Download LA ACCION DEL ALGEBRA DE STEENROD SOBRE LAS

LA ACCION DEL ALGEBRA DE STEENROD SOBRE LAS
VARIEDADES DE STIEFEL PROYECTIVAS
BY ENRIQUE ANTONIANO
1. Introduction
Vn,s is the variety of Stiefel orthonormal s-frames in Rn . The projective Stiefel variety
Xn,s is obtained by identifying at Vn,s a s-frame (v1 , . . . , vs ) with negative (−v1 , . . . , −vs ).
Baum and Browder [2] studied the structure of Hopf algebra cohomology modulo two
of the groups P O(2n) = X2n,2n−1 and determined the action of the Steenrod algebra on
it. On the other hand, Gitler and Handel [5] calculated the action of the Steenrod algebra
over H ∗ (Xn,s , Z2 ) leaving a small indeterminacy.
This paper completely determines the action of the Steenrod algebra on the Stiefel
projective varieties (see Theorem 2.1).
These results are applied to find obstructions to the existence of sections of multiples
of the Hopf bundle over real projective spaces.
The content of this article is part of the thesis [1] written under the direction of Samuel
Gitler, whom I express my gratitude. I also want to thank Pedro Armendariz and Carlos
Rodriguez for their help.
2. The action of the Steenrod algebra
Henceforth cohomology groups will
be with coefficients in Z2 . Let s < n,
n
N = min{j | n − s + 1 ≤ j and j ≡ 1(2)}. Let V (yn−s , . . . , ŷN −1 , . . . , yn−1 ) the vector
n−1
Y
space generated by monomials
ykǫ with ǫk = 0, 1 and k 6= N − 1.
k=n−s
We remind you [5] of
H ∗ (Xn,s ) = Z2 [y]/y N ⊗ V (yn−s , . . . , ŷN −1 , . . . , yn−1 )
with y = p∗ (x), where p : Xn,s −→ P ∞ classifies the double cover π : Vn,s −→ Xn,s and
x ∈ H 1 (P ∞ ) is the generator.
let t = ν2 (N )= exponent of the maximum power of 2 dividing N . We define
n
Ak = A(q, i, k) = q−1−k
q−1−i k
q−N n
Bk,j = B(q, i, k, j) = nq NN−1−k
k
i−j
−1−i
(
q+2t−1 −N n
if t ≥ 3
i−1
q+2t−1 −N
ǫ=
0
if t < 3
We have then
1
Theorem 2.1. The action of the Steenrod algebra in H ∗ (Xn,s ) is determined by the
formula
i
Sq yq−1 =
i
X
X
Ak y k yq+i−1−k +
Bk,j y q+k+i−N −j yN +j−k−1 + ǫy q+i−1
0≤k<j≤i
k=0
Gitler and Handel [5, Theorem 2.8] proved Theorem 2.1 leaving indeterminate the value
of ǫ.
We will prove the Theorem 2.1 only for n even, where the result follows easily overall,
taking in an context [5, Theorem 3.3]. For this study the cohomology homomorphism
induced by the natural action ψ : P O(2n) × X2n,s −→ X2n,s which makes it necessary to
see the selection of generators for H ∗ (P O(2n)) made by Baum and Browder [2] coincides
with that of Gitler and Handel [5]. This has been tested by Gitler [4, Proposition 1.2]
when n is a power of two.
Often make reference to the transformations ρ : Xn,s −→ Xn+m,s induced by the inclusion
of an s-frame under Rn in Rn+m and f : Xn,s −→ Xn,s−k induced forgetting of k vectors of
s-frame in Rn . generators of H ∗ (P O(2n)), selected by Baum and Browder, are denoted
with uk , which are characterized by the formula
k−1 X
k
∗
uk−j ⊗ y j
[2, Theorem 8.7]
ψ uk = 1 ⊗ uk + u k ⊗ 1 +
j
j=1
matching the selections of generators is of the following three lemmas
Lemma 2.2. If
2n
k+1
≡ 0(2) then in H ∗ (P O(2n)), yk = uk + ǫy k with ǫ = 0, 1.
