sPx Sq)l # (sP sq)2
q
S
of
If
(xl, Yl
0
Sp
#
0+i
generated by
(f) p E-4.
Hence
Z4
it then follows that
(0(Mp, q))
We need to show that the generators
Z c
4
can be realized as the image of
4 Z4
p
S
I-0+I +_i (+_.i -0
1
(f)q 4
(n0(Mp, q)) c Z4 , -4
6
i=q
We now show that
(R)
GL(2, Z)
is an element of a subgroup of
P
This subgroup has elements
e
S
q
4
We shall adopt the notation
where the subscripts i and 2 denote the first and second sUn.hands
q
p
be reflections of S
and R
and S
R
respectively.
P
p
we define f 6
XS
and (x2, Y2 6
q
and let
E (S pXS
q(s
q)l
q)2
Mp,
(Rp(X2),Rq(Y2))
f(xl, Yl)
(xl, Yl
f((xl, Yl)(x2, Y2))
f(x2,y2
In other words
p
6 (S XS q) l
(xl, Yl
E Auto
(f)p
For
f
takes
(f)
P (e2)
e
e
2o
maps
f
2
0
I
g
4
4
f
if
I
e
-e
I
lo
p
-
el, e 2
are the generators of the first and
Z
e
to
2
-e2o
2
0 1
0
(-i
q
e
and
e
I
e
2
I
and
eo
which generates
to
-I 0
e
Z
x
(f)p(el)
and so
I
takes
f
since
then it is easily seen that
in dimension
e
I
4
lo
e
e
lo
1
-e
Rp(X2)
to
I
-e
2
and
-e2o-e 2
0
-I
Hence
2
Similar argument shows
which generates
Then
Z4
maps
hence the proof.
PROPOSITION 2.2
If
p+q
is even but
p,q
1,3,7
then
(0(Mp, q))
GL(2, Z)
GL(2, Z)
PROOF:
0 1
-I0
maps f
x
e
in dimension
maps
Z
to
Hence
I
e
el
2
that
onto
e
x
Hp(Mp,
(sPxsq)2
(x2’Y2)
H (sP S qs p S q)
P
second summands of
and
and
((Rp(X2),Rq(Y2)) (xl, Yl))
From
1
[5, Appendix B]
1)
and
[6] one sees that GL(2, Z)
Since p,q
and
1,3,7 it follows by [7,
01
q
p
S ---> SO(p+l) and g S m> SO(q+l) such that f
is generated by
1] that there
and
g
exist
have index +I.
S.O. AJALA
22
We then define h 6 M
P, q
E (S p x sq) 1
(Xl, yl
(Xl, yl
h(x2, Y2) (f(xl)’X2 g(yl)’Y2
(x2, Y 2)
h((Xl, Yl) (x2, Y2)) ((xl, Yl) (f(xl).x2, g(yl)’Y2)
h(x I, yl
i.e.,
f
and
+i
h
takes
6
(sPxsq) 2
and x
to f(x l)’x
to x
2
2 then it
I
I
follows by an easy application of [7, Prop. 1.2] or [6, Prop. 2.3] that
@(h)p is
i
also since g has index +I and h takes
and
to
to g(yl)-y
Yl
Since
has index
x
I)
(h)q
then
6M
(0
is
1
P, q
(xl, Y I)
(Rp(x2),Rq(Y2))
(x2,y2
(xl, Yl
((xl, Yl),(x2, Y2)
Since
C
takes
that. (Ct)p
that
x
maps
Rp(X2)
to
I
c
x
x
to
2
o
(-I 0)’ (-io}
0)
to
then it follows that for
GL(2, Z)
PROPOSITION 2.3
If
H @ H
l(rr
0 1
(sPxsq) I
6
p
p+q
@
(S XS
q)2
it follows from Proposition 2.1
I
()q
GL(2, Z)
Since
I)
0
-1 0
is
This means
is generated by
(0
1
1,3,7
p,q
GL(2, Z)
is even but
p
and
are odd but
q
p,q
+ 1,3,7
o (p, q))
By using Proposition 2.1 and [8, Lemma 5] it is enough to produce a
PROOF:
diffeomorphism in M
+i
and
and by similar reasoning
(rro(Mp, q)
P
(xl, YI)
(x2, Y2
((Rp(X2),Rq(Y2)),(xl, Yl)
0 1
-1 0
is
0 1
-I 0
and
1)’
0
2
by
i.e.,
then
Y2
Yl
and
q
0
+i
P’q
whose image under
is
0 +i
+I 0
I--n +l)’I +I -’n)’
because-v
This is
is trivially the image under
coordinate
0 +I
(i
IO 210
I
1
2
I. in each of the dimensions
generate H. However
of identity map and reflections on each
is by Proposition
2.1 the image under
of an element
p
M
there exists a map
S ---> SO(p+l) of index 2 by [8] so
However,
P, q
S q --> SO(q+l) of index 2 and then we can define f 6 M
also is a map
thus.
P, q
p xsq)
f(xl, Yl (xl, Yl
(Xl, Yl 6 (S
1
while.
