Kyushu J. Math.
Vol. 57, 2003, pp. 371–382
ON THE INDEX AND CO-INDEX
OF SPHERE BUNDLES
Ryuichi TANAKA
(Received 20 February 2003)
Abstract. We study the general properties of the index and co-index in the sense
of Conner–Floyd for sphere bundles and determine these invariants of some sphere
bundles over projective spaces.
1. Introduction
Let X be a topological space with a free Z/2 -action. Following [Y, CF1], the index
of X, denoted ind X, is the largest integer m for which there is a Z/2-map from S m
to X. The co-index of X, denoted co-ind X, is the smallest integer m for which
there is a Z/2-map from X to S m . Here S m is equipped with a Z/2 -action via the
antipodal map. The Borsuk–Ulam theorem shows that ind S n = co-ind S n = n and
that ind X ≤ co-ind X in general.
The co-index was originally called the B-index [Y]. Shifting the number by one,
co-ind X + 1 is called level and denoted by s(X) [DL, PS, S]. It is also called the
Z/2-genus [B].
In [S], the co-index of the real projective space RP 2n−1 (n ≥ 2) was determined
as s(RP 2n−1 ) = n + 1, n + 2 or n + 3, accordingly as n ≡ 0, 2, n ≡ 1, 3, 4, 5, 7, or
n ≡ 6 (mod 8). In [B], estimates for the co-index of the lens space L2n−1 (2k ) were
obtained: s(L2n−1 (2k )) ≥ ](n − 1)/2k−1 [ +1, where ]x[ means −[−x]. For an odd
prime p, the smallest integer m such that there is a Z/p-map from L2n−1 (p) to S 2m−1
is discussed in [M] and very sharp estimates are obtained there.
Let α be a finite-dimensional real vector bundle over a CW complex B. We choose
a Riemannian metric for α and consider the sphere bundle S(α). S(α) has a Z/2-action
by the antipodal map on each fibre. We define ind α by ind(S(α)) + 1 and co-ind α
by co-ind(S(α)) + 1. For two such metrics, the corresponding sphere bundles are Z/2
homeomorphic and the definition does not depend on the choice of the metric.
In this terminology, s(RP 2n−1 ) can be interpreted as co-ind η2 and s(L2n−1 (2k ))
k
as co-ind η2 . Here η is the canonical line bundle over the complex projective space
CP n−1 .
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R. Tanaka
In this paper, we calculate ind α and co-ind α for some families of vector
bundles over projective spaces and study the properties of index and co-index.
We are particularly interested in two points. The first is how these numbers will
reflect the bundle structure (e.g. triviality), the other is how they will behave under
stabilization.
We first discuss in Section 2 the properties of index. We will be interested in
comparing ind α with dim α, the dimension of the fibre of α. Obviously, dim α
serves as a lower bound to ind α. Under some cohomological conditions, it is
shown that ind α coincides with dim α (Proposition 2.4). Specifically, ind α is
obtained there when α is the m-fold sum of the canonical line bundle over RP n
(Corollary 2.6).
In Section 3 we discuss the properties of co-index. When dim α ≥ dim B,
co-ind α is related to the notion of the geometric dimension. It actually gives an upper
bound to co-ind α (Proposition 3.2). If α is the tangent bundle over RP n , it is shown
that both co-ind α and co-ind(α ⊕ 1) are equal to these upper bounds and the‘stability’
equality of the co-index holds—co-ind(α ⊕ 1) = co-ind α + 1 (Theorem 3.7). On the
other hand, dim α is a lower bound to co-ind α. Over RP n , it seems that co-ind α is
equal to this lower bound if and mostly only if α is trivial (Theorem 3.4).
In Section 4 we calculate both the index and the co-index of the ‘stable family’
α ⊕ m(m ≥ 0) for two α. The first is the canonical line bundle over the projective
space FPn (F = R, C, H) (Theorem 4.1), the other is ηk with k odd (Theorem 4.3).
Through our calculation, we will see that the ‘stability’ for index, ind(α ⊕ 1) =
ind α + 1, does not hold in general, even if α lies in the stable range as a vector
bundle (see the second remark after Theorem 4.1). On the other hand, the stability for
co-index does hold in these examples and, moreover, the co-index is seen to be equal
to the upper bound mentioned in the above paragraph. It would be interesting to ask if
these points are true in general.
Throughout this paper, we write dim α = m if α is an m-dimensional real vector
bundle. The m-dimensional trivial bundle is denoted simply by m. It is to be assumed
that all cohomology has coefficients Z/2 unless otherwise stated.
2. General results for index
In this section, we discuss the properties of index. For α, an m-dimensional real vector
bundle over a CW complex B, we recall that the index of α is, by definition,
ind α := max{k | ∃f : S k−1 −→ S(α), Z/2-map}.
On the index and co-index of sphere bundles
373
We first note the following lemma.
L EMMA 2.1.
(1) dim α ≤ ind α ≤ co-ind α.
(2) ind α ≤ ind(α ⊕ 1).
Proof. In (1), the inclusion of the fibre implies the first inequality and the Borsuk–
Ulam theorem implies the second. From the inclusion α → α ⊕ 1, (2) follows.
2
Remark. For the tangent bundle τM of a closed manifold M, Conner–Floyd showed
that ind τM = dim M by making use of the Wu formula [CF1, Theorem 6.11]. So the
equality dim α = ind α holds if α is a tangent bundle of manifold. However, this does
not hold in general. Also, it does not hold that ind α + 1 ≤ ind(α ⊕ 1). Such examples
are given in Section 4 (see the second remark after Theorem 4.1).
Let P (α) denote the associated projective bundle of α and let e(∈ H 1 (P (α)))
denote the Euler class of the line bundle α → P (α). The fibre of the bundle P (α) →
B is RP m−1 and its cohomology is the truncated polynomial ring Z/2[t]/(t m ).
Here t (∈ H 1 (RP m−1 )) is the Euler class of the canonical line bundle over RP m−1 .
The inclusion of the fibre i : RP m−1 → P (α) satisfies i ∗ (e) = t, which implies that
the fibre of P (α) is totally non-homologous to zero in P (α). Also, the cohomology of
the fibre is Z/2-free and finitely generated. By the Leray–Hirsch theorem, it follows
that
H ∗ (P (α)) ∼
= H ∗ (B){1, e, e2 , . . . , em−1 }
as a H ∗ (B)-module.
Let W (α) be the total Stiefel–Whitney class of α; W (α) = 1+w1 +w2 +· · ·+wm
(∈ H ∗ (B)). Then em can be written as em = wm + wm−1 e + wm−2 e2 + · · · + w1 em−1 .
P ROPOSITION 2.2. Let α be a real vector bundle over B with dim α = m.
If W (α) = 1, then ind α = m.
Proof. Since ind α ≥ dim α in general, we prove that ind α ≤ m. Assume that
there is a Z/2-map f : S m −→ S(α). The Z/2-map f induces a map f˜ :
RP m −→ P (α). By the naturality of the Euler class we have f˜∗ (e) = t, so that
f˜∗ (em ) = t m = 0 ∈ H m (RP m ). On the other hand, the assumption W (α) = 1
i
˜∗ m
implies that em = m−1
i=0 wm−i e = 0. Hence f (e ) = 0, which is a contradiction.
Thus we have ind α ≤ m.
2
In the case that the base space B is the sphere S n , only the nth Whitney class wn
(∈ H n (S n )) is concerned. Thus we have the following.
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R. Tanaka
C OROLLARY 2.3. Let α be any real vector bundle over S n with dim α < n.
Then ind α = dim α.
The next proposition relates the cohomology of the base space with ind α.
P ROPOSITION 2.4. Let α be a real vector bundle over B with dim α = m.
∗ (RP k )) = 0 for some k ≥ m, then m ≤ ind α
∗ (B), H
(1) If B satisfies Hom(H
≤ k.
∗ (RP m )) = 0, then ind α = m.
∗ (B), H
(2) If B satisfies Hom(H
Remark. In the above, Hom(·, ·) consists of all homomorphisms (of degree zero) as
graded algebra over the Steenrod algebra mod 2.
Proof. Statement (2) is the case that k = m in (1). We prove (1). Assume ind α > k.
Then there is a Z/2-map f : S k −→ S(α) and it induces f˜ : RP k −→ P (α).
As in the proof of the previous proposition, we have f˜∗ (ek ) = t k = 0 ∈ H k (RP k ).
Let f¯ : RP k −→ B be the composition of f˜ with the projection p : P (α) −→ B.
∗ (B), H
∗ (RP k )) = 0. We have
Then f¯∗ is the zero homomorphism since Hom(H
m−1
m−1 ∗
∗
m
∗
i
i
f˜ (e ) = f˜ ( i=0 wm−i e ) = i=0 f¯ (wm−i )t = 0 and so t m = f˜∗ (em ) = 0.
2
This contradicts t k = 0 since m ≤ k.
We apply the above proposition to the case that the base space B is the projective
space RP n . Let ξn be the canonical line bundle over RP n . We obtain the following.
C OROLLARY 2.5. Let α be a real vector bundle over RP n .
(1) If dim α > n, then ind α = dim α.
(2) If dim α ≤ n and α contains ξn as a subbundle, then ind α = n + 1.
Proof. Let m = dim α.
∗ (RP m )) = 0. Hence ind α = dim α by (2)
∗ (RP n ), H
(1) Since m > n, Hom(H
of the above proposition.
(2) We first show ind α ≤ n + 1. We take k = n + 1 and apply (1) of the
∗ (RP k )) = 0, we
∗ (RP n ), H
above proposition. Since dim α ≤ n < k and Hom(H
get dim α ≤ ind α ≤ k = n + 1. Next we show ind α ≥ n + 1. The inclusion
S(ξn ) → S(α) is a Z/2-map and S(ξn ) is identified with S n . Hence ind S(α) ≥ n;
that is, ind α ≥ n + 1.
2
Remarks. (1) The tangent bundle τ (RP n ) is a typical example which does not
satisfy either of the assumptions in this corollary. By the Conner–Floyd theorem,
ind τ (RP n ) = dim τ (RP n )(= n). From (2), it follows that τ (RP n ) cannot contain
the canonical line bundle ξn , whereas τ (RP n ) ⊕ 1 ∼
= (n + 1)ξn does contain it.
On the index and co-index of sphere bundles
375
(2) Analogous results hold for complex and quaternion projective spaces.
Let F denote C or H, d = dimR F and let ηF be the canonical F line bundle over
FPn . For a real vector bundle α over FPn , the equality ind α = dim α holds if
dim α > d(n + 1) − 1. If dim α ≤ d(n + 1) − 1 and α contains ηF as a subbundle,
then ind α = d(n + 1). Here ηF is regarded as a real bundle.
For mξn , the m-fold sum of ξn , we have the following which is immediate from
Corollary 2.5.
C OROLLARY 2.6. ind mξn = max{m, n + 1}.
As remarked above, we also have ind mηF = max{dm, d(n + 1)} for F = C
or H.
3. Co-index and geometric dimension
In this section, we discuss the properties of co-index. The co-index of α, where α is
an m-dimensional real vector bundle over a CW complex B, is
co-ind α := min{k | ∃f : S(α) −→ S k−1 , Z/2-map}.
Throughout this section, we assume that the base space B is a finite complex.
The following is an analogue to Lemma 2.1.
L EMMA 3.1.
(1) ind α ≤ co-ind α ≤ dim α + dim B
(2) co-ind α ≤ co-ind(α ⊕ 1) ≤ co-ind α + 1
Proof. (1) If X is a Z/2-space and X/Z/2 is paracompact, it is known that co-ind X ≤
cat X/Z/2 [CF1], where cat is the reduced Lusternik–Schnirelmann category. Also,
it is well known that cat Y ≤ dim Y for a finite-dimensional complex Y . Thus
co-ind α = co-ind S(α) + 1 ≤ cat P (α) + 1 ≤ dim P (α) + 1 = dim α + dim B.
(2) The inclusion α → α ⊕ 1 implies the first inequality. Let f : S(α) → S k−1
be a Z/2-map. Then the fibrewise suspension f of f is a Z/2-map S(α ⊕ 1) → S k .
This implies co-ind(α ⊕ 1) ≤ co-ind α + 1.
2
Remark. Both the equalities in (2) are possible. The first equality, co-ind α =
co-ind(α ⊕ 1), holds for example when α ⊕ 1 is trivial but α is not (see the remarks
after Proposition 3.2 and Theorem 4.2).
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R. Tanaka
When B is a manifold, denoted M, and α is τ (M), the tangent bundle of M, an
immersion M ⊆ Rk yields a Z/2-map S(τ (M)) → S k−1 , so that co-ind τ (M) ≤ k.
This suggests a relation of the co-index with the geometric dimension in a general
setting. For a virtual vector bundle β, we write g-dim β for its geometric dimension.
P ROPOSITION 3.2. Let α be a real vector bundle over B with dim α = m and suppose
that dim B ≤ m.
(1) If α is stably trivial, then m ≤ co-ind α ≤ m + 1.
(2) If α is not stably trivial, then m ≤ co-ind α ≤ m + g-dim(−α).
Remark. When B = S m and α = τ (S m ), α is stably trivial and co-ind α = m
(m = 1, 3, 7) or m + 1 (m = 1, 3, 7) [CF2, Theorem 3.1]. Thus both the equalities
are possible in (1).
Proof. (1) Let α ∼ m (stably equivalent). Since dim B ≤ m, we have α ⊕ 1 ∼
=m⊕1
(equivalent) by the stability theorem. The composite of the inclusion map α → α ⊕ 1
with this isomorphism yields a Z/2-map S(α) → S m . Hence co-ind α ≤ m + 1.
(2) Since α is not stably trivial, g-dim(−α) > 0. Put k = g-dim(−α).
By the definition of geometric dimension, there is a k-dimensional vector bundle β
such that −α + m + k ∼ β. Then α + β ∼ m + k. By the stability theorem, we have
α⊕β ∼
= m ⊕ k since m + k ≥ dim B + 1. The composite of the inclusion α → α ⊕ β
with this isomorphism yields a Z/2-map S(α) → S m+k−1 . Hence co-ind α ≤ m+k. 2
P ROPOSITION 3.3. Let α be a real vector bundle over B with dim α = m, and
suppose that dim B ≤ m and α is not stably trivial. If co-ind α ≥ m + 12 (dim B + 1),
then co-ind α = m + g-dim(−α).
Proof. Put co-ind α = k. By the definition of co-index, there is a Z/2-map f :
S(α) → S k−1 . Since dim B ≤ 2(k − m) − 1, f is fibrewise homotopic to a restriction
of some fibrewise monomorphism g : α → Rk [HH, Theorem 1.2]. Consider g as the
bundle map g : α → B×Rk and let β be the co-kernel of g. Then α⊕β ∼
= k. Hence we
have g-dim(−α) ≤ dim β = k − m, and so, k ≥ m + g-dim(−α). The opposite
inequality follows from Proposition 3.2(2).
2
Now we consider the case B = RP n . In the rest of this section, we investigate
the co-index of certain vector bundles over RP n .
First we take interest in when the equality co-ind α = dim α holds. In general, if
α is trivial, then co-ind α = dim α. As stated in the remark following Proposition 3.2,
the converse is true for α = τ (S n ) [CF2, Theorem 3.1]. This was shown by making
use of the Milnor–Spanier construction to a Z/2-map τ (S n ) → S n−1 to get a map
On the index and co-index of sphere bundles
377
S n × S n → S n and then reducing the problem by the Hopf construction to the Hopf
invariant one problem. It seems to be a difficult problem whether, in general, the
equality co-ind α = dim α holds only if α is trivial. Regarding this problem, we have
the following theorem. Let ϕ(n) be the number of integers k such that 0 < k ≤ n and
k ≡ 0, 1, 2, 4 mod 8.
T HEOREM 3.4. Let α be a real vector bundle over RP n with dim α = m.
(1) If co-ind α = m, then α is stably trivial.
(2) If co-ind α = m and m > n, then α is trivial.
(3) If α is of the form k1 ξn ⊕k2 with k1 , k2 non-negative integers, then co-ind α = m
if and only if α is trivial; that is, k1 ≡ 0 mod 2ϕ(n) .
Proof. We first prove (1). Let co-ind α = m, then there is a Z/2-map f : S(α) →
S m−1 . By a theorem of Borsuk, f is of degree odd on each fibre. Let k denote
the degree. By the mod k Dold Theorem by Adams [A, Theorem 1.1], we have
n ) and
J (k t α) = 0 in J (RP n ) for some integer t. Since J (RP n ) = KO(RP
n
n
) has no odd torsion, we have [α] = 0 in KO(RP
). Thus α is stably trivial.
KO(RP
The assertion (2) follows from (1) by the stability theorem.
We proceed to (3). Let α = k1 ξn ⊕ k2 and suppose co-ind α = m. By (1),
n ). Recall that
α must be stably trivial, so we have [k1 ξn − k1 ] = 0 in KO(RP
n
ϕ(n)
) is generated by [ξn −1] and isomorphic to Z/2 . Therefore 2ϕ(n) divides
KO(RP
k1 . Since 2ϕ(n) ≥ n+1, it follows that k1 = 0 or k1 ≥ n+1. In the case k1 ≥ n+1, we
have dim α ≥ n + 1 and this implies that α is trivial by the stability theorem. Note that
we have seen α is trivial if and only if k1 ≡ 0 mod 2ϕ(n) . So the converse is obvious. 2
C OROLLARY 3.5.
(1) co-ind(n + 1)ξn = n + 1 if and only if n = 1, 3, 7.
(2) co-ind τ (RP n ) = n if and only if n = 1, 3, 7.
Proof. (1) By the above theorem, co-ind(n + 1)ξn = n + 1 if and only if n + 1 ≡ 0
mod 2ϕ(n) . Since 2ϕ(n) ≥ n + 1, it is equivalent to that 2ϕ(n) = n + 1 and therefore to
that n = 1, 3, 7.
By Lemma 3.1, dim(τ (RP n ) ⊕ 1) ≤
(2) Let co-ind τ (RP n ) = n.
n
n
co-ind(τ (RP ) ⊕ 1) ≤ co-ind τ (RP ) + 1 = n + 1 and so we have co-ind(τ (RP n )
⊕ 1) = n + 1. Since τ (RP n ) ⊕ 1 ∼
= (n + 1)ξn , we have co-ind(n + 1)ξn = n + 1 and
from (1) we get n = 1, 3, 7. The converse is obvious.
2
Next we are concerned with the relation of the co-index and axial maps. A map
RP n ×RP n → RP m is called axial if both of the restrictions to RP n ×∗ and ∗×RP n
are (homotopic to) the inclusion RP n → RP m .
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R. Tanaka
T HEOREM 3.6. For n ≥ 1, co-ind(n + 1)ξn is equal to the smallest integer m such
that there exists an axial map RP n × RP n → RP m−1 .
Proof. Let g : RP n × RP n → RP m−1 be an axial map. Then g can be lifted to a map
g̃ : S n × S n → S m−1 . Since g is axial, it is easy to see by the unique-lifting property
that g̃ is a bi-Z/2-map in the sense that g̃(−x, y) = −g̃(x, y) = g̃(x, −y). We define
ḡ : S n ×Z/2 S n → S m−1 by ḡ([x, y]) = g̃(x, y). Since g̃ is a bi-Z/2-map, ḡ is welldefined and is a Z/2-map. As a Z/2-space, we can identify (n + 1)ξn canonically with
S n ×Z/2 S n . Thus we obtain a Z/2-map S((n + 1)ξn ) → S m−1 . Therefore, if m(n)
denotes the least integer m such that there exists an axial map RP n ×RP n → RP m−1 ,
we have shown that co-ind(n + 1)ξn ≤ m(n).
Conversely, let f : S((n + 1)ξn ) → S m−1 be a Z/2-map. Then f can be
regarded as a Z/2-map S n ×Z/2 S n → S m−1 as above. Let π denote the projection
S n × S n → S n ×Z/2 S n . The composite f ◦ π : S n × S n → S m−1 is a bi-Z/2-map.
Then f ◦ π clearly induces a map g : RP n × RP n → RP m−1 . The restriction
g|RP n ×∗ : RP n × ∗ → RP m−1 is covered by the Z/2-map (f ◦ π)|S n × ∗ :
S n × ∗ → S m−1 . Hence, considering the Euler class, (g|RP n ×∗ )∗ is non-trivial
on the cohomology, which means the map g|RP n ×∗ is essential. If n < m − 1,
it follows that g|RP n ×∗ is homotopic to the inclusion RP n → RP m−1 because
[RP n , RP m−1 ] ∼
= Z/2. Similarly, so is g|∗×RP n . Thus g is axial if n < m − 1.
If n = m − 1, it follows that co-ind(n + 1)ξn = n + 1 and so we have n = 1, 3, 7 by
Corollary 3.5. For n = 1, 3, 7, we obviously have axial maps RP n × RP n → RP n .
2
Therefore we have obtained co-ind(n + 1)ξn ≥ m(n).
T HEOREM 3.7.
(1) For α = τ (RP n ), we have co-ind(α ⊕ 1) = co-ind α + 1.
(2) For α = τ (RP n ) and for α = τ (RP n ) ⊕ 1, we have co-ind α = dim α +
g-dim(−α).
Proof. In view of Corollary 3.5, the statements clearly hold for n = 1, 3, 7 because
g-dim(−α) = 0 in these cases. So let n = 1, 3, 7 and let α = τ (RP n ). Let l(n) denote
the least integer m such that RP n can be immersed in Rm . By the Hirsch theorem on
immersions, l(n) = n + g-dim(−α) [H]. Then we have co-ind α ≤ l(n). Let l (n)
denote the least integer m such that there is an axial map RP n × RP n → RP m .
Since α ⊕ 1 ∼
= (n + 1)ξn , we have co-ind(α ⊕ 1) = l (n) + 1 by Theorem 3.6. Thus,
by Lemma 3.1, we have l (n)+1 = co-ind(α⊕1) ≤ co-ind α+1 ≤ l(n)+1. In [AGJ],
it was shown that the number l (n) coincides with l(n). Therefore, the statement (1)
follows. Also, (2) follows because g-dim(−(α ⊕ 1)) = g-dim(−α).
2
On the index and co-index of sphere bundles
379
4. Some examples
In this section, we compute the index and the co-index of certain bundles over
projective spaces. Let F be R, C or H and let d = dimR F (which is equal to 1, 2
or 4 accordingly). Let ηF denote the canonical line bundle over FPn . We regard ηF as
a real bundle. Note that dim ηF = d.
T HEOREM 4.1. Let m be a non-negative integer.
(1) ind(ηF ⊕ m) = max{d(n + 1), m + d}.
(2) co-ind(ηF ⊕ m) = d(n + 1) + m.
∗ (FPn ), H
∗ (RP d(n+1) )) = 0 and
Proof. (1) Case (i): m < dn. Since Hom(H
dim(ηF ⊕ m) ≤ d(n + 1), we have ind(ηF ⊕ m) ≤ d(n + 1) by Proposition 2.4(1).
On the other hand, since S(ηF ) can be identified with S d(n+1)−1 as a Z/2-space, we
have ind(ηF ⊕ m) ≥ ind ηF = d(n + 1). Hence, we obtain ind(ηF ⊕ m) = d(n + 1).
∗ (FPn ), H
∗ (RP m+d )) = 0 and dim(ηF ⊕ m)
Case (ii): m ≥ dn. Since Hom(H
= m + d, we have ind(ηF ⊕ m) = dim(ηF ⊕ m) = m + d by Proposition 2.4(2).
(2) Since S(ηF ) = S d(n+1)−1 , we have co-ind(ηF ⊕ m) ≤ co-ind ηF + m =
d(n+1)+m. Next we show co-ind(ηF ⊕m) ≥ d(n+1)+m. Put d(n+1)+m = k and
suppose there is a Z/2-map f : S(ηF ⊕ m) → S k−2 . Let f˜ : P (ηF ⊕ m) → RP k−2
be the induced map. By the naturality of the Euler class we have f˜∗ (t) = e,
so that ek−1 = f˜∗ (t k−1 ) = f˜∗ (0) = 0 in H k−1 (P (ηF ⊕ m)). Here, as in
Section 2, e is the Euler class of the line bundle associated to S(ηF ⊕ m) →
P (ηF ⊕ m) and t is the generator of H ∗ (RP k−2 ). Recall that H ∗ (P (ηF ⊕ m)) ∼
=
H ∗ (FPn ){1, e, e2 , . . . , em+d−1 }. Let b denote the generator of H ∗ (FPn ). Since
W (ηF ⊕ m) = W (ηF ) = 1 + b, we have em+d = bem . Therefore, we have
em+dn = bn em and so em+dn+d−1 = bn em+d−1 . Thus we have ek−1 = bn em+d−1
and ek−1 = 0 in H k−1 (P (ηF ⊕ m)). This contradicts ek−1 = 0.
2
Remarks. (1) In the case F = R and n = 1, S(ηF ⊕ m) is known as the generalized
Klein bottle of dimension m + 1 which we denote K m+1 . Since (co-)ind S(α) =
(co-) ind α − 1, we have obtained that ind K m+1 = m and co-ind K m+1 = m + 1
(m ≥ 1).
(2) For α = ηF ⊕ m, we have obtained that if m < dn then ind α > dim α and
ind(α ⊕ 1) = ind α, while if m ≥ dn then ind α = dim α and ind(α ⊕ 1) = ind α + 1.
Thus for F = C, H and for m with dn − d < m < dn, neither ind α = dim α nor
ind(α ⊕ 1) ≥ ind α + 1 holds, although α in this case is in the stable range as a real
vector bundle over FPn .
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R. Tanaka
(3) For α = ηF ⊕ m, co-ind(α ⊕ 1) = co-ind α + 1 holds for any m.
In addition, co-ind α = dim α + g-dim(−α) holds for any m. In fact g-dim(−α) =
dn as shown below. Since W (−α) = (1 + b)−1 , we have wdn (−α) = bn = 0.
Hence g-dim(−α) ≥ dn. On the other hand, g-dim(−α) ≤ dim FPn = dn.
Thus g-dim(−α) = dn.
For the next example, let ηn denote the canonical line bundle over CP n and
consider its k-fold tensor product (as complex vector bundles) ηnk . We regard ηnk as
a real vector bundle.
T HEOREM 4.2.
(1) For k odd, we have ind ηnk = co-ind ηnk = 2n + 2.
(2) For k even, we have ind ηnk = 2, and co-ind η1 k = 3.
Remark. In the case where k is even, it seems to be difficult to determine co-ind ηnk in
general. Note that co-ind ηn 2 = s(RP 2n+1 ) as mentioned in Section 1.
Proof. Since ind ηnk ≤ co-ind ηnk , it suffices to prove the following.
(i) For k odd, ind ηnk ≥ 2n + 2.
(ii) For k even, ind ηnk = 2.
(iii) co-ind ηnk ≤ 2n + 2.
(iv) For k even, co-ind η1 k = 3.
Let f : S(ηn ) → S(ηnk ) be the k-fold tensor product map, f (z) = zk . If k is
odd, f is a Z/2-map. Since S(ηn ) is identified with S 2n+1 as a Z/2-space, we have
assertion (i).
To show (ii), we look at the Whitney class. We have w2 (ηnk ) = c1 (ηnk ) mod 2 =
kc1 (ηn ) mod 2 and this is zero if k is even. So if k is even, we have W (ηnk ) = 1 and
by Proposition 2.2 we get ind ηnk = dim ηnk = 2. Thus assertion (ii) follows.
By Lemma 3.1, we have co-ind ηnk ≤ dim ηnk + dim CP n = 2 + 2n, which shows
(iii).
Finally, we prove (iv). We first note that, if n = 1, then CP n = S 2
and recall that m-dimensional sphere bundles over S i are classified by elements of
πi−1 (O(m + 1)). The sphere bundle S(η1 k ) corresponds to the characteristic element
k in π1 (O(2)) = Z. Let i denote the natural inclusion O(2) → O(3) and consider
the homomorphism i∗ : π1 (O(2)) → π1 (O(3)). Since π1 (O(3)) = Z/2, we have
i∗ (k) = 0 for k even. It follows that S(η1 k ⊕ 1) is trivial. Hence co-ind η1 k ≤
co-ind(η1 k ⊕1) = 3. On the other hand, assume that there is a Z/2-map S(η1 k ) → S 1 .
Let i : S 1 → S(η1 k ) be the inclusion of the fibre. Then deg f ◦ i is odd by the Borsuk
theorem. Since S(η1 k ) = L3 (k), we have H 1 (S(η1 k ); Z) = 0 and this contradicts
On the index and co-index of sphere bundles
381
that deg f ◦ i is odd. Thus we have shown co-ind η1 k ≥ 3 and so the assertion (iv)
follows.
2
Remark. If k is odd, g-dim(−ηnk ) = 2n because w2n (−ηnk ) = 0. So co-ind α =
dim α + g-dim(−α) holds for α = ηnk with k odd. In the case where k is even and
n = 1, η1 k is stably trivial and co-ind η1 k = co-ind(η1 k ⊕1) holds (cf. Proposition 3.2).
T HEOREM 4.3. Let α = ηnk ⊕ m with m ≥ 1.
(1) For k odd, we have ind α = max{2n + 2, m + 2} and co-ind α = 2(n + 1) + m.
(2) For k even, we have ind α = m + 2.
Proof. We proceed in a way similar to the proof of Theorem 4.1.
(1) First we look at the index.
Case (i). m < 2n. Since ind α ≥ ind ηnk and ind ηnk = 2n + 2 for k odd
(Theorem 4.2), we have ind α ≥ 2n + 2. The opposite inequality follows, just as
∗ (RP 2n+2 )) = 0
∗ (CP n ), H
in the proof of Theorem 4.1, from the fact that Hom(H
and dim α ≤ 2n + 2. Therefore we have ind α = 2n + 2 if m < 2n.
Case (ii). m ≥ 2n. In this case, we have ind α = m + 2 because
∗ (RP m+2 )) = 0 and dim α = m + 2.
∗ (CP n ), H
Hom(H
Next, we look at the co-index. Since co-ind α ≤ co-ind ηnk + m and co-ind ηnk =
2(n + 1) for k odd (Theorem 4.2), we have co-ind α ≤ 2(n + 1) + m. The opposite
inequality follows, just as in the proof of Theorem 4.1, because W (α) = W (ηnk ) =
1 + b for k odd.
(2) Since W (α) = W (ηnk ) = 1 for k even, the assertion follows from
Proposition 2.2.
2
Remark. The equality co-ind α = dim α + g-dim(−α) holds for α = ηnk ⊕ m with
k odd. In the case where k is even and n = 1, η1 k ⊕ m (m ≥ 1) is trivial and
co-ind(η1 k ⊕ m) = m + 2.
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Ryuichi Tanaka
Department of Mathematics
Faculty of Science and Technology
Tokyo University of Science
Noda
Chiba 278-8510
Japan
(E-mail address: tanaka [email protected])