Proof. Consider the following commutative diagram
g
H ∗ (P O(2n))
ψ∗
O
/ H ∗ (P O(2n)) ⊗ H ∗ (P O(2n))
O
f∗
H ∗ (X2n,2n−k )
1⊗π ∗
1⊗f ∗
ψ∗
/ H ∗ (P O(2n)) ⊗ H ∗ (X2n,2n−k )
)
/ H ∗ (P O(2n)) ⊗ H ∗ (SO(2n))
O
1⊗h∗
1⊗π ∗
/ H ∗ (P O(2n)) ⊗ H ∗ (V2n,2n−k )
where h is induced by the vector miss a framework in R2n . It is not difficult to verify that
g(f ∗ yk + uk ) = 0 but kerg = T = {y, y 2 , . . . , y N −1 } [2, Theorem 8.2] and f ∗ (yk ) = yk
[5, lemma2.6], thus ending the demonstration (proof).
Lemma 2.3. Let 2T −1 < 2n < 2T ,
yk = uk + ǫy k with ǫ = 0, 1.
2n
k+1
≡ 1(2) and 2T −2n+1 ≤ k, then in H ∗ (P O(2n)),
Proof. Consider the following commutative diagram
ψ′
H ∗ (P O(2n))
ψ∗
O
/ H ∗ (P O(2n)) ⊗ H ∗ (P O(2n))
π ∗ ⊗1
)
/ H ∗ (SO(2n)) ⊗ H ∗ (P O(2n))
O
ρ∗
H ∗ (X2T ,2n−1 )
i∗ ⊗ρ∗
ψ∗
/ H ∗ (P O(2T )) ⊗ H ∗ (X T
2 ,2n−1 )
π ∗ ⊗1
/ H ∗ (SO(2T )) ⊗ H ∗ (X T
2 ,2n−1 )
5
ψ′
where i is the usual inclusion. Then ψ ′ (ρ∗ yk + uk ) = 1 ⊗ (ρ∗ yk + uk ), this is possible if
ρ∗ yk = uk but ρ∗ yk = yk [5, Theorem 3.3], thus ending the demonstration.
Lemma 2.4. Let 2T −1 < 2n < 2T ,
with ǫ = 0, 1 in H ∗ (P O(2n)).
2n
k+1
≡ 1(2) and 2T − 2n + 1 > k, then yk = uk + ǫy k
Proof. Suppose true the assertion of the lemma in dimensions greater than k and keep
uk + yk = y s a, a not divisible by y, then
0 = Sq k−s (uk + yk ) = y s a2 + terms that do not cancel the previous.
So a2 = 0 but 2(k − s) < 2k ≤ 2n − 1 so a = 0.
Let 2T −1
3. P O(2n) action on X2n,s
2n
≡ 0(2), then
< 2n ≤ 2T and k+1
Theorem 3.1. For a selection of yk ∈ H ∗ (X2n,2n−k ) modulo y k ,
k−1 X
k + 2T − 2n
∗
uk−j ⊗ y j
ψ (yk ) = 1 ⊗ yk + uk ⊗ 1 +
j
j=1
The theorem is true if k < 2ν2 (2n) and therefore when n is a power of two,
since in that case
f ∗ : H ∗ (X2n,2n−k ) −→ H ∗ (P O(2n))
is a monomorphism. Note that ψ ∗ has the following generic way
k−1
N
−1
−1
hk−i−1
i N
X
X
X
X
k−i
i
i
k−i−j
∗
ψ (yk ) = 1 ⊗ yk + uk ⊗ 1 +
ai ⊗ y +
(y ⊗ 1)
bj ⊗ y
+
αi y i ⊗ y k−i ,
i=1
i=1
j=1
i=1
where ai , bij are sums of monomials in us with dimension of index lower and αi = 0, 1.
The Theorem 3.1 is a consequence of the following three lemmas:
Lemma 3.2. With the notation above bij = 0 for all i, j.