+I
--0
of
f(x2, Y2) ((Xl)’X2, 8(yl)-y 2) (x2, Y2) 6 (sPxsq) 2
f((xl, Yl) (x2, Y2)
((xl, Yl) ((Xl).X2, 8(yl).Y2)
i.e.,
It easily follows that since
f
takes
x
I
to
c having index 2 then it follows by applying
1 2
Similar argument shows that {(f)
0
q
i)
PROPOSITION 2.4
{(w0(Mp, q))
GL(2, Z)
-1
since
R
is reflection of
P
13,7
is
even,
p=
I
and takes
[7 Lemma 5]
is
(10 21
1,3,7 but q
x
that
to
2
(f)p
hence
c(xl)’X2
with
is
{(0(Mp, q))HeH.
is odd and
+ 1,37
then
@ H
0)’ (0 1)
H
q
If p+q
x
(-1 0)’ (0 1)
GL(2, Z) while
and by [8] there exists :S q ---> SO(q+l)
Sp
generates
then we define
h 6 M
P, q
generates
of index 2.
If
DIFFEOMORPHISM GROUPS OF CONNECTED SUM
h
Since
takes
R
Now if
f
define
and takes x
R _(x
2
P 2)
similarly h takes
to
I
is a reflection on S
q
q
6 (S
x
to
Yl
pxsq)I
(sPxsq) 2
E
(h)q
it follows that
2,
has index
since
(Xl,(yl).Y2)
1
(-I0 0)
(h)p
that
x
(xl, Y I)
(x2, Y2)
(Rp(X2),y I)
h(Xl, Yl)
h(x2, Y2)
it follows by Proposition
I
y, and
to
12)1
ct(Yl).y 2
to
Y2
2.1
and
0
sP --->
and
SOp+ I
is of index
+i
then we
EM q
P,
then it is easy to see that
REMARK.
For
i i
(0 I)
(f)p
o(Mp, q))
then it follows that
(xl, Yl)
(x2, Y2)
(Xl, Rq(Y2)
((Xl)-x2, Yl
f(xl, Yl)
f(x2, Y2)
sequently one manifold) comes in here,
Combination of Propositions
(0(Mp, q)) 4
PROOF:
p
q
p+q
is odd and
we have the same result as
< q only
one dimension (con-
7
5
7
S XS
i.e
is even and
p
# S5
XS
7
odd
q
+ i, 3, 7
then
H
01
(-I0)generate
Since
Z
01
12}
(-I0)’(01)
and
4
then we only need to find the diffeomorphism in M
P, q
Similar to Proposition 2 4, we define f 6 M
erators
f(xl, yl
f(x2, Y2)
PR-
the image of
2.1, 2.2, 2.3, and 2.4 proves Theorem 2.1(i).
Suppose
PROPOSITION 2.5
5
p
sO
Hence the proof.
H
viz
q)2
1
(-I0 O)
1,3,7 and q=1,3,7
odd but
p
E (Sp XS
(f)q
and
GL(2, Z)
(sPxsq) I
E
.above using the same method but since by assumption
where
23
(Xl, C(Yl)-y2)
Sp
is the reflection on
and
cx:
Sq
maps to these gen-
that
by
p, q
p
(S x S q) 1
(Xl, yl
(x2, Y2)
(Rp(x2, yl
generate
(
-->
(sP x sq) 2
is of index 2
SOq+I
which exists
0 1
p
g 6 M
P, q
Also we define
thus
i.e.,
(Xl, Rq(Y2)) (xl, YI) E (sPxsq) I
g(x 2,y2
(x2, Yl
g((xl, yl), (x2, y2)) ((Xl, Rq(Y2)), (x2, yl))
where
R
then
q
(g)p
is the reflection on
identity
i 0
(0 1
sq
and since
it follows that by applying Proposition
these matrices generate
O(Mp, q))
Z
4
and
Z
4
g
takes
O(g)q
x
Yl
x
to
I
I
and
Rq(Y2)
to
0 1
-i 0
x
2
and
Hence
to
Y2
f
x
2
to
(0(Mp, q)) -4
Suppose
GL(2, Z)
p+q
respectively then it follows that
is odd and
p
is even
q
is odd and
Yl
is mapped
H
PROPOSITION 2.6
Then
H
2.1,
takes
g
Since
(sPxs q) 2
(x2, Y2)
g(xl, Y I)
I, 3,7.
S.O. AJALA
24
PROOF:
q=
Again since
0 1
Since
(-i 0)
of index I.
we define elements of
M
P, q
1,3,7 by [6, Prop. 2.4] there
Z
generates
I I
q
-> SO q+l
generate GL(2, Z)
(0 I)’ (-I i)}
and
:S
exists a map
0
4
0
that are mapped onto these generators.
Let
h 6 M
P, q
be defined thus
(Rp (x2) yl)
h(Xl, yl)
h(x 2,y2
(Xl,(yl).Y2)
(x2, Y2
((Rp(X2),Yl) (Xl,(yl)-Y2))
h(Xl, Yl) (Xz, Y2)
i.e.,
Rp is
(h)q
where
while
6 (S pXS
(x I,yl
where
the reflection of
S
p
6 (S
p
q)l
XS q)2
Then it is easy to see that
Also one can define
f 6 M
-i0 II/
(h)p
0
as
P, q
(Xl, Rq(X2) where (xl, Y I) 6 (sPxsq)I (x2, Y2) 6 (sPxsq)2
f(x 2,y2
(x2, Yl
i.e., f((xl, Y I) (x2, Y2))
((Xl, Rq(Y2)) (x2, Yl)) where Rq is a reflection
f(xl, Y I)
and so it is easily seen that
(0i0i)while
(f)p
(-IOl0)so
(f)=
q
and., since these sets of matrices generate GL(2, Z) and
(0(Mp, q)) Z4 GL(2, E) Combining Propositions 2.5
Z
h
S
of
q
is
respectively then
4
and
2.6,
we obtain Theorem
2.1 (ii).
REMARK.
+
If
p is odd but
i, 3,7 and q is even, we get the same result
as in Proposition 2.5 using equivalent method. Also if p i, 3, 7 and q is even,
we obtain" the same result as that of Proposition 2.6.
/
Mp, q
Since
denotes the subgroup of M
consisting of diffeomorphisms of
P, q
which induce identity map on all homology groups, it follows that
sPx sq# sP sq
M+
is the kernel
P, q
f
We now compute M+
p, q
the homomorphism
We define a
homomorphism
G:
p
i(S X
where
p
S
X
H
P
[p0])
P0
[p0
0 (M, q) --> pSO(q+l)
is the usual identity embedding of
S
is a fixed point in
in
S
p
q
is homologous to
Z
Z
p
i(S X p)
[p0
p
S X Sq
into
far away from the connected
x S q # S p x S q represents
(sPxsq#s PXS q)
p
S X
Since
sum, then
the sphere
a generator of the homology
(f)
and since
p
S X Sq
p
f(s
is identity, it follows that
< q
and by Hurewicz theorem,
f
pxp0)
and
are homotopic and in fact with the dimension restriction, they are diffeotopic.
f
is diffeotopic to a map say
tubular neighborhood theorem,
p
q
S D
where f"(x,y)
f"(sPxD q)
(x,(f")(x)-y)
SO(q) --> SO(q+l) be the inclusion map and
map on the homotopy groups. Then we define
i
G[E}
LEMMA 2.7
PROOF
f.h
-16M+P, q
G
Let
i,
f"
and
(f")
7DSO(q)
-->
such that
p
S
---> SO(q)
17DSO(q+I
is well-defined.
such that
f
and
h
is pseudo-diffeotopic to the identity.
are pseudo-diffeotopic then
If
G[f]
i,(f")
By
Let
the induced
i,((f’))
+
f,h 6 MP,q
i
and
DIFFEOMORPHISM GROUPS OF CONNECTED SUM
Sp
(x, y) 6
D
q
(x,(f")c(h")-l(x).y)
SO(q+l)
G(h)
i.(h") 6 P
(x, i.(f") (x)" y)
Diff(s pXS q) defined by
fl(x,y)
Hence
-I
fl-h[
i.c(h")
i.(f")
We wish to show that
f.h
(x,(h")(x)’y)
h(x,y)
and
for
then it follows that
f.h-l(x,y)
Since
(x,(f")(x)-y)
f(x,y)
where
i.(h")
G(h)
25
(x,y) 6
fl, hl
Diff(sPxs q)
glDiff(sPxs q)
p+I
fl’h[
I
by
fl.h hence
p
--> SO(q+l).
S
i. (f,,) i. (h,,)
6
it extends to a
p+I
denote the q-sphere bundle over
$8
-i
flhl
q
its definition.
g 6 Diff(D
there exists
i.e
Let
S
characteristic map
q
and
thus
p
(x,y) 6 S S
is diffeotopic to the identity
S
D
of
g
diffeomorphism
X S q)
then consider
flhl-l(x,y) (x,i.(f") i.(h")-l(x)’y)
I
is pseudo-diffeotopic to identity so is
1 6
E Diff(Sp
(x, i.(h") (x) y)
hl(X,y
and
i.c(f") 6 pSO(q+l)
G(f)
Since
then we can define maps
sPD q
XS
q)
such that
p+l-sphere with
Then we have
p+I X S q
-l- D
v_)i Dp+I
sq
X
flhl
so this gives a q-sphere bundle over a p+l-sphere with the characteristic class of
i.c(f")-i.(h")
the equivalent plane bundle being
g 6
Diff(DP+Ix S q)
-I
However
S
f"
i.
Hence we define a map
i. (h")
H
S
D__
-I
p+I
q
S
-->
S
f
(x,y)D
if
(x,y)
H(x, y)
g(x,y)
I X
S
flhl
i. (f") i. (h")
-I
s
DIXS
This means that
is well-defined and is a diffeomorphism.
Si.(f,,).i.(h,,)-i -I is
p+I with characteristic class
a trivial q-sphere bundle over S
It then follows from [I Lemma 3.6(b)] that i.(f")
i.(h’’)
defined.
It is easy to see that
i.(pSO(q))
G
By the definition of
PROOF:
c
G(o(M,q)6 M
f
then we can define
a 6
i.((SO(q)))
In fact since p <
q
i.p(SO(q))
if
p
(x,y) 6 (S X S q) l
(x,y)
if
(x,y) 6 (s p x s
f 6
then
"vn(M+-q
q)
P
(S
np+ISq---> npSOq ----> npSOq+ I --> ripS q
G
[a]
and
(x,a(x)’y)
i.
easily seen that
G
is well-
where
we then show that
a:S p
-> SO(q+l)
by
P, q
then
Hence
G(o(M.,q)
) c i.(.SO(q))
G
6
If
i.(f")-i.(h")
is a homomorphism.
f(x, y)
since
extends to
sq=
p+I
s
H
flh-I
i
then we have
is surjective.
and so
0
q)2
G(f)
6 i.(.SO(q))
hence it follows from the exact sequence
> that i.
Hence the proof.
is an epimorphism and so it is
S.O. AJALA
26
[6, Lemma 3.3].
