S.
SASAO
KODAI
MATH.
J.
8 (1985), 285—295
ON SELF-HOMOTOPY
EQUIVALENCES
OF
S-PRINCIPAL BUNDLES
OVER
S
n
Dedicated
to
Professor Nobuo Shimada
on his
60-th birthday
BY
SEIYA SASAO
§
1.
Introduction.
The
purpose
of
this paper
is to
study
the
group
ε(X) of
homotopy classes
of self-homotopy equivalences
for the
total space
of a
S
3
-principal bundle over
S
n
(n^6).
Since this group acts
on the set of all
homotopy invariants
it
must
be useful
in the
homotopy theory
to
clarify
the
group structure
and it's
action.
However,
as
stated
in [1], any
finite group
can be
realized
as a
subgroup
of
ε(X)
for
suitablly chosen space
X, so it
seems
to be
difficult that
we
clarify
them
in
general. Many authors have computed
the
group
ε(X) for
various type
of spaces. Especially
J. W.
Rutter
has
determined
ε(X) in our
case
of n—Ί in
[4]
and
also
N.
Sawashita
and M.
Mimura treated
our
case under some addi-
tional
conditions
in [6] J. W.
Rutter's results were complete except
one
sub-case
because
he
could
use the
speciality
of
n—Ί, however,
our
results
are
weaker
compared
with
his
ones because
of
generality.
In our
theorem
the
group
structure
of ε(X) is
only clarified
up to
extension,
and to
determine extensions
is left
as
problems.
§2.
Method
and
Theorem.
Let
p: X-^S
n
be a
S
3
-principal bundle over
S
n
with
the
characteristic class
£(<Ξπ
n
-i(S
3
)),
and let Y
x
be the
space
of
continuous maps from
X to Y
with
compact-open
topology. Then
we
have
a
fibring
p
x
:
X
x
->S
nX
in the
sence that
the
map p
x
satisfies
the
homotopy lifting condition
for
CW-complexes.
If we
take
the
identity map
l
x
and the
projection
p as the
base points
of X
x
and S
nX
respectively then
the
following exact sequence
can be
obtained
as
usual
n,{X
x
, l
x
) —>
^(S
nX
f
p) —>
π
o
(p
χ
-\p),
lχ) —>
π
o
(X
x
,
l
x
) —>
π
o
(S
nX
,
p).
Since,
using
the
free action
of S
3
on X, we can
easily obtained
a
homeomorphism
φ
:
(S
3X
, •)
—>
(p
x
'\p),
l
x
)
(*(X)=1)
defined
by
φ(f)(x)=xf(x)
for
f:X->S\
the
pair (p
x
~\X),
l
x
) may be
identified
Received August
27, 1984
285
286 SEIYA
SASAO
with the pair (S
sX
, *). Now it is clear that the space X
x
forms a topological
semi-group with unit element l
x
under the multiplication defined by composition
of maps and also we can
give
the space S
sX
the same structure with unit ele-
ment
* by defining multiplication:
ftg(χ)=f(χg(χ))g(χ).
Since these structures are inherited by the set
π
o
(S°°
x
,
*)=π
o
(p
x
~
1
(P), lχ) and
π
o
(X
x
,
1*), which we may regard as semi-groups, and then the map
π
o
(S
3X
,
*) —>
π
o
(X
x
,
l
x
)
is homomorphic by definitions. For any semi-group G we denote by reg. G the
group consisting of regular elements of G. Since it
follows
from definitions that
ε(Z)=reg.
π
o
(X
x
,
l
x
) and that the boundary
Wl
(S
wX
,
P) —> π
o
(p
z
-\P), lχ)^πo(S*
x
, *)
is homomorphic, the preceeding sequence can be transformed into the exact
sequence
π
1
(X
x
>
lχ) —>
π
x
{S*
x
,
p) —> reg. π
Q
(S*
x
, *) —* ε(X) —> π
o
(S
nX
, p).
Thus
our purpose is to study
the
image of the map ε(X) —> π
o
(S
nX
, p) and
the
image of the homomorphism d
x
: π
1
(S
nX
, p) —> reg. π
o
(S*
x
, *).
Off course these require describing π
o
(S
nX
, p) and reg. π
o
(S*
x
, *) by comparablly
well-known concepts. These problems shall be treated in § 3 and § 4. Let
f: K-+L be a map and suppose that H
n
(K) and H
n
(L) are both isomorphic to
integers Z. Since the degree of /# : H
n
(K)-+H
n
(L) is defined as usual we denote
by d
n
(f) the degree of /*. Since our space X has H
3
(X)=H
n
(X)=Z d
z
(f) and
d
n
(f)
are defined. Clearly if / is a homotopoy equivalence we have d
s
(f)=±l
d
n
(f)~±l.
We denote by ε+(X) the kernel of the homomorphism d=(d
3
, d
n
):
ε(X)->Z
2
xZ
2
, i.e. we have an exact sequence:
d
1
—> ε
+
(X) —> ε{X)
—>Z
2
XZ
2
.
Remark, d is equivalent to the usual representation ε(X)->Aut//*(Z). Let
τ
be the Blaker-Massey map S
6
^S
3
. Then we have
THEOREM
A.
Assume
that τ-E
3
ξ=0 modf^^+gCS
71
"
1
),
then
we have
d : ε(X) —> Z
2
xZ
2
zs
onto
if
2£=0
and
d : ε{X) —> {(-1, -1)} is
onto
if 2f^0.
ON
SELF-HOMOTOPY EQUIVALENCES 287
Remark. If the order of ξ is odd it can be shown that d is trivial in the
case of ro£
3
fξέ0 modf^n^S
71
"
1
).
Let η be the essential map of π
n+1
(S
n
) (nΞ>3),
THEOREM
B. // ξ
°-η—{)
there
exists
an exact
sequence
0 —> π
n
(Sηxπ
n+s
(Sη/H
ξ
—+ ε
+
(X) —-» G
ξ
—^0,
and if ξ°ηφθ we have an exact
sequence
0 —^ Z
2
—^ π
n
(S*)Xπ
n+z
(Sη/H
ξ
—> ε
+
(Z) —> G, -—> 0
except
the
case
ξ°η—η°Eξ
where
Gξ and Hξ are
defined
in §5.
Remark. We could not get similar results in the exceptional case, but /
think
that the cases must be determined by τ°E*ξ°η=0 or not. Indeed, / can
prove that if τ°E
z
ξ °η=0 (=^y«£^r»f£
4
9 we have the
first
sequence in the
exceptional case.
§
3. The set π
o
(S
nX
, p)
X may be regard as a CW-complex of a form S
z
Ue
n
\Je
n+
\ so we denote by
A
the subcomplex S
z
\Je
n
. Since S
3
can be considered as a fibre we have the
fibring which is obtained from the restriction of maps
r:S
nX
->S
nS
(r(j5>)=*).
On
the other hand, using dim X=n-\-3 and the homotopy cellular approximation,
we know that any map X->S
n
\/S
n+z
can be uniquely determined up to homotopy
by a pair of maps: X-+S
71
and X-+S
n+
\ Let q:X-^S
n
VS
n+B
be the map cor-
responding to the pair, p: X->S
n
and the collapsing X-*S
n+s
, and define the map
ψ
:S
nsnvsn
+
s_^
SnX
by
ψtf^f.g
f
0Γ a map
f
: s
»
V
S
re+s
->S\ Then it is easy
that
Ψ
gives
rise a homeomorphism from the space
S
nSnvsn+
*
onto the fibre
r'X*) of the above fibring. Since it
follows
from the assumption n^β that
TΓ^S^
3
,
*)=π
o
(S
nS
\ *)=0 we have the bijection
IS
71
,
S
n
lxlS
n+s
, S
n
l^π
0
(S«
snvsn+3
, (1, 0))—>*o(S
n2Γ
, ί) = [X, S
n
].
*
Now
we define a multiplication in the set [5
n
, S
n
] X [S
n+3
, S
71
] by the formula:
(m, Λ)7(M,
β)—(mn, mβ+noi). Clearly this multiplication
gives
an abelian semi-
group structure with unit element (1, 0). Especially we have
LEMMA
1. reg. {[S
π
, S
n
~]xS
n
+\
S
n
~\)
^Z
2
®π
n+3
{S
n
)
Proof.
Since (m. ά)l{n, β)=(l, 0) implies mn—\ and mβ+na—0 we have m—
±1,
w=±l
and
α+/3=0.
Conversely
(±1,
α) clealy regular for any a and the
imbedding α->(l, a) is an injective homomorphism and then the decomposition
is trivial. These show the
proof.
288 SEIYA SASAO
Let
/, g be
maps
X—>X.
Then,
by the
above bijectivity,
we
have
pof—(m\/a)°q
and
p°g=(nV
β)°q
for some integers
m,n and a,
β(<Bπ
n+3
(S
n
)).
LEMMA
2.
p(g°f)=(mn, (na+d
3
(f)mβ))°q
Proof.
Consider
the
following diagram
X
> X >X
s
n
vs
n+3
>
s
n
>s
n
vs
n+3
V
?nVa
nVβ
Since
we
have
p(g°f)=(nVβ)°q°f
and ?
/={(mVα),
(0,
rf
B+8
(/))}
(?
the
proof
follows
from
(nVj8){(mV«),
(0,
d
n+3
(f))}=(mn, (na+d
n+3
(f)β))
=
(mn, (na+d*(f)mβ)).
Now,
from lemma
1 and 2, we can
obtain
the
commutative diagram:
1
—>
ε
+
(ΛT)
—>
ε(Z) -^> Z
2
xZ
2
\
\ v
0
—>
,τ
n+s
(S
n
)
—>
Z
2
Xπ
n+Ά
(S
n
)
—>Z
2
—>0
Let
iA : A-+X and i
3
:
S
3
-+A
(Cl) be
inclusion maps.
LEMMA
3. Let λ be the
attaching
map of the
(n+3)-cell
of X. For a map
h:A-^A
of
type
(—1,-1),
i.e.
d(h)=(—l,—l)
we
have
where
s is an
integer
satisfying
2sE
4
ξ=0.
Proof.
Using
the
commutative diagram:
ON
SELF-HOMOTOPY EQUIVALENCES
289
n+
2(Sη
—^
7Γ
n+
,G4)
—>
τr
n+2
(Λ
S
3
)
> π
n+3
(S
4
)
—> ^
n
3
z*
and
Hopf
map v
A
:
S
7
—>S
4
,
we
have
h*{λ)—λ+isSΓ)
for
some element
Γ of
π
n+2
(S
3
)
and
H(v^E'ξ))
by (3.1) of [5]
These show that i*(EΓ)=ι*E(τ°E*ξ),
i.e.
E(Γ—τ°E
z
ξ)
is
contained
in the
3-image.
Since
the
we have,
for
some integer
s and
γ<=π
n
+
2
{S
n
~
ι
),
On
the
other hand,
we
know
the
decomposition:
π
i
(S
i
)=Eπ
ι
.
1
(S
s
W^
πι
(sη.
Hence
we
have that 2s£
4
f=0
and
r=(s+l)r°£
3
f+f°r,
i.e.
Thus
the
proof
is
completed.
Next,
let k : A^A be
another map
of
type
(—1,-1)
and let
Q:A-+AVS
n
be
a
map
collapsing
the
equator
of the
n-cell
of A to a
point. Then
it is
well-
known that
k can be
represented
as a
composion:
A
>
AVS
71
>
X
Q
hVσ
By using
Q*(λ)=λ+[σ,Cs],
we can
know that