217
Internat. J. Math. & Math. Sci.
Vol. i0 No. 2 (1987) 217-226
DIFFERENTIABLE STRUCTURES ON A GENERALIZED
PRODUCT OF SPHERES
SAMUEL OMOLOYE AJALA
School of Mathematics
The Institute for Advanced Study
Princeton, New Jersey 08540
and
Mathematics Department
University of Lagos
Akoka-Yaba
Lagos-Nigeria, West Africa
(Received April 21, 1986)
In this paper, we give a complete classification of smooth structures on
a generalized product of spheres. Ihe result generalizes our result in 1] and
R. de Sapio’s result in [2].
ABSTRACT.
KEY WORDS AND PHRASES. Differential structures, product of spheres.
1980 AMS SUBJECT CLASSIFICATION CODE. 57R55
I.
INTROJCTION
In [2] a classification of smooth structures on product of spheres of the form
where 2 _< k _< p, k+p _> 6 was given by R. de Sapio and in [I] this author
sks p
extended
R. de Sapio’s result
to smooth structures on
sPXSqXS r
where
2<p_<q_<r.
The next question is, how many differentiable structures are there in any arbitrary
product of ordinary spheres. In this paper, we give a classification under the
relation of orientation preserving diffeomorphism of all differentlable structures of
k
k
I
denotes the unit
spheres S x
_<
x... xS r where 2 _< k I < k2 _<
n+l
R
(n+l)-space
the
Euclidean
in
structure
differential
the
usual
n-sphere with
sk2
O
n
sn
kr
denotes the group of h-cobordism classes of homotopy n-sphere under the connected
sum operation.
n
will denote an homotopy n-sphere.
p
H(p,k)
denotes the subset of
k
is diffeoS
P
P
such that
which consists of those homotopy p-sphere
k
and it is not always
S
By [2], H(p,k) is a subgroup of
p
O
then H(p,k)
zero and in fact in [I], we showed that if k _> p-3
O
morphic to
sPx
By Hauptremutung
morphism.
@P
3], piecewise,
Consider two manifolds
S
linear homoemorphism will be replaced by homeo-
I
X
sk2 X Sk3
S
k4
and
shall denote the connected sum of the two manifolds along a
k2 +k4
skl X Sk3
k2+k4 -1
we
cycle by
218
S.O. AJALA
k
k
k
k
(S I x S 2 x S 3 x S 4 )
k +k
2 4
(
’k 4 k
)XS I X S 3
k2+k42
Int(D
removing
k
XS
i
k
3
XS
This is geometrically achieved by
from both manifolds and then identify their_ coumon
e
a_nd id.entify the manifolds along their co.non boundary
S
S4
k2
k2+k4
S
a
k2+k4-1
old
to obtain
This is a well-defined operation
We then take the cartesian
k+k
2 4
)=
ko
k.
k
S S XS S
is dlffeomorphic
k2 k
I k 3 to have S k i k 3
product with S xS
XS x (S xS 4
ko+k,
kl ko k k.
kl k
kl
k
S OS S kk
4
k I k 2 k3
to S xS xS XS k42 then
XS S
S
(S
But
kl xS k2 XS k3 XS k4
k2+k4 xSkl XSk3 =S I XSk3X(S k2 xS k4 k2+k4 ).
)k2k,
We will then prove the following.
CLASSIFICATION THEOREM
S
klxsk2
n=k
kr
XS
X...
where
M
2 < k
n
M
is a smooth manifold homeomorphlc to
I <
< K r-i
and
k4-3
then there exists homotopy spheres
+kr
l+k 2+...
n
such that
If
<
k
<
l+-k
,...
k
and
n-k
r
n-k
r,... I ,n
is diffeomorphic to
k
k
tk
k
k
rXS ZX...xS r-) (Y.k2+kr XS 5 XS k3 X...XS r- i)
k
n-k
k 2’r
OxS X...xS r) # "#-r (y" rxs r)
k.+k k^
(
k,
"’l’"r k.+k^+k
z
sklx...XS r)
(
+k
I 2 n3_k. k
(Y. ’XS
’)J n
n.kl
We shall use the above classification theorem to give the number of differentlable
kr We shall lastly compute the number of structures in
structures on
sklxsk2x...xS
some simple cases.
2. PRELIMINARY RESULTS
We shall apply obstruction theory of Munkres 4]. Let M and N be smooth nmanifolds and L a closed subset of M when triangulated. A homeomorphlsm
(M-L) is a dlffeomorphlsm and each
f :M --> N is a diffeomorphism modulo L if
fl
simplex t
By
[4]
of
L
has a neighborhood
if two n-manifolds
M
and
N
V
such that
is smooth on
f
V-L near
M is
(I
are combinatorially equivalent then
diffeomorphic modulo an (n-l)-skeleton L onto N
n
n
is a diffeomorphism modulo m-skeleton m
If f :M --> N
showed that the obstruction to deforming
f
<
to a diffeomorphism
n-m
n
then Munkres
n
--> Nn modulo
g :M
n’m
is a group of
Where F
r
(m-l)-skeleton is an element Xm (f) 6 Hm(M,Fn-m)
n’m’l
modulo the dlffeomorphisms that are extendable to diffeodiffeomorphlsms of S
n-m
0 then g
If k (f)
We call g the smoothing of f
morphisms of D
q
is
exists. Recall that in ([I], Lemma 2.1.1) we proved that if q _> p then P X S
p
q
In Remark (i) following that
for any homotopy sphere Z
diffeomorphlc to sPx S
lema,
we showed further that even when
n
p-3 < q
sklxsk2x
the result is still true.
k
...xS
Suppose f M -->
modulo
which is a diffeomorphlsm
(n-ki)-skeleton
LEMMA 2. I
r
is a piecewise linear homeomorphism
i
_<
i
_<
r
then there exists an
DIFFERENTIABLE STRUCTURES ON A GENERALIZED PRODUCT OF SPHERES
k
i
homotopy sphere
k
n
-->
h :M
S
sk2
IX
... ski’l
219
and a piecewise linear homeomorphism
k
k
r
I
S
XY.
X
which is a diffeomorphism modulo
k
n
I
Since f :M --> S
PROOF.
[4],
skeleton then by Munkres
...
iXS ki+
(n-ki-I) k skeleton.
r
S
is a diffeomorphism modulo
.f
the obstruction to deforming
(n-kl)-
to a diffeomorphism
modulo
[]
k
ki
ki
D
k
U
i
D
k
and a homeomorphism
and so
So
2
i--->S
S
where
i-
is a
ks
ki k i
D UD
D21
kid
I
i
(f) 6
i
j
S
We define
ki
ki
S
where we have
Int(D I
is identity map on
j
->
diffeiomorphlsm.
ks
-I on
and radial extension of
is a piecewise linear homeomorphism by the definition and the ob-
j
[-i] .kki(f So consider
k.-i
k.
k.+l
k.-i
z
kl X...XSkr) X.k i
z
XS
X...xS kr) XS z_>(S klx ...xS
xS
idXj: (sklx ...XS
struction to deforming
to a diffeomorphism is
j
the map
i
The map is a piecewise linear homeomorphism and the obstruction to_deforming it to a.
kI
kr
k..-I
Notice that the manifold (S X...XS
X...XS
diffeomorphism is [
k
k
k
k
kI
ks i ki+1
i+l
i
and
S I X... xS i-I x
XS
X
(S X...xS ’XS
X...XS r x
XS
-I]
.
sklx... XSki
kl
.kki(f)
Lkixski+l X... XSkr
Consider the composite
-kki(f
0
Aki(f)
diffeomorphism modulo
LKMMA 2.2
skeleton
< i,j _<
i
(n-ki-l)
Let
skeleton .is k k (.h)
Xk. ((idx.)" f)
i k
z
n
--> S IX...XS i i Xy.Ki X
f
n
:M ->
skeleton.
sklx...xS
linear homeomorphism
k
I
n
f :M --> (S X...XS
.
be a diffeomorphism modulo
ki+kj
n-(ki+kj)
and a piecewise
i’Ixski+Ix...xSkj.lskj+Ix ...XSkr)
k
k
(.ki+k.JXS IX...XS
kr)
is a
Hence the lemma.
kr
then there exists homotopy sphere
r
to a
k- (idxj) + kki(f)
si+Ix...Xkr
h’M
hence
h
the obstruction to deforming
(idx j).f=h
(n-ki-1)
di f feomo rphism modulo
+
kr
skix?
-I X
ki+kj
n-(ki+kj)-I
which is a dlffeomorphism modulo
PROOF.
Since
skeleton.
is a diffeomorphism modulo
f
f
that the obstruction to deforming
n-(ki+kj)
skeleton,
to a dlffeomorphism modulo
it follows
n-(ki+kj)-I
skeleton is
k f) 6 Hn_
where
We define
S
(ki+k j
ki+k,’l
j
S
ki
Fki+kj
n,rki+kj
--->
S
ki+k,-I
k.
k i k.
xS 3 _> S XS 2
and radial extension of
-1
on
.
[]
is a diffeomorphism and
ki/kj to
ki+kj
Int(D
morphism and the obstruction to deforming
Then consider
Let
j
k(f) 6
ki+kj Dki+kJ
be identity map on
hence
j
?i+kj
skiskj
ki+k j
D
ki/k j
Int(D
is a plecewise linear homeo=
to a diffeomorphism is
[-I] =-k(f)
S.O. AJALA
220
k.
XS
S
j xid
(S
k
XS
X
X
k
and
k
j
ixS j #
k
i
S
k
ki+k (S IX...XSki" lski+lX...XS J-lxskj+ix...XSkr)
that
kj (sklx...xS i’lxski+lx...XSkj_ixskj+ix" XSkr)
(sklxsk2x...XS X...xS JX...xSkr)
kj
1
(S X...XS i-lxski+Ix...XS kr)
xS # .ki+k
(sklx...xSkr) # .ki+k xsklx...xSki- ixski+l X...XSkj. Ixskj+Ix. ..xSkr)
k
k.
No te
(sklx...xski’ixski+ix...XSkj_ ixskj+ix ...XSkr) -->
k.
I X
k
i
(S
k
i
k
j
k
X
j
ki+kj
hence the above map is
id X j
(s
(S
kl x ...XSkr)
kl x...xskr)
----->
(j X id)’f
the obstruction to deforming the composite
1
skeleton is zero.
f’
Hence if
xS
-k(f)
which is piecewise linear and its obstruction to a diffeomorphism is
n-(k.+k.)-i
I
k
ki+kj xskl x... xSki" lxski+l X" xskj Ixskj+i X"
#
r
hence
to a diffeomorphism modulo
(j X id)’f
Mn
f’k:
.xS r)
then
kl X...XSkr )k (ki+k"IXSkl X...xski’Ixski+ix...xskj’Ixskj+Ix
tk
modphism modul J n-(ki+kj)-I skeleton.
(S
---’>
is a dlffeo-
3. CLASSIFICATION
THEOREM 3.1
k
n
S
is a smooth manifold homeomorphic to
M
If
kl+k
I XS k 2
k
x...xS
n-k
k2+kr, ..., Ekl+k2+k ,...,
3
r
then there exists homotopy spheres, Y.
n-k
n
1
is diffeomorphic to
such that M
and
E
r
r
2oeo
n
(s
klx...xskr)
kl+krxsk2x ...xsk-l) # k2+krxsklxsk3x
k2+k
kl+k
kl+k2+k3xsk4X... XSkr)
@
r
#
r
@
k
XS
r-
I)
r
kl+k2+k 3
(n’krxskr)#... # (--"klxskl) #n
#
n-k
2
where
_< k I <
PROOF.
[4],
h
k
2
n-k
r
<
<
k
h
n
Suppose M
kr-3
r
>
S
kI
X...xS
is a diffeomorphism modulo
n-k
appears in dimension
I
I
_< kr_ 1 _<
kr
and
n=k
I +
k2+... +k r
is the homeomorphism.
(n-l) skeleton.
By Munkres theory
Since the first non-zero homology
(apart from the zero dimension) it then means that h
(n-kl) skeleton.
(n-kl-l) skeleton is
is a diffeomorphism modulo
kr
The obstruction to
Hn_kln, Fkl)
defoing
1
h
to a
Lena
By
F
k(h) E
1
h’ and a homotopy sphere
2.1, there exists a plecewise linear homeomorphlsm
k
r
n
which is a diffeomorphism modulo (n-kl-i)
such that h’ :M -->
X... x S
diffeomrphism modulo
k ix Sk2
221
DIFFERENTIABLE STRUCTURES ON A GENERALIZED PRODUCT OF SPHERES
S
kI
to
k2
S
kl
S
... skr
k <k
since
I
kI
It then follows that
2
n
h’
hence
kl
it was proved that
M
-> S kI
kr
XS
(n-k2-1)-skeleton.
is diffeomorphic to
S
kr
is diffeomorphlc
is a diffeomorphism modulo
h’
(n-kl-I)
to a diffeomorphlsm until
This is because
n, Z)
H (M
i
0
assume that h’
< i < n-k I
The obstruction to de_forming h’
(n-k)
skeleton.
z
n k2
skeleton is k(h
Fk2
(n-k-l)
(M ,I"
6 H
z
n-K2
k2
for
sk2
x Sk2 x
There is no other obstruction to deforming
skeleton.
the
I] Lemma 2.1. I
In
skeleton.
is a diffeomorphism modulo
So we can
n-k2+l
to a diffeomorphism modulo
2.1, there exists
kl k2
M --> S E
Again by Lemma
n
a_ homotopy sphere Y.
and a piecewise linear homeomorphism h
k3
kr
which is a diffeomorphism modulo (n-ko-l) skeleton. By the same
XS
k2 k
is diffeomorphic to
xS
argument as above since k 2 <kq we see that
S
k
k
S
k
S
S
hence
.
k-
k
XS
n
h"" M -->
This shows that
.
k
x... XS
sklx...xSkr
k.
h"
n-k3-1
(n-k3)-skeleton.
is a diffeomorphism modulo
using the same argument we can construct a homeomorphism say
which is a diffeomorphism modulo
(n-kr)-skeleton. However,
k.
k
S
k
...xS
XS
(n-k2-1)-skeleton.
is a diffeomorphism modulo
n
By the same argument since M has no homology between
assume that
S
is diffeomorphic to
and
n-k2-1
we can
SIx...xSwar
Proceedinguthis
-->
n
h":M
to deform
h"
.
to a
(Mn’lkr
diffeomorphism modulo (n-kr-l)-skeleton there is an obstruction ") 6 Hn k
r
kr
k
and a
It also follows by Lemma 2.1 that there exists a homotopy sphere
I" r
kr which is a diffeomorphism
n
piecewise linear homeomorphism f :M -->
sklx...xskr’ix
(n-kr-l)-skeleton.
r
modulo
p-3 < r
k -3 <
r
S kl X
Now in Remark (i) of i] it was shown that even when
r
p
and so by our assumption that
is diffeomorphic to S x S
xP
it
_<
kr. 1kr’lk r kr
XS
X
sklx..,
S
xS kr
to deforming
follows that
kr’l x kr
is diffeomorphic to
f
k(f) 6 H
n_(kl+kr
(Mn, Fkl+kr)
f’ "Mn -->
(sklxs k2
...
which is a diffeomorphism modulo
XS
kr)
X S kr’l X S kr
f
and so
(n-kr-l)-skeleton.
n-(kr+kl)-i skeleton
x S kr
n
M
Hence
--->
The next obstruction
to a diffeomorphism is on
Fkl+kr
S kr’l
is diffeomorphlc to
S kl X
is a diffeomorphism modulo
homeomorphism
l+kr
S
and it is
By Lema 2.2, there exists a plecewise linear
# (kl+krxS k2x... S kr-l)
kl+k r
n-(kl+kr)-i
skeleton for some homotopy sphere
kl+kr
At this point, we want to remark that if
%(f) 6
and
suppose
.)
k.
max(k_...,k r-.) then it follows from
kl+kr-3 max(k2 .k rk
that I Kr X S J is dlffeomorphlc to
Remark (i) of I] since k.+k -3 < k
r
defined using
<
Skl+kr Skj
+
and so
J kr’l
l+kr X sk2x...ES
is dlffeomorphlc to
S
k l+kr
X
sklx...XS kr’l.
(sklx’"xskr)k ikr(l+krxsk2x’"skr’l) is dlffeomorphic
(sklxsk2x...xskr)k.k( SkI+krxsk2x...xskr’l) and this is diffeomorphic to
This then implies that
sklxsk2...Skr
because
skIskr # Skl+kr sklxskr
to
So this means that the factor
S.O. AJALA
222
S kr’l will
kl+kr sk2
kl+kr-3 _< max(k 2, ...,kr. I) n
Anyway, we have
dlffeomorphism modulo
<
disappear in the above sum if we have the condition
-->
f’ "M
n-(kl+kr)-I
n-(kl+kr)-I
n-(k2+k r) skeleton and
modulo n-(k2+kr)-I skeleton is
i
(skl...skr)k1+k_(.kl+krsk2...Skr-l)
skeleton.
H
Since
i(Mn Z)
modulo
the obstruction to deforming
%(f’) E
Hn_(k2+kr
0
f’
then there is no obstruction to deforming
(Mn, "k2+kr)
which is a
n-(k2+k r)
for
<
to a dlffeomorphism
f’
to a diffeomorphlsm
"k2+kr
Using the
sme technique as in the proof of Len-na 2.2 it can be easily shown that there exists
:S k2+kr’l
S k2+kr’l
--->
Dk2+kr
2+kr
an homotopy sphere
Dk2+kr
%(f’) E Ik2+kr
and
is a diffeomorphism and a piecewise linear homeomorphism
(2+krxskl...skr "I)
ski... S kr ---> (sklx...S kr)
j
=
where
k2+kr
-%(f’)
where obstruction to a diffeomorphism is
We now define a map
(kl+krsk2x...Skr’l)
kl+k r
(kl+krxsk2...Skr-l)
----> (sklx...Skr) # (2+krsklsk3...Skr’l)
kl+k r
k2+kr
kl+kr)
(sklx...XSkr)
sk2...Skr-I and identity on
Ind(D
where j’ j on
k2
kr-I
k2
.kl+kr X S x
S kr-I
S
Int(Dkl+kr) S
(sklx...skr)
j’
j’
Clearly
-%(f’)
is piecewise linear and its obstruction to a diffeomorphism is
hence the obstruction to deforming the composite
g
j’- f’
where
r
+ %(f’)
j’" f’
is a diffeomorphism modulo
n-(k2+kr)-I
X(J’.f’)
%(j’)
skeleton.
Proceeding in this way we see that the next obstruction to a diffeomorphism
0
Hence
Using the above technique continuously we can con-
(n-(k3+kr))-skeleton.
will be on
g
struct a piecewise linear homeomorphism
(sklx...XSkr)
g’ :Mn -->
(kl+kr
kl+k
(2+krxsklxsk3...xskr-l)
r
#
sk2x...xskr’l)
...
k
k2+k r
kil+...+k i 2 skjl...$kj P)
#
iI
(
+... +
ki
skeleton. The obstruction to
n-(k r_l+...+kl)
is k(g’) q
skeleton
modulo
extending g’ to a diffeomorphism
(kr-l)
"n’kr
By using the same technique as in the proof of Lemma 2.1 there exists a
kr
which is a diffeomorphlsm modulo
plecewlse linear homeomorphism
j
j
and homotopy sphere
sklx...xSkr ---> (sklx...XSkr)
Hkr(Mnln’kr)=
n-kr
such that
(n’krxS kr)
n-k
r
has an obstruction to a diffeomorphism to be -l(g’)
From this we define the map
223
DIFFERENTIABLE STRUCTURES ON A GENERALIZED PRODUCT OF SPHERES
j "(
SkIx... X skr) kl+#k r(l+kr x sk2 X
k2kr( 2+krxsklxsk3x... xskr_l) #
skr-l)
k
kil+..+kixskj lx...xs jp)
#
k. +...+k.
jpil, i 2. .i
k
j’
where
+...+k i
i
(I; l
k. +...+k.
#
#
on
j
(sklx...XS kr)
It is easily seen that
j’
j’
deforming the composite
k
xs Jlx...xskip)
(Int(Dkl+kr)xsk2x...xSkr’l)
and identity elsewhere.
is piecewise linear homeomorphism and the obstruction to
g’
h’: j"g’
Hence the map
to a diffeomorphism is zero
where
k2+kr
[k2+krxsklxsk3x. xskr.1) #
is a diffeomorphism modulo
k
r-I <
i
<
(kr-l)
skeleton
r
However,
n, Z)
H i(M
since
there is no more obstruction to deforming h’
k -i
r
To deform h’
kr_l-Skeleton.
modulo
n-kr x Skr)
#
n-k
to a diffeomorphism
(kr_l-l)
Fn’kr-I
to a diffeomorphism modulo
there is an obstruction and this equals
A(h’) 6
for
0
Hkr.10n,ln’kr’l)
n’kr-I
Applying the above technique again, we can get an homotopy sphere
skeleton,
and a
plecewise linear homeomorphism
n
h" :M
kr) t [kl+krxsk2x xskr-1)
kl+kr
l+k2+k3 x S k4 X... X S kr)
k2+krxsklxsk3...
#
S kr’l) #
k2+kr(
kl+k2+k 3
(sklx
->
XS
n-k
n
r
which is a dlffeomorphism modulo
kr_2-1
skeleton
r-
kr_l-I
The next obstruction will be on
skeleton.
Proceeding this way gradually down the remaining
skeleton,
we can
construct a map
n
g:M -->
(sklx...XS kr-)kl+kr -l+kr X skZx xskr-l)
l+k2+k3 X sk4x.., xSkr)
k2+k r
# (’krxskr)
n.kr
n#_kr.l(n’kr’ixskr’l) # ..n.kl# (n’klxskl)
Hi(Mn
which is a diffeomorphism modulo
then
g
kl-Skeleton.
is a diffeomorphism modulo one point.
an homotopy sphere
n
n
such that M
Since
Z)
0
for
0
<
i
<
k
It therefore follows that there exist
is diffeomorphls to
224
S.O. AJALA
xskr
l+krxsk2x xskr-1) $ 2+krxsklxsk3x. xskr-1)
k2+kr
kl+kr
(l+k2+k3xsk4x...xskr)
(n’krskr)
n-k
r
kl+k2+k 3
#
# (n-klxskl) ] # n
n-k
1
Hence the theorem.
H(pk)
Recall that
P
P
such that
X S
TBEOREM 3.2
2
_< k I <
k
2
p
consisting of homotopy p-spheres
The number of dlfferentlable structures on
<
0
p
denotes the subgroup of 0
k
p
is diffeomorphic to S X S
<
kr_l
kr-3 _< kr_ I _< kr
0 k2+k r
and
kl+kr
X
H((kl+kr),(k3,...,kr.l))
e
H((kl+k2+k3,(k4,...,kr))
e
X
.-k
x S kr where
equals the order of the group
X
H(k2+kr,(kl, k3,...,kr_l)
X
sklx
X
r
H(n.kr,k’r
X
X
H(n.kl, kl)
n
x O
Let (0(k.+k),0(k+k
z rk+_
r
the
trivial elements o f e I kr
represent
PROOF
),...,0(k.+k.+k.),...,0(n-kr),...,0(n-kl),0(n))
o2+r,..., ekl+k2+k3, ,0n’kl On then
we define a map
x0n’krx. X 0n’klM0n, 0 (kl+kr),..., 0 (n-k l) 0 (n))
----> (Structures on sklx.., xSkr, 0)
l+kr 6 0kl+kr
represents the usual structures on sklx .XS kr
If
k
8 O l+krx0k2+krx... X0kl+k2+k 3x.
where
0
l+k2+k3
6
ekl+k2+k3
n’kr
6
on’kr
n-kl
6
on-kl
znE en
and
then
we define
l+kr
[(sklx.
kr
l+k2+k3
"kr
xskr)# (el+kr x sk2 X xskr-l)
kl+k r
( l+k2+k3xSk4x...xSkr)
kl+k2+k 3
$ (’kxsk)
E"
n-k
"kl Z n
k2kekrxsklxsk3
( "krxSkr)
n-kr
E
-
l+kr, l+kr 6ekl+kr ;kr, kr6ok2+kr
..
...;
h-coSordt respectively
en ey
are differphic.
l+k2+k3, l+k2+k3
;-,-It then follows
sk2x. xskr is differphic to 2 l+kr x Sk2X. XSkt-1
1
s
x
zk- k x s3x...xsr- , aeo o +kr x sk x skx...xs
X
xskr-l)
1
is ell-defned because
are
x
=-
225
DIFFERENTIABLE STRUCTURES ON A GENERALIZED PRODUCT OF SPHERES
i l+k2+k3kr
ll’kr
2 -kl
S k4 X
skSx... S kr
is
diffeomorphic to
2-kr
S
is diffeomorphic to
S kI
and so this means that
x S kr
and
2 l+k2+k3kI sk4...
l-kl
S
S kr
Also
is dlffeomorphic to
(sklx xskr)
r
3# 21+k2+k3sk4x.
(2"klxskl)] # 2
#
kl+k2+k
#
Hence 8 is well-defined map.
Clearly 8 takes the base points
...,O(n-kl), O(n) to the base point 0
s
k2+kr
skr) #
0(kl+kr)
.n#_k2 2-E2xskr)
0(k2+krl),...,0(kl+k2+k3),...,O(n-kr),
This is because if all the homotopy spheres
is
are standard spheres, then all the summands involving
sklx...xSkr
will vanish leaving only
I+kr 6 H((kl+k_)
Suppose
By Theorem 3.1,
(k2, ...,k_ i))
in the image of
is onto.
8
k2+kr 6H((k2+kr)
(k1,k., ...,k I)),
l+krH((k.+kr ),(k,..,k
.)) mis mes l+krxsk2x. XSkr’l is
rz
k
kr-I
hence sklx...xS r#l+krxs 2X...xS kr-I is
diffehic to skl+rxsk2x...xS
diffeorphic to (ski... XS kr) # skl+krxsk2x...XS kr-l= sk2xsk3x.., xskr-Ix(sklxskr#s kl+kr)
men
for
ich is differphic to
sklxskr
sk2xsk3x...xskr-Ixsklxskr
l+kr
is means at for
6
(sklx’’’xskr)k k (l+krxsk2x"’xskr-l)
e
sud
I+rxsk2x...XSkr-I
show at all
oer
in
sds involving
sklxskr #skl+kr
since
H((kl+kr),(k2...,kr_l)) en
is
diffehlc
e age
e Zs
to
vanishes.
in
H(in-i)
sklx...XSkr
Silar
and so
arments
vanish hence in
cas
en
nduces a map,
8kl+kr
8kl+k2
@kl+k2+k 3
H(ky+k2+k3, (k4,...’,kr))
X
@n)-->(stctures
ich is onto since
If
X
8
X
on
@n-k r
H(n2kr, kr) X
sklx...xSkr)
X
@n-kI
H(n_kl, kl)
is onto.
(l+kr, o(k2+kr),...,O(kl+k2+k3),...,O(n-kr),...,0(n-kl),O(n))
l+kr 6H((kl+kr) (k,...,kr_l))
d
0
then
eor 2.2.1 of I] at
by e se meod if (0(kl+kr),0(k2+kr)
it follows by an easy generalization of
S.O. AJALA
226
[5],
A],
Theorem
Reinhard Schultz showed that the inertial group of product of any
number of ordinary spheres is trivial.
0(k2+kr),...,0(n_kr ), ...,O(n-kl
then follows that
0kl+kr
H (kl+kr,
n)
then
0
n
(O(kl+kr)
n
S
is diffeomorphic to
It
is one to one and onto hence the number of differentiable
sklxsk2x...xSkr
structures on
Tis result implies that if
(k2,... kr_ I)
xH
is equal to the order of
0k2+kr
kl+k2+k3
(k2+kr, el, k3,... kr_l )x’’"
0kl+k2+k3
xH((kl+k2’+k3) (k4,...,k’r)) x
XH
((kl+k2+k3) (k4,... kr)
0n-kl
x(’n_kr, kr) x
0n-kl. n
XH(n_kl, kl) X O
EXAMPLES
We recall that in Table 7.4 of [5], O n denotes the number of homotopy spheres
k
n+k
which do not embed in R
We shall use the values computed in that table in some
of the examples given here.
S2S2XS2xS2
structures on
H(8,2)
0
Since
Fi
0
is the order of
for
i
_<
0 8= 2
and so the number of smooth structures on
similar reasoning, the number of smooth structures on
Since
0
12
0
and
H(9,3)
4
i
<6
then the number of smooth
Also ,=
2=
S2XS2XS2XS2X2
S2XS2xS2xS4
1081
12.
By
12.
then the number of smooth structures on
S3XS3XS3XS3 2 whereas since 015 16256 and 09 8 combined with the fact that
012 0 and H(9,3) 4 it follows that the number of smooth structures on
S3XS3XS3XS3XS3 is 32512. From 3] we see that 058 I and H(8,4) 08 and FI2=0,
then the number of smooth structures on S4XS4XS4XS4
2. By a similar argument, it
is easily seen that the number of smooth structures on S4XS4S4XS4XS4 is the order
016
Also since H(10,5)
X 020
010 then the number of smooth structures on
8(16, 4)
5
20
015
S5XS 5XS 5xs
is the order of
X O
H(15,5)
I am grateful to Professor P. Emery Thomas, University of California, Berkeley,
for very useful commmnications.
I.
Samuel Omoloye Ajala. Differentiable structures on product of spheres, ouston
Journal of Mathematics Vol. i0 No. I (1984) 1-14.
2.
R. de Sapio. Differentlable structures on product of spheres,
Hal. (Vol. 44), 1969.
3.
C.omment Math.
D. Sullivan. On the Hauptvanmuting for manifolds, Bulletin American Math. Soc.
(1967), 598-600.
J. Munkres. Obstruction to smoothings of plecewlse linear homomorphism, Annals
of Math. 72 (1960), 521-554.
J. Levlne. Classification of DifferentLable knots Annals of Math. 86 (1965), 15-50.
73
4.
5.
6.
Reinhard Schultz. On inertial group of a product of spheres, Transactions of
American Math. Soc. 156 (1971) 137-157.