The next lemma is similar to
LEMMA 2.9
such that
f
Let
u 6 ker G
S
is identity on
PROOF:
If
< q-I
p
then there exists a representative
pXD q
p+I(S q)
then
0
and also
P
(S q)
0
f 6 M+
P’q
of
u
and so it follows
from the exact sequence
Tp+ I (S q)
--->
that
is an isomorphism hence if
i,
(P’)
f(x, y)
i,
np (SOq) --> Wp (SOq+l) --> np (S q)
--->
0
(x,y)
be defined
hence
f
p
q
S XD
sPx D+
connected sum in M
g(x, y)
g’
and
M
G(u)
f
P, q
and clearly by the definition of
and represents
u
because it is pseudo-diffeotopic to
We now wish to compute
0 (Diff+ (Sp X S q))
Given
u
f
is also pseudo-
f
is pseudo-
keeps
sPx D+q
fixed
Hence the proof
f
and
Diff+(sPx S q)
Here we adopt the notation
SPX S q
is surjective.
the set of all diffeomorphisms of
homology groups.
g
g
then
i,c(f")
To do this, we define a homomorphism
ker G
---->
g
0
g
Then the composition
diffeotopic to
N
P, q
xsq)2
is pseudo-diffeotopic to the identity and so follows that
show that
g 6 M
away from the
(sPxs q) i
G
u 6 ker
are diffeotopic and since
Ker G
general let
6sPxD+q and SPxDqc(SPxSq) I
p
(x,y) 6 (S
if
diffeotopic to the identity in
N
implies
we define
(x,y)
g
(x,y)
for
g’ 6 M q by
P,
(x,i,(P’)-l(x)’y) if (x,y) 6
Tp(SO(q+l))
g’ (x, y)
then
However in
(sPx Sq)l
0
p
(S X S q) 2
(x, y)
g’
i,(P’)
then it means
are subsets of
(x,(f")’l(x)’y)
i,c(f") 6
G(u)
then
(xy) 6 S p XD q
we then define
pq
since
6 ker G
for
p
q
S xD
is identity on
q
thus, if
[f
u
(x,(f’)(x).y)
f(x,y)
Since
--->
to mean
to itself which induce identity on all
6 Ker G
let
follows from Lemma 2.9 that we can take
+
be its representative then it
f 6 M
P, q
q
f to be identity on
D
So we have
SPX
a map
f
f
(S
is identity on
p
xsq) I# (Sp xsq)2
q
p
S pXD c (S xsq)
"-’>
(Sp XS q)3 # (Sp xsq) 4
I
Using the technique introduced by Milnor
modification on the domain
[9] and 3],
we perform the spherical
(sPxsq)I# (SP xsq)2
that removes
p
Clearly we obtain "(S x S q)
p+l S q-I
and replaces it with D
X
p
p+I S q’l is
S x
X
diffeomorphic to
Dq
id
such that
S
p+q
Since
f
sPxDqc(SPxSq)I
2
since
is the identity on S
p
x Dq
p
S X DqC (sPx S q)
and then perform the corresponding
3
p
p
to obtain (S X S
spherical modification on the range (S X S q) # (Sp X S
3
After this modification we are then left with a diffeomorphism say f’ of (sPx S q)
p
onto (S
pXS
ioe., f’ 6 Diff(S XS q) since f 6 M+
then f’ 6
P, q
So we define N[f}
f’]
we can assume that
f(S p X D q)
q)4
pXSq)4
q)4
Diff+(S
)
DIFFEOMORPHISM GROUPS OF CONNECTED SUM
N
LEMMA 2.10
is well-defined.
is identity on
Sp D
diffeotopic to
g
q
and
p
xsq#s p XS q) XI)
FI(sPsq#sPxsq)
and
Sp D
q
is pseudo-
f
Since
f
then
g
is pseudo-diffeotopic to
f
is also identity on
g
then there exists a diffeomorphism
F 6 Diff((S
sPxDqXI
such that
f,g 6 Ker G
Let
PR/)OF:
27
is identity on
FI(sPsq#sPxsq) xl g If
p
(S x S q)1 # (Sp x S q)2 X I
replacing it with +I sq-I Xl
while
f
0
F
such that
of
we now perform the spherical modification on the domain
Xl and
F by removing SpXD qXl c (S pS
q
p
p
then we obtain the manifold (S X sq) 2 X I and since F is identity on S X D X I
# (Sp X sq) 4 X I
we then perform the corresponding modification on the range (sP
q
p sq)
p
I>+l
D
I to obtain
c
it
with
and
replacing
I
I
(S
by removing S X D
3
(sP sq) I We then obtain a diffeomorphism
q)l
sq)3
sq-Ix
4
(S pxS
F’
p
F’
Diff+(S p S ql)
F’ 6
i.e.,
S XS
q
N
It is easy to see that
LEMMA 2. II
Let
PROOF"
such that
N
h’ 6
is
p+q then
Diff+(sPx S q)
we need to find a diffeomorphism
we have
h’ 6
(x,y)
if
h’(x,y)
h
+
6M
Sp
Diff+(SPxSq)I (S
P’q
Since
XD’cq(sp XS q) 1
h
apply this result of
S q)
P, q
is
+
h 6 M
P, q
We then define
(SPX
S q)
is well-deflned and
h
as earlier stated,
then it is identity on
I
N(h)
and clearly
h’
and so
---->
pSO(q+l)
N
is surjectlve.
which is similarly
and where Sato gave a computation of
G
We will
Ker B
to the next lemma.
Ker B
Ker N
Let
h’
then we can assume
[6, 3] the homomorphism
defined as homomorphism
PROOF:
h 6 Ker G
0Diff+(Sp
LEMMA 2.12
sq)2
is identity on
hence
We recall from
B
x
+
h 6 M
p+q
p
(x,y) 6 (S XS q) I -D
p+q
(x,y) 6 (S p XS q) 2 -D
if
p
q
Diff+(SPx S q- Dp+q)
h (x, y)
where
SPX S
is a disc in
thus
+
M
and
is a homomorphism.
p+q
If D
D
identity on
F’ I(S p XS qX0)
f’ and
pseudo-diffeotopic to g’ and so N is well-defined.
F’
is surjective.
h’
N(h)
q)4 I
N(F)
hence
g’ hence f’
i
(SpS
q)2 I -->
Ker B
is in one-to-one correspondence with
f 6 Ker B
f’ 6 M+
we will produce a diffeomorphism
+(S
p
Since f 6 Ear B then f 6 Diff
f 6 Ker N
XS q) and
p
p
XS
We define a diffeomorphism f’ (S xS
--’> (Sp XS
p
if
f(x,y)
(x,y) 6 (S x S q)
q)l#(S
fl
q)2
SpxD q
p
such that
P, q
identity.
by
q)3(S xsq)4
i
f’(x,y)
(x,y)
if
-’q
(x, y) 6 (S x S q) 2 -Dp
p
f’ 6 M+
Since f’
f on (S p X S q)
and since
P,q
Dq
c
sq)
X
(S X
I is identity then it follows that f’IsPDq=Identity and so
q
f’ 6 Ker G
D c (SPX S q)
to erform spherical modification
However using
f’
flS
is well-defined and
p
p
sPx
on 5oth sides of the domain and range of
on
(sPxsq)2
we clearly see that
N(f’)
f’
and the fact that
identity 6
f’
Diff(sPxsq)2
is the identity
hence
f’ 6KerN.
28
S. O. AJALA
f 6 Ker N
Conversely let
f’ 6 Ker B
that
B:
where
i
(x,b(f’) (x) .y)
SO(q) m> SO(q+I)
pxs qxl
for
and
f’
<
p
q
(x,y) 6 SpxD
then
q
We want to show
f
under
N
is
We now cosider
identity.
is defined in
[6]
similar to our
f’IS pxD q=S pxD q
b(f’) Sp ---> SO(q)
If
7
is
the
SO(q+l)
--->
induced
i, :WpSO(q)
P
is the inclusion map and
B(f’)
However since
S
Diff+(Sp XS q)
f’
Since
homomorphism then
H
is pseudo-diffeotopic to the
omiff (sPxs q) --> p(SOq+l)
f’(xy)
_Diff+(S pxS q)
then it means the image of
/
G
homomorphism
where
f’
N(f)
trivial hence
B(f’)
f’ 6
N(f)
then
Ker N
f
Since
i,b(f’) 6 SO(q+l)
and
is pseudo-diffeotopic to the identity then let
m> S pXS qxl
be the pseudo-diffeotopy between
f’
and
identity
id
Then
p+I
q
D
D
Sp
p+l
XS q=Dp+IxSq_
p+IxSq_D
id
S
q
S pxS
q
qxl_ Dp+IxS q
id
p+IxS q
id2(x,y)
2
DI>+l X sq_Dp+I X S q
is the required diffeomorphism between
=S
id
and
Dp+l X
f’
where
id l(x,y)
id(x,y,O)
(x,y,l)
However
id(xyl)= (xy)
consider
sq%2Dp+l
X S q=
id
(x,y)
fl(x’y’0)
f’(x,y) and
the q-sphere bundle over
Si,b(f,
)
(p+l)-sphere whose characteristic class of the equivalent normal bundle is
p+I S q
q
hence
Dp+I x
S
D
X
by the above
i,b(f 6
diffeomorphlsm and since p < q it follows by
Prop 36] that i,b(f’)
0
Hence f’ 6 Ker B and so Ker N is in one-to-one correspondence with Ker B
Since N is surjective by Leuna 2.11 then we have
a
rrpSO(q+l)
LEMA 213
sq_,
[,
Si,b(f,
The order of the group
Ker G
sP+Ix
equals the order of the direct sum
group
Ker B
Also since
G
0Diff+(Sp X S q)
is surjective by Lesna 2.8 then it is easily seen that
LEMMA 2.14
70(M, q)
The order of
is equal to the order of the direct sum
group
oDiff+(sPx S q)
SO(q+l) O Ker B @
p
However one can easily deduce from
LKMMA 2.15
Also from
[6,
ker B
q
Thin. II] and
Thin. 3.10] we have
[I,
70Diff+(sP X S q) pSO(q+l) @ 7qSO(p+l) @ O p+q+l
LEMMA 2.16
Combining Lmmmas
SO(p+l)
[6, S4]
O V+q+l
2.12, 2.13 2.14, 2.15,
THEOREM 2.17
order of the group
For
p
<
q
2.16,
we obtain
the order of the group
p+q+l
pSO(q+l) qSO(p+l)
O
and
e O
o(M; q)
equals twice the
3. CLASSIFICATION OF MANIFOLDS
Consider the class of manifolds
[M, II, k 2
where
M
is a manifold of type
DIFFEOMORPHISM GROUPS OF CONNECTED SUM
-
.
(n,p, 2)
where
and
n=p+q+l
p=
3,5,6,7 (rood 8)
29
Xl, i 2
and
are the generators of
q+l
H (M)
of Theorem I.i we have an embedding 0
By
the
proof
-> M
p
which represents tile homology class X. i 1,2
If we then take the connected sum
q+l
D
along the boundary of the two embedded copies of
we have an embedding
i:SpXD
sPx
i
S pD
q+l#S
pD q+l
[M, lI, 12]
Two of such manifolds
--->M
and
[M’,I,
II
i=
1,2
I
q+l #S p X D q+l
XD
iI,[SP] ll+l 2
l(x)
M’
which takes
%1
be an orientation preserving embedding such that
since there is an orientation reversing diffeomorphism of
to itself (because
i2, [S p]
{+ k We
p
p
i I(S XD q+l#S XD q+l)) U
such that
i
---> M
Sp X D
q+l
with
i
2(x)
for
x 6
p
S
now obtain
q+l-disc bundle over
is a trivial
then we have an orientation reversing embedding
(M- Int
onto
{M’,k,k]
kl, k2]
and
sP x D q+l #sP X D q+l
S p)
M
be the equivalent class of manifolds satisfying these
The operation is connected sum along the boundary
structure.
q+l
p
S XD
i
I+%2
This equivalent class which is also the diffeomorphism class has a group
q
p
p
S X S q# S x S
of
q+l
For if [M,
then
let
D
6
conditions.
p
S
4n
Let
h
will be said to be equivalent if
there is an orientation preserving diffeomorphism of
to
i,[S p]
such that
M#M’
i
2
S
p
xD
q+l
# Sp
XD
q+l
---> M’
from the disjoint sum
2p
p
(M’ -Int i 2(Sp XDq’l#S XD q+l))
xS
q#SpXS q
connected sum along double p-cycle.
by identifying
We will call this operation the
2p
Where the
in
MM’
means that we are
2p
identifying along the boundary of embedded copies of connected sum along the boundary
q+l
SpD
It is easy to see that H (MM’)
Z Z
Since we
of two copies of
have identified
i
p
I(SpXs q#S XS q)
iI*[SP] kl#%{+12 #%
with
i
2(S
p
P 2p
p
XS q#S xS q)
we can define
H (M#M’) then we see that M#M’ 6
P
2p
The connected sum along the double p-cycle is well-deflned and
LEMMA 3.1
the generators of
n
associative.
PROOF:
We need to show that the operation does not depend on the choice of
p
q+l
the embeddings
Suppose there is another embedding
---> M which
i: S X D
represents the homology class
i 1,2 and gives a corresponding embedding
p
S XD q+l#S pXD q+l ---> M
By the tubular neighborhood theorem 0
i(S pXDq+l)
.
i
and
P
0i( S
SO(q+l)
p
XD
q+l)
0
differ only by rotation of their
since
p=
3,5,6,7 (rood 8) hence
fiber, i.e., by an element of
the two embeddlngs are isotopic and so
the corresponding embeddings
i
I
S pxD
q+l#S pxD q+l
if:
e
----> M
and
-->
are isotopic.
definition does not therefore depend on the choice of
ment it does not depend on
i
2
Associativity is easy to check.
i
With similar arguI
The connected sum is therefore well-defined.
S.O. AJALA
30
LEMMA 3 2
If
n
[M’, X, X2’]
PF:
preserving
[M, AI, X2]
If
e
i.l (SpXD I)
1,2
e
i=
f :M--> M
bedding
and so
XD
I)
carries
e
i(S
f
I
p
k
kll
to
I
k2
and
en
M
Xll
to
).
orientation
and
to the corresponding edding
p
pxD
embedding
hence
1
(M12p#M’
there exists
X1
ich carries
x 1 S p X 1) --> 1
Int i(S p
which carries
n
are equivalent in
such that they are equivalent.
k) is equivalent to
2
2p
il(SP XDI sPx D1)
bedding
’:
M ,k I kl,
then
Since M,M
1
diffeohism
hence it crries
k12k 6 n
[MI, A I I
Int
f
i(S XD
I)
I#S
c M
to
induces a
[I(S
X I Sp X
p
I)
k12
to
Trivially we have the identity map
p
id:S’ -Int i’(S p
XD
--> M’
XDIs I)
which
k I to k
I’ and k 2 to 2’
eir boundary S p X SS p X S q to have
Int i(S
carries
We
M#M’
pXDIS p XD I)
en
take
e
cosec ted sum along
which is disjoint sum of
2p
i(sPxDI#sP X D I)
M-Int
x 6
for
M-Int
i’ (x)
for
:M#M’ -->
g
q
sPxssPxs
p
p
i I(S XDI#s XD I) U
i’(x)
d
M’-Int i’ (S p
2p
x zx{
p
x 6 S
MI#M’
2p
alent to
q
x SS x S
d
ere
i,[ S p]
we have
e
and identify
e
two copies on
e mifold
pxD
(SXDI#s I) U (sPxDIsPxDI)
d
p
--"n
,XIX
is a grip wire identity elent
6
2)
’
of hology groups.
I
sPx DIsP X D 1
SpxSI#SpxS I
v
p
S
XD I
H (S
i
p
Since
e
inverse
eor later,
in me case of
we need
(-M,-Xl-k2)
n0Diff+(S XDI#s XDI)
p
p
0Diff(S X D S X D I)
S XD
l(x)
identity map we will obtain
To be able to prove our main
p
p
p
i
following.
3.3
and for
e
by identifying
XSSp XS q --> Sp XSSp XS q If 0 ’k0 are e gen2
1
p
of Hp(S XSI#spXS I)
Z E d -k I+ (-2) 6Hp(-M)
E
where
d i:M--> -M is e orientation reversing dlffeorphi en
+
-AI -k2
S
id=identity:
erators
disjoint s of
at proves the la.
in
If we now take o copies of
and
v{M’,XZX{,X2
2p
n
xz2x.
to
2p
i.e.,
e
is
i(x)
Clearly we have a diffeorphlsm
on M and identity of M’ and g carries
f’
ich is
S pXS I#S pXS I
I#M’
2p
by identifying
M’-Int i’(S pXDI#S pXD I)
p
eir cn boundaries by
DI#s p X D I)
Silarly M
xz/x{ x2x
I#M"AII*X{’kI2A]
to
X
is
--> to H,(S
p
S X
DI#s
p
p
X DI
has
e
ISpxD I)
Z
[ ZeZ
in0
if
i=p
i
,102)
investigate
we define a hrphi
by induced autorphi
hotopy te of
if
l,k0
elent.
0p,q)
p
X DI#s X D I)
then
x D
(S pxSI#S pxS
DIFFEOMORHISM GROUPS OF CONNECTED SUM
2,
Using similar ideas in
31
it is easy to prove the following.
LEMMA 3.4
Z4
if
p
is even
GL(2, Z)
if
p=
1,3,7
H
if
p
is odd but
’(0(Diff(sP XDq+l#sp XDq+l)))
Diff+(S p X Dq+Is X D q+l) c
morphisms of sPx D IsP X D -I
p
Let
p
Diff(S x
Dq+Is p X D q+l)
be the set of all diffeo-
which induce identity automorphisms on its homology.
p- Dq+I#s p
D q+l)
oDiff(S
Then it follows that
’
is the kernel of
a homomorphism
G’
6
If
P
SOq+l
p
SO(q+l) ----> ^Diff’ (S p X
u
and
Sp
ga
=
[a]
1,3,7
We define
Dq+l#s p X D q+l)
5
then we define a map
Dq+I#sP X D q+l
----> S p
X
Dq+ls p X D q+l
by
ga (x, y)
ga
is
for
p
(x,y) 6 (S XD
(x, a(x)" y)
for
(x,y) 6
g-a
clearly well-defined and it is a diffeomorphism and since
G’
LEMMA 3.5
PROOF:
sPxI---->
SO(q+l)
P
SO(q+l)
such that
F
of
fixed,
XI
and let
H(SPXI)
a’
by
s xD )
p
(x,y) 6 S XD
and
and
q+l)2
ga’
is surjectlve.
If]
Let
homology groups.
a
(sP Dq+I#sp X D q+l)
(x,H(x,t)’y)
ga
a’
homotoplc to
(, )
This is the diffeotopy which connects
PROOF:
is
H(sPx0)
(x,H(x, ). y)
F(x, y, t)
G’
a
be the homotopy such that
then we construct a diffeomorphism
LEMMA 3.6
p
is well defined.
a’ 6
If
S
keeps
ga6Diff+(sPXDq+I#sPDq+I)
it induces identity on all homology groups hence
H:
q+l)l
(S pD q+l)2
(x, a(x)’y)
6
Dq+I#sp D q+l)
(S pxDq+I#s pD q+l)
Z
(S p
oDiff
However H
P
then
Z
f induces identity on all
and so if
kI
k2
and
represents the generators of the first and second summand and the embeddings
pxD q+l and i S
pD q+l
S
i
-----> S
----> S
2
I
px [po
px [po
pxDq+l#s
kI
represents the homology class
on homology then
f(sPx [p0
Hurewicz theorem
i
I
and
and
f
o i
I
pxDq+I#S
k2
respectively, since f induces identity
are homologous. Since p < q and by
are homotoplc, by Haefliger [10] and by the diffeoand
iI(SPx [p0
topy extension theorem and tubular neighborhood theorem, there exists f’ in the
(S p x D
dlffeotopy class of f such that f’ (x,y)
(x,a(x).y) for (x,y)
q+l
p
p
p
is the tubular neighborhood of S X
and a S --> SOq+l
where S x D
p
q+l
Similar argument applies to the embedding
and
D
] --> S X
q+l)l
[p0
12 sPx [p0
Dq+IsPx
S.O. AJALA
32
F’
so we have a map
f’
f"
and so
in the diffeotopy class of
must be of the form
hence in the dlffeotopy class of
6 p
where
f
(x,a(x).y)
f"(x,y)
(x,y)
(S XD
q+l)2
It follows that
(x,y) 6 (S pXD q+l) I
.y)
f (x, y)
p
(x,y) 6 (S XD
(x,a(x) y)
G’
Hence
q+l)2
is surjective.
=+ p XDq+IsP Dq+l
0Difz(S
One can easily deduce from Lemma 3.6 that
is a factor
p(SOq+l)
group of
be an n-dimensional closed simply connected manifold of
Let M
HEOREM 3.7
p= 3,5,6,7 (mod 8) then he number of differentiable manifolds satisfying the above conditions up to diffeomorphism is twice the
(np, 2)
type
n=p+q+l
where
with
C.
We define a map
PROOF:
mo rph i sm.
Let
+
70 (Mp, q)
f
have
q+l)2
p
(S x
q
D
S
p
XD
q
C
is an iso-
sqs p x S q which
p"
p Dq+Is
Dq+l) I and
(S
and attach them on the boundary by
(sP’xD q+lS p XDq+l )IU (S p XDq+lS p XDq+l ) 2
compatible with
n
and show that
p
S
is a diffeomorphism of
f
XDq+Is pXDq+l
Sp
of
0(M+_, q)----->
then
We then take two copies
induce identity on homology.
S pXDq+I#S pXD
@n
SO(p+l)
order of the direct sum group
An orientation
f to
is chosen to be
and the manifold obtained belongs to the group
)i
he generators of the p-dimensional homology group is fixed to be the one
c (s p X Dq+l # S p X
represented by the usual embedding
---> (sPx D
i
I
pp
c (S
We then define
and S
---> (S
2
I
p
p
We now show that C is well-defined
C[f] (S p x
X
XD
X
f
+
Let fo,
then there exists
M
such that
is pseudo-diffeotopic to
P, q
p
and
H (S pxSS pS q) I ---> (S pxSS xS q) I such that H(x,y,O)=
p
pxD
U (S p
then we wish to show that (S
is
H(x,y,l)=
D
sPx [po]
px -[p0
XDq+I#s XDq+l)
pxDq+I)
Dq+I#s Dq+I)u(sP Dq+ls q+l-)
fl
(S p x
diffeomorphic to
Dq+lsp
Dq+l)
(S pDq+I#S pDq+l) US pxDq+I#S Dq+l-S
p
1
p
U (S p
pXDq+I#S q+l)
Dq+I#s Dq+l)
t.d
t
($PDq+I#$ pDq+l) U ($PDq+I#$ pXDq+l) -$PDq+IDq+l
(x,y,l),
fo
idl(X,y,O)=
(x,y),
q+l)
XD+I#s
fOw
e then define a map
xDq+l#S pDq+IU (S pSS p S q) I
fO
id0(x,y)
Dq+l)
fl
f0
fl
where
q+l)
IH
U Sp
’do
Dq+I#S pDq+l
[
d
U ($P$$P$q) IosPgDq’tq’$pvq+l
f6(x,y, 0)- fo(x,y)
and
fi(x,y) ffl(x,y,l).
This is a well-defined map and is he required diffeomorphism from
(sPXDq+I#sPXDq+I) U (sPXDq+I#sPxDq+l)
C
to
5
fo
is well-defined and it is easy to see
it follows that
C
and
is surjective.
C(f)
(sPXDq+I#sPxDq+l) (sPXDq+I#sPxD q+l)
5
f
5
that C is a homorphlsm. By Theorem
We now need to show that
(M, XI, X 2)
is
trivial, then
C
is injectlve.
it follows that
Hence
i. 1
Suppose
DIFFEOMORHISM GROUPS OF CONNECTED SUM
M
(sPDq+IsPxDq+l) (sPDq+I,SPxDq+l) 2
iUf
(S p X Dq+l # S p
Dq+I)IU(SP
id
k01’k02
homology generators
k2
k02
to
p
3,567 (rood S) then
d
sPSq#sPxs q
Dq+l) 1
U (S p
q+l)
(S
D
It is easy to see that since
that
# S p D q+l) 2
d
lid
p
kI
carries
C
77
d
P
(SOq+l)= 0
f
is just
Hence
f
It then follows that
is injective.
to
is the identity on
S
q+l
d
# Sp
XS
q+l
the order of the group
is proved.
p-dimensional
kI
which carries
to
k01
and
Dq+l)2
p
Dq+t)2
zP x sq+Is p x S q+l
k01
02
(sPDq+IsPDq+!)
and
k
I
d
Since
WqSO(p+l)
@n
and because
to
2
On the
is a diffeomorphism it follows
group of
n(SOq+l
but since
p=
by Lenna 3.5,
3,5,6,7,
is pseudo-diffeotopic to the identity and so
C
is an isomorphism.
By Theorem 2.17 and
p= 3,5,6,7 (rood 8) it follows that the order of the group
since
with
(sPXDq+I#sPXDq+I) 2 which means
p
p
to Diff+(S X Dq+Is X Dq+l)
but
+(S p XS q#s p XS q) is extendable
"oDiff+(sPXDq+I#sPxDq+l) is a factor
then
Dq+l#s
Dq+I#s p
f 6 Diff
rood 8
p
f
extends to a diffeomorphism of
f
S
by a diffeomorphism
(S p D q+lS px
5
p
boundary
is diffeomorphic to
i.e.,
(S D q+l#S
p=
q+l
D
33
and since
C
70(
q)
is twice
is an isomorphism the theorem
The methods used here if carefully applied can be used to obtain a general
result.
THEOREM 3.8
(n,p,r)
where
If
M
n=p+q+l
is a
and
smooth, closed simply connected manifold of type
p= 3,5,6,7 (rood 8) then the number of differentiable
manifolds up to diffeomorphism satisfying the above is equal to
SO(p+l) O
of
7
I.
TURNER, E.C.
q
r
times the order
@n
Diffeomorphisms of a Product of Spheres, Inventlones Math. 8_
(1969) 69-82.
Diffeomorphism groups and classification of manifolds, J. Math. Soc.
2. SATO, H.
apan, Vol. 2_!i No. I, (1969) 1-36.
Groups of Homotopy Spheres I, Ann. of Math. 77
3. MILNOR, J. and KERVAIRE, M.
(1963) 504-537.
On the structure of manifolds, Amer. J. Math. 8.4 (1962) 387-399.
4. SMALE, S.
KUROSH, A.G. Theory of Groups Vol. II, Chelsea, New York, 1955.
SATO, H. Diffeomorphism groups of sPx S q and Exotic Spheres, Quart. J. Math.,
Qxford (2), 2__0 (1969) 255-276.
n
7. LEVINE, J.
Self-Equivalences of S x
Trans. American Math. Soq.
(1969) 523-543.
Killing the middle homotopy groups of odd dimensional manifolds,
8. WALL, C.T.C.
Trans. American Math. Soc. I03 (1962) 421-433.
A procedure for killing homotopy groups of differentiable manifolds,
9. MILNOR, J.
Proc. Symposium Tucson, Arizona, 1960.
Plongements differentiables des varits. Commentari Helvetlci
i0. BAEFLIGER, A.
47-82 (1961).
Math____.
sk