Proof. Let i0 be the maximum value of i such that bij0 6= 0 for some j and let j0 be the
maximum of such values of j. Of the equation (ψ ∗ ⊗ 1)ψ ∗ yk = (1 ⊗ ψ ∗ )ψ ∗ yk withhold
terms containing the projective class ”y” to the maximum power in the first inning and
not of the form y i ⊗ y j ⊗ y k−i−j ; have
!
!
k−i
k−i
0 −1
0 −1
X
X
(y i0 ⊗ 1 ⊗ 1)
ψ(bij0 ) ⊗ y k−i0 −j = (y i0 ⊗ 1 ⊗ 1)
bij0 ⊗ ψ(y k−i0 −j )
j=1
j=1
but the sum of y i0 is not canceled any other such expression, so bij00 = 0.
Lemma 3.3. For a selection of yk (module y k ), αi = 0 for all i.
∗
k
Proof. As ψ (y ) =
N
−1
X
k i
y ⊗ y k−i , suffices to prove that
i
k
a)αi =0 if
=0
i
k
k
b)αi =αj if
=
=1
i
j
i=1
Gives the equality (ψ ∗ ⊗ 1)ψ ∗ yk = (1 ⊗ ψ ∗ )ψ ∗ yk retain the terms containing only the
projective class. From the resulting equation can be derived the following system of
linear equations which is the motto
k−i
i+j
αi
+ αi+j
=0
∀ i, j ∈ [1, N − 1].
j
j
Lemma 3.4. With the previous notation ai =
k+2T −2n
k−i
ui
Proof. Choose yk ∈ H ∗ (X2n,2n−k ) to be worth the Lemma.Simplify the equation
(ψ ∗ ⊗ 1)ψ ∗ yk = (1 ⊗ ψ ∗ )ψ ∗ yk to obtain
k−1
X
∗
ψ (ai ) ⊗ y
i=1
∗
k−i
=
k−1
X
∗
ai ⊗ ψ (y k−i )
i=1
where it is clear that ψ (ai ) ∈ H (P O(2n)) ⊗ T, T = {y, . . . , y N −1 }, then ai = ǫi ui with
ǫi = 0, 1 [2, section 2] besides it
i+j
k−i
≡ 0(2), i ∈ [1, k − 1], j ∈ [1, k − i − 1] ∩ [1, N − 1].
+ ǫi+j
(1)
ǫi
i
j
now defined enough to study ψ ′
∗
H ∗ (X2n,2n−k )
ψ′
/ H ∗ (SO(2n)) ⊗ H ∗ (X2n,2n−k )
UUUU
O
UUUU
UUUU
π ∗ ⊗1
UUUU
ψ∗
U*
H ∗ (P O(2n)) ⊗ H ∗ (X2n,2n−k )
First demonstrated the motto for even values of k. Assuming a slogan on H ∗ (X2n+2,2n−k )
and in dimensions greater than k, simply choose yk so that under
ρ∗ : H ∗ (X2n+2,2n−k ) −→ H ∗ (X2n,2n−k ), ρ∗ (yk+2 ) = yk+2 + y 2 yk
[5, Theorem 3.3] and check what the result set by analyzing the equation says
ψ ′ ρ∗ yk+2 = (1 ⊗ ρ∗ )ψ ′ yk+2 .
for odd values of k, choose yk ∈ H ∗ (H2n,2n−k ) so that Sq 1 yk = yk+1 [5, Theorem 2.8].
Of the equation ψ ′ Sq 1 yk = Sq 1 ψ ′ yk and (1) is not difficult to end the demonstration. 4. Demonstration of Theorem 2.1
Note first that Theorem 2.1 true when n is a power of two as in this case f ∗ is a
monomorphism. Hereinafter,
2T −1 < 2n < 2T and R = ν2 (2n).
Theorem 2.1 is an immediate
consequence of the following two lemmas. 2n
≡ 1(2)}.
Let N such that N ≡ 1(2), t = ν2 (N ) and M = max{j | j < N and 2n
j
Lemma 4.1. If N − M = 2R , the conclusion of Theorem 2.1 is valid in H ∗ (X2n,2n−M ).
Proof. Note that f ∗ : H ∗ (X2n,2n−(M +S) ) −→ H ∗ (X2n,2n−M ) is a monomorphism if
S ∈ [0, 2R − 2]. It is this range, take yM +S ∈ H ∗ (X2n,2n−(M +S) ) as in Theorem 3.1 and
yM +S = f ∗ (yM +S ) ∈ H ∗ (X2n,2n−M )
From equation Sq i ψ ∗ yq−1 = ψ ∗ Sq i yq−1 not difficult to verify the assertion of Lemma. Lemma 4.2. If N − M > 2R ,the conclusion of Theorem 2.1 is valid in H ∗ (X2n,2n−M ).
Proof. Let 2m = 2n + N − M − 2t . Choose generators of H ∗ (X2m,2n−M ) as in lemma 4.1
and choose the generators of H ∗ (X2n,2n−M ) such that if M < q < N under
q
X
2n − 2m q−l
∗
∗
∗
∗
y yl−1 [5, Theorem 3.3]
ρ : H (X2n,2n−M ) −→ H (X2m,2n−M ), ρ yq−1 =
q
−
l
l=q−2n+2m
It is not difficult to verify the assertion of the slogan of the equation
ρ∗ Sq i yq−1 = Sq i ρ∗ Yq−1 ((I think it should be ’y’ not ’Y’))
5. Sections on projective spaces
Let ξk the Hopf bundle of lines on the real projective space P k , nξk has s linearly
independent sections if and only the fibration Vn,s −→ Xn,s −→ P ∞
has a section on the k- skeleton of the base, P k . That is, if there is a transformation q
by the following commutative diagram
Xn,s
z=
zz
z
p
zz
z z
/ P∞
Pk
q
Aware of the action of the Steenrod algebra on Xn,s (Theorem 2.1) measures some obstructions to the existence of such q. The results obtained are as follows.
Let s(n, k) be the maximum number of linearly independent sections which admits nξk .
The geometry of codimension((Geometric codimension)) n xik defined by cg(n, k) =
s(n, k) − n + k. Let n = 8l and k = 8m, then
Theorem 5.1. If ml ≡ 2(4) and m > 1, cg(n + p, k + q) ≤ j, j depending only on p and
q according to the following table
p 0 1 2 3 4 5 6
q
1
2
3
4
5
6
7
3
3
4
5
6
7
8
4
5
6
4
5
6
7
3
4
5
3
4
5
6
3
4
5
5
3
4
5
4
5
6
5
5
6
4
3
4
5
4
4
5
3
3
3
4
3
3
4
3
Proof. See the proof of Theorem 5.2, followed
Theorem 5.2. If
l
m−1
≡ 2(4) then cg(8l + 6, 8m) < 4.
Proof. According to Theorem 2.1 in H ∗ (X8l+6,8(l−m)+10 )
Sq 4 y8m−4 =y8m + y 2 y8m−2 + y 4 y8m−4
Sq 3 y8m−3 =y 2 y8m−2 + y 8m ((check this term))
Sq 2 y8m−2 =y8m + y 2 y8m−2
which makes impossible the existence of a commutative diagram as above.
Theorem coincides with [6, Theorem 3.1 (B)] K. Lam, but the demonstration presented
here, besides being homogeneous, it is conceptually more elementary.
W. Iberkleid [3] has informed me also have obtained the theorem 5.2, although much
more complicated techniques involving functional operations in MU.
References
[1] E. Antoniano, Sobre las variedades de Stiefel proyectivas, Tesis, Centro de Investigacion del IPN,
Maxico, D. F. (1976)
[2] Baum, Paul F.; Browder, William, The cohomology of quotients of classical groups. Topology 3 1965
305336.
[3] Feder, S.; Iberkleid, W. Secondary operations in KK-theory and the generalized vector field problem.
Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq,
Rio de Janeiro, 1976), pp. 161175. Lecture Notes in Math., Vol. 597, Springer, Berlin, 1977.
[4] Gitler, S., The projective Stiefel manifolds. II. Applications. