701
Internat. J. Math. & Math. Sci.
VOL. ii NO. 4 (1988) 701-712
SMOOTH STRUCTURES ON SPHERE BUNDLES OVER SPHERES
SAMUEL OMOLOYE AJALA
Institute for Advanced Study
School of Mathematics
Princeton, New Jersey 08543
USA
and
Department of Mathematics
University of Lagos
Akoka Yaba
Lagos Nigeria
West Africa
(Received May 30, 1986)
ABSTRACT.
In [I] R. De Sapio gave a classification of smooth structures of a p-sphere
bundle over a q-sphere with one cross-section and p
<
q.
In [2] J. Munkres also gave a
classification up to concordance of differential structures in the case where the bundle
has at least two cross-sections.
p
>
q.
In [3] R. Schultz gave a classification in the case
Here we will give a classification of the p-sphere bundle over a q-sphere
without any cross-section and p
KEY WORDS AND PHRASES.
<
q.
Smooth structures, differential classification, internal groups.
1980 AMS SUBJECT CLASSIFICATION CODES:
I.
57R55.
INTRODUCTION
Let E represent p-sphere bundle over a q-sphere with
Nq_ISO(p+I)
B
characteristic class of the corresponding p+1-disc bundle over the q-sphere.
De Sapio gave a complete classification of the special case where B
O.
the
In [4] R.
In [5] and [6]
Kawakubo and Schultz respectively also gave a classification of E for this special case.
This author in
[7] gave a generalization of this special case to product of three
ordinary spheres.
In [1] a classification of E was given for p
<
q
I and where E has
In [3] Schultz gave a classification of E for p
a cross-section and B
O.
without cross-section.
We shall here remove the fact that E has a cross-section so that
not every element of
pulled back to the element
Xq_iSO(p+l) can be
Xq_ISO(p) q_iSO(p+l)
q and E is
q_lSO(p)
in the
homomorphism S,
induced by the inclusion s
SO(p) SO(p+1).
n
S denotes the unit n-sphere with the usual differential structure in the Euclidean
S.O. AJALA
702
n denotes an
homotopy n-sphere and o n denotes the group of homotopy
n-spheres. H(p,k) denotes the subset of o p which consists of those homotopy p-sphere
S
By [4, Lenna 4], H(p,k) is a subgroup of
such that z p S k is diffeomorphic to S p
p
0 and it is not always zero and in fact in [7] we showed that if k -’_- p-3H(p,k)
o p.
(n+l)-space R n+l
Z
k.
We shall adopt the notation E(. q) to represent the total space of a p-sphere bundle over
a homotopy q-sphere
THEOREM.
7.
q.
We will then prove the following:
If M is a smooth, n-manifold homeomorphic to a p-sphere bundle over a
p+q _> 6 and p < q then there exists homotopy
q-sphere with total space E where n
n
q
spheres z and z such that M is diffeomorphic to E(. q) # z
We shall define a pairing
n.
pSO(q) q_lSO(p+l)
G
h.omotopy
p+q
q_ISO(p+I) is the characteristic class of a p-sphere bundle
7. q, then
G(pSO(q),B) equals the inertial group of E(zq).
and show that if B
an
0
q-sphere
over
The
above theorem together with the latter will give us the following.
THEOREM.
Let E be the total space of a p-sphere bundle over a qsphere then the
diffeomorphism classes of (p+q)-manifolds that are homeomorphic to E are in one-to-one
correspondence with the group
on
Image" G B
p+q
n
where
>_
6
and
p
<
q.
CLASSIFICATION THEOREM
2.
In this section, we will prove the classification theorem for any manifold M n
homeomorphic to E.
We will apply the obstruction theory to smoothing of manifolds
developed by Munkres in [8].
Since p+q
6 and 2 < p < q then E is simply-connected and
the homology of E has no 2-torsion, hence the "Hauptvermutung" of D. Sullivan [9]
applies
and
this
means
that
piecewise
homeomorphism
linear
can
be
replaced by
homeomorphism, we shall not distinguish the two.
DEFINITION. Let M and N be smooth closed n-manifolds and L a closed subset of M of
dimension less than n.
Let f
M
N be a homeomorphism such that for each simplex
L, y and f() are contained in coordinate systems under which they are flat.
to be a diffeomorphism modulo L if
fI(M-L)
f is said
is a diffeomorphism and each simplex y of L
has a neighborhood V such that f is smooth on
V-L
near y.
By [8, Theorem 2.8], if M and
N are homeomorphic then there is a diffeomorphism modulo (n-1)-skeleton of M.
f
M
y of
N is a diffeomorphism modulo m-skeleton m
<
If
n then the obstruction to deforming
SMOOTH STRUCTURES ON SPHERE BUNDLES OVER SPHERES
703
f to a diffeomorphism modulo (m-1)-skeleton g
M N is an element (f)
n-m)
n-m
n-m-1
where r
is a group of diffeomorphism of S
modulo those that extend to diffeo-
Hm(M,r
morphisms of D
n-m.
g is called the smoothing of f.
If k(f)
0 then by [8, {}4]
smoothing g exist.
THEOREM 2.1.
If M is a smooth n-manifold homeomorphic to E where E denotes the
total space of a p-sphere bundle over a q-sphere, 2 _< p < q and n
p + q then there
n
q
exist homotopy spheres z and z such that M is diffeomorphic to E(7. q) # z n where E(7. q)
z q.
denotes the total space of a p-sphere bundle over the homotopy q-sphere
PROOF.
E is the total space of a p-sphere bundle over a q-sphere with
characteristic class [b]
Sp
S q-1
q_ISO(p+1)
then
Sp
Dq
E
,) D q
S p where
fb
S p is a diffeomorphism defined by
fb(x,y)
(x,b(x).y), (x,y)
e
fb Sq-1
S q-1
Sp
=I oZ efr]sewherei
O,p,q,p+q
Hi(E)
n
Since M is homeomorphic to E where n
p+q
_>
6 2
<
q, then M n is simply connected
p
H3(M,Z)
has no 2-torsion, then "Hauptvermutung" of D. Sullivan [9] implies
that there is a piecewise linear homeomorphism h
M n E which by [8, {}5] is a diffeo-
and since
morphism modulo (n-1)-skeleton.
Since
Hi(M,Z)
0 for n-p+1 <
assume that h is a diffeomorphism modulo n-p
q skeleton.
_<
n-1 then we can
The obstruction to a
"p) r p. If [] (h) "p where
diffeomorphism modulo q-I skeleton is ),(h)
S p-1 S p-1 is a diffeomorphism that represents .(h) and let 7. p denote the homotopy
Hq(M,I
p-sphere where
sP
D @U D.
j"
We define a map
sP P
where
Sp
D
L)
id.
D
such that
X
if
X +
DD
j(x)
x
x
SO j is an homeomorphism which is identity on
DIP
and the radial extension of
and so the first obstruction >,(j) to deforming j to a diffeomorphism is
We then define id
j
Dq
Sp
Dq
[-I]
p
7. where id is the identity, then id
-I on D
-x(h).
j is a
homeomorphism and it follows from [8, Def. 3.4] that the first obstruction >,(id xj) to
S.O. AJALA
704
deforming id xj to a diffeomorphism is also -(h). We can form a manifold E’ by
identifying two copies of D q
along their common boundaries S q-1
z p by the diffeoSq-1 sP S q-1 sP where fb(x,y) (x,b(x) y) and
morphism
zP
fb
q_iSO(p+l).
[b]
g
E
(DqxSP)
(Dqxz p) 1’
sP)2 (DqxzP)l U
fb
(Dqxs p) 2’
and
(DqxSP)
E
z p. We define a map
Dq
U
fb
.} (Dqx
fb
on both
zp
Dq
So E’
(DqxzP) 2
the map looks like
U (DqxSP) 2
(DqxSP)
fb
idxj
U (Dqxr.P)2
(DqxzP)
U sq-Ixs p
fb
,g
E’
E’
(DqxsP)
fb
by
g(x,y)
.
(DqxSP) 2
id
idxj
dxj
L)
fb
j(x,y)
id
sq-Is p LJ (DqxzP) 2
id
g is an homeomorphism and the first obstruction to a diffeomorphism is ,(id xj)
It follows that the obstructions to smoothing the composition g
>,(g.h)
},(g) + (h)
-(h) + (h)
O.
It follows that g.h.
E’
M
h
-,(h).
is
E’ is a diffeo-
M
morphism modulo (q-1)-skeleton. However in [7, Remark 1] we showed that D q
z p is
S p if p < q + 2 and so by our hypothesis p < q then it follows
diffeomorphic to D q
r. p is diffeomorphic to D q
that D q
S
This implies that E and E’ are diffeomorphic
p.
hence g’
p +
M
<
E is a diffeomorphism modulo (q-1)-skeleton.
Since
Hi(M,Z)
0 for
q-l, there is no more obstruction to deforming g’ to a diffeomorphism until
we get to (p-l) skeleton.
We can then assume that g’ is a diffeomorphism modulo
The first obstruction to deforming g’ to a diffeomorphism modulo (p-1)q) r p. Let [] (g’) r q where
skeleton is >,(g’;)
S q-I
S q-I is a
diffeomorphism which represents (g’)
S q-1
Sp
S q-1
Sp
r q. We define (id)
p-skeleton.
Hp(M,r
((x),y) and if B
where (xid)(x,y)
S q-1 S P where
fb(x,y)
morphisms of S q-1
S q-1
S p where
(xid)
[b]
q_lSO(p+l)
we also define
fb Sq-1
Sp
(x,b(x).y). We then have two orientation preserving diffeop
Sq-1 S p
S unto itself which we can compose to get (xid)
fb
fb(x,y)
((x),b(x)-y).
We then construct a manifold by
p
q
S along their common boundary S q-1
attaching two copies of D
S p using the diffeoDq S p. Notice that this manifold is
L)
Sp
morphism (xid)
fb to have
2
D
(xid).fb
a p-sphere bundle over a homotopy q-sphere
.q
Dlq U D2q
whose characteristic map is
SMOOTH STRUCTURES ON SPHERE BUNDLES OVER SPHERES
[b]
B
q_lSO(p+l).
705
We define a map
u
fb
by
(x,y)
if
h(x,y)
(.-(ll),y)
Hence h is identity on
Sp
D
(x,y)
f
D
(x,y)
S p and a radial extension of
D
Sp
D
L)
(xid)’f b
sP
-I
D.
on
It then follows
that h is an homeomorphism with the first obstruction to a diffeomorphism being
[-1]
-(g’). Then by [8, 3.8] the first obstruction to deforming the composition g’
Sp
D
M
g
(xid)-f b
->,(h) + >,(h)
Hi(M,Z)=
Since
Sp
<
J
(xid)’fb
;(g’) + x(h)
;(g’oh)
0 and hence g is a diffeomorphism modulo (p-1)-skeleton.
0 for 0 <
D
S p into a diffeomorphism is (g)
D
U
h
Since
p then we can assume that g is a diffeomorphism modulo one point.
D
S p is a p-sphere bundle over a homotopy q-sphere
q
L D with characteristic map [b]
Xq_lSO(p+l),
we shall denote it by
E(zq).
Since g is a diffeomorphism modulo one point then it is known that there is an homotopy
n-sphere Z n such that M is diffeomorphic to E(}: q) #
Hence the proof
.n
3.
INERTIAL GROUPS
by Theorem 2.1, every manifold homeomorphic to E is diffeomorphic to
n
# z for some homotopy spheres z
z classification of such manifolds reduces to
Since
E(Z q)
q, n,
sn.
classification of manifolds of the form E(s q) #
To complete this classification, we
then need to investigate what happens when we vary the homotopy spheres and in particu-
lar we need to investigate the Inertial group of
E(sq). We
will investigate these in
this section.
LEMMA 3.1. Let
then
E(s)is
PROOF.
D?
Sp
i
id
[b]
U
(ixid)’fb D2
S q-1
sq-1
fb
B
diffeomorphic to
Suppose
Sp
Sp
q_iSO(p+l)
E(s)
D
be homotopy q-spheres such that
and
E(Z)if and
only if
is diffeomorphic to
S p is diffeomorphic to
z?
E(Z)).
D
Sp
+/-
z
U
i
D
1, 2
H(q,p).
This means that
U
(2xid).fb
D
S p where
S p is the diffeomorphism defined by @i(x,y)
(i(x),y) and
S q-1
S p is defined by fb(x,y)
(x,b(x).y) where
S q-1
is the characteristic map of the bundle.
be regarded as the boundary of the (p+l)-disc bundle over
The manifold
z 2 which
E(z))
is denoted by
can
S.O. AJALA
706
D p+I
D
)
D p+I=
D
(2xid) "fb
E(r)it
can be embedded in
embedds in
E(z2q)and
D(s2q ).
follows that
z z
so we have
E(r.)is
So if
and
z
diffeomorphic to E(z 2) then since
embedds in
E(r.),
sitting in
But
if we translate
r.
we can run e tube between them to obtain an embedding
E(z).
#
(-r)/ E(r.)
r.
r.
naturally
away from
so that the
embedding is homotopically trivial and so by the engulfing result of [10, chapter 7] it
r.
means that
(-r.)
#
can be embedded in the interior of a (p+q+l)-disc in
E(z)
and by
[11, 3.5] the embedding is isotopic to a nuclear embedding into the interior of
D p+I is an homotopy equivalence, it
D p+I is diffeo#
then follows by Smale’s theorem [12, Theorem 4.1] that
Sp
S p is diffeomorphic to S q
D p+I and so it follows that z q # (-r.)
morphic to s q
Sq
Sq
However the embedding z # (-r)
r.
(-r)
q2
hence
(r. qI, #
r.
r.
(-r.)
#
H(q,p). Conversely suppose
p
S is diffeomorphic to S q S p
(r.q2))
trivial normal bundle then it follows that r.
This shows that each r.
normal bundle.
(-r.)
#
Since S q
r.
#
H(q,p) then this implies
p
S embedds in Rp+q+ with
embedds in
with trivial
Rp+q+
2 embedds in
for
with trivial
r.
into the
normal bundle and by [11, {}3.5] the embedding is isotopic to an embedding of
p+I
q
q
Dp+ However for
is an homotopy
S
D
1, 2 the embedding r.
interior of S
equivalence hence it follows from [12 Theorem 4.1] that r,
Sq
D p+I which implies that
D p+I is diffeomorphic to
z
r.
D
L,
D p+l--i
along S q-1
(i
zq2
r.
sq-l/ Sq-1
D where i"
D Dp+I ixid
Dlq
r.
represents
"q
e
z
Now since
1, 2, then we can write
S q-1
id
i
D p+I
r.
id)(x,y)
(i(x),y) where (x,y) e S q-1 D p+I. So
D p+I is diffeomorphic to
D p+I implies
D p+IL
D
lXid
Now consider the manifold D(S q)
D p+I.
D p+I where we identify two copies of Dq
D p+I by the diffeomorphism
D
D p+I is diffeomorphic to
D p+I
Dq+
L_
over a q-sphere with characteristic map [b]
D
q_
D p+I
D p+I defined by
S 1-1
D p+I is diffeomorphic to
D
DP+I,Xid D2q
D p+I.
2
D p+I which is a (p+l)-disc bundle
’b q_lSO(p+l).
We then form the quotient
space
D(Sq) L,
zq 1
by identifying D
(x,y)(x D q D
e
Since
r.
_=
D p+I
q_
D p+I
y
e
L9
fb
c
D p+I)
D(S q) and
1Xid
D
x
D p+I
c
r.
The manifold D(S q) u r.
D p+I is diffeomorphic to
r.
D p+I.
Let d"
D p+I by the relation (x,y)
D p+I is similarly constructed.
};
D p+I
r.2q
D p+I be the
SMOOTH STRUCTURES ON SPHERE BUNDLES OVER SPHERES
707
diffeomorphism and since any diffeomorphism fixes a disc, we can assume that d is
p+q+l
D p+1 then we can define a diffeomorphism.
identity on the disc D
,
D
D(S q)
g
I :ql Dp+I D(Sq) U
s
D p+I
where
d(x)
g(x)
x
x
for
:
:
x
for
.
D p+I
D(Sq).
D p+I and identity on D(sq). g is well defined because d
p+I and g is a diffeomorphism. The
x D
is identity on the disc connecting D(S q) and
S q-1
D p+I can be clearly seen as follows. Let (ixid).fb
manifold D(S q) U
D p+I be the diffeomorphism defined by
S q-1
D p+I
d on
This means that g
z?
(i(x),b(x)-y),
((ixid).fb)(x,y)
q_
D p+I by the
D p+I and D
D
D p+I we get a (p+l)-disc
D p+I then attaching two manifolds D+q
(x,y)
q
J
D p+I
diffeomorphism (i id)’fb we have D+
S q-1
(ixid)-fb
sq
bundle over the homotopy q-sphere
D(S q)
Dq+
U
LJ
i
D
I
is constructed it is easily seen that
11
D p+I
D
q_
U
(ixid).fb
then it follows that
Dqx_ D p+I=
D(s?)hence
a(D(;)) E(z)is
However from the way
2
g is the diffeomorphism of
diffeomorphic to
D p+I
D(S q) tJ s
D(l:q1) onto
aD(;2q)= E(.).
Hence the theorem is proved.
REMARK 1.
and
r.
This theorem implies that
E(Slq)
are equivalent in the quotient group
is diffeomorphic to
E(};)
if and only if
oq/H(q,p).
To complete this classification, we need to determine the inertial group of E(l;q).
The inertial group ,(M) of an oriented closed smooth n-dimensional manifold M is defined
to be the subgroup of o n consisting of those homotopy n-spheres l; n such that M #
diffeomorphic to M.
Let E 6 represent the total space of a p-sphere bundle over a real q-sphere with
p+n
characteristic class 6
q_iSO(p+l). In [13] we defined a map G 6 pSO(q) o and
showed that the image of this map equals the inertial group of E
no cross-section.
We shall similarly define a map
6
where p
Op+q
G .B pSO (q)
<
q and E
has
6
and show that
the image of this map equals the inertial group of E(S q) where E(s q) is the total space
of p-sphere bundle over a homotopy sphere
G
O.B()
S q-1
D p+I f
_l(@Xid).f
Dq
b
zq
D
U D2q.
S p where [a]
Let
and [b]
pSO(q)we
B
e
define
q_ISO(p+I)
and
S.O. AJALA
708
S q-I
fa_l(Xid).f b
(a-l(b(x).y)
S q-I
Sp
(x),b(x)-y).
S p is a diffeomorphism defined by
One can easily show that
G. B
its image is an homotopy (p+q)-sphere as similarly shown in
q-sphere
is well-defined and that
[13].
Let E(r.q) denote the total space of a p-sphere bundle over an homotopy
LEMMA 3.2.
D D
zq
fa_l(Xid)-fb(x,y)
D
with characteristic class
l(E(r.q)).
PROOF. If z p+q I(E(sq)) then
E(z q) # }i p+q E(llq), that is,
Xq_lSO(p+l)
B
then
G.Bp(SO(q))
d
(DqIxSP
d
since p
<
q then
this means there is a diffeomorphism
p) # r.P+q
DxS
(xid)’f b
p(E(sq))
Sp
Dq
L
U
D
(xid)’f b 2
Sp
d(oxS q) represents a generator and
is infinitely cyclic and
E(zq). By Haefliger’s theorem [14], dlO S p
Sp
so is homotopic to the inclusion 0
Sp
E(r. q) are isotopic and by isotopy extension theorem and
the inclusion 0
and’
tubular neighborhood theorem, d is isotopic to a map which we shall again denote by d
q S p Dq
Dq
and (x,y)
(a(y)-x,y) for [a]
S p where d(x,y)
such that diD
Dqxs p) # P+q by
U
(Dqxs p
S p. We now remove D q x S p from E(z q) # }:P+q
(xid)-f b
D p+I. After this operasurgery away from the connected sum and replace it with S q-1
q-1 x D p+I
tion on the summand E(z q) of the connected sum we have the manifold S
S p extend to
S q-1
S p S q-1
U
Since the diffeomorphism (xid)-f b
Dq S
pSO(q)
p.
(xid).fb
D p+I
Dp+I onto itself then Sq-1
S q-1
the diffeomorphism of
Dq
Sp
is
(id)-f b
diffeomorphic to S q-1
x
D p+I
L)
Dq
S
p,
the diffeomorphism g is defined thus
id
D p+I
sq-1
(xid).fb
S q-1
!
Sp
Dq
U
id
U
p+l
q
sp
(xid).f b
where
(x,y)
Dq
(x,y)
if
Sp
g(x,y)
((xid).fb)(x,y)
(p+q)-sp.here S
p+,
is reduced to S
p+q
D p+I
S q-1
However by [7 Lemma 2 1 2]
if
U
Dq
(x,y)
Sp
S q-I
D p+I.
is diffeomorphic to the standard
id
hence after this surgery E(F. q) is reduced to S p+q and so E(Y q) # E
Z
p+q
Z
p+q
Z
p+q.
p+q
SMOOTH STRUCTURES ON SPHERE BUNDLES OVER SPHERES
709
We perform the corresponding modification (under d) on E(E q) to remove the
p) in
S p with product structure
obtain a manifold S q-1
D p+I l] Dq S p where
d(DxS
p-sDhere 0
sp
D p+ 1 L] Dq
this manifold S q-1
(d’llsq-lxSP).(xid).fb and this
B()_ by the definition of G
dl(DxS p) which gives pSO(q)
G
pSO(q) (namely)
p+q
and so z
G.B(pSO(q)), hence
suppose p+q %.B(pSO(q)) then for some
exists an element
G.B(e)
S p where
this modification we
is
z p+q because of the way we performed the surgery using d. However,
diffeomorphic to
Dq
E(zq). From
I(E(zq))C_G.B(pSO(q) ).
pSO(q), P+q
thus there
such that
Conversely
D p+I
S q-1
l.]
(xid). fb
f -1"
a
is a diffeomorphism of S q’l onto itself representing
sP+q
sq
D
U
D
and
Notice that G
) is thus the obstruction to the
p+q
donstruction of a diffeomorphism S
s P+q. To construct a diffeomorphism from S p+q
.P+q, we map sq-1 D p+I
S p+q to itself using (xid).fb to have
f -I and
a
fb
are as defined earlier.
Sp+q
sq-lx D p+I
l.)
!
id
(x id)" fb
.P+q
and try to extend it to
f
b_ f
(0 -Ix id). fa" (x id).
Dqx S p
diffeomorphism
fa
DP+I
Dq
f
Sp
1" (x id). fb
On the boundary S q-1
So this means that r. p+q
fb"
extending the diffeomorphism f
diffeomorphism of
S q-1
Sp
Dq
S P of D q
G. B ()
(0 -lx Id). fa" (x id). fb
b -I"
Sq’l
Sp
the map is
is the obstruction to
Sp
Sq’l
S p to a
Dqx S p onto itself. We can then define a map E(z p) E(zq) using the
S p where fa(X,y)= (a(y).x,y)(x,y)e
S p we then
Dqlx sP/
Dx
Dx
have
E(Z q)
E(Zq)
On the boundary S q-1
S p of
Sp
Dx
Dlqx
D1qx S p,
I
Dq
Sp
(x id). fb
2
L]
D2q
Sp
U
fa
Sp
(x id).
this map is
fb
fb_ 1. (-lxid).fa. (xid). fb
obstruction to extending this to a diffeomorphism of
E(Zq)
onto itself is the
and the
710
SoO. AJALA
.(@’lid).f a
obstruction to extending the map f
b_
S p onto itself which is r. p+q.
D
diffeomorphism and so r. p+q
E(r. q) # z p+q is a
It then follows that E(r. q)
I(E(r.q))
hence
E(E(zq))
REMARK 2. We note that if p
(xid).f b to the diffeomorphism of
G.Bp(SD(q)
2, 4, 5, 6 (mod 8) and p
<
IIpSO(q)
q-1 then
0 and
so the image of G is trivial and hence in this particular case, the inertial group of
E(r. q) is trivial and this cofncides with the result of [4, Proposition I].
REMARK 3.
By [15], intertial group I(M) of a smooth manifold M is a diffeotopy
invariant of M.
So if 2p
_>
p-sphere bundle over an homotopy q-sphere
:q
q-sphere where E(z q) is the boundary of
has the homotopy type of S
and
q,
D(zq). r. q has
IIq_iSO(p+l)
are simply connected and from [12-
follows by [15] that
the
I(E(r.q))
I(EB).
>
5 and p
D(r.q).
>
q
Since
3 then D(r. q)
Theorem 4.4], it follows that D(r. q) is
over the q-sphere S q hence the boundary
of D(Z q) is diffeomorphic to the boundary BD(S q)
E(r.q)
classifies
the homotopy type of D(z q) and
3q + 3 and since p + q
diffeomorphic to a (p+l)-disc bundle D(S q)
aD(z q)
of a
q
it follows that S has the homotopy type of
q+l then it follows that 2(p+q+1)
E(r.q)
B)
Let D(Z q) be the associated (p/1)-disc bundle over the homotopy
associated disc bundle.
>_
of a
is equal to the inertial group I(E
p-sphere bundle over the standard q-sphere, where B
2p
I(E(r.q))
q+1 then we can deduce that the inertial group
E B of
D(sq). It
This means that the inertial group of S
B
then
[13]
in
coincides with Lemma 3.2.
Combination of Lemmas 3.1 and 3.2 give the following.
THEOREM 3.3.
Let E be the total space of a p-sphere bundle over a q-sphere with
characteristic map B
IIq_ISO(p+l)
then the dlffeomorphism classes of p+q-manifolds that
are homeomorphic to E are in one-to-one correspondence with the group
eq
where p+q
n
_>
6
and
p
en
q.
REFERENCES
to Sphere Bundles Over Spheres.
1.
De Sapio, R., Manifolds Homeomorphic
American Math. Soc. 75 (1969), 59-63.
2.
Munkres, J., Concordance of Differentiable Structures
Math. J. 14 (1967), 183-191.
Two Approaches.
Bull.
SMOOTH STRUCTURES ON SPHERE BUNDLES OVER SPHERES
3.
4.
Schultz, R., Smoothing of Sphere Bundles of Spheres in Stable Range.
Math 9 (1969), 81-88.
De Sapio, R., Differential Structures on a Product of Spheres.
Vol. 44,
5.
6.
711
Inventiones
Comm. Math. Helv.
(1969), 61-69.
Kawakubo, K., Smooth Structures on S p
Schultz, R., Smooth Structures on S p
S q.
S q.
Proc. Japan Acad. 4__5 (1969), 215-218.
Annals of Math. 90 (1969), 187-198.
Houston Journal of
7.
Ajala, S. 0., Differentiable Structures on Product of Spheres.
Math., Vol. 10,
(1984), 1-14.
8.
Munkres, J., Obstruction to Smoothing of Piecewise-Differentiable Homeomorphisms.
Annals of Math. Vol. 7__2,
(1960), 521-554.
9.
Sullivan, D., On Hauptvanmutung for Manifolds.
598-600.
10.
Bull. Amer. Math. Soc.
73 (1967),
Zeeman, E., Seminar on Combinatorial Topology (mimeogrpahed notes), Inst. Hautes
Etudes, Sci. Pub]. Math. (1965).
11.
Levine, J., Classification of Differentiable Knots.
Annals of Math. 82 (1965),
15-50.
Amer. Math. J. 84 (1962), 387-399.
12.
Smale, S., On the Structure of Manifolds.
13.
Aja]a, S. 0., Inertia] Group of p-sphere Bundle over a q-sphere
Cross-section. Nigerian Journal of Science Vol. 18, (1984), to appear.
14.
Haefliger, A., Plongements Differentiables de Varietes dans Varietes, Comm. Math.
Helv. 36 (1961), 47-81.
15.
Kawakubo, K., Inertial Group of Homology Tori.
(1969).
16.
without
J. Math. Soc. Japan Vol. 2._1, 1_
Hsiang, W. C., Leving, J., and Szczarba, R. H., On the Normal Bundle of a Homotopy
Sphere Embedded in Euclidean Space. Topology 3 (1965), 173-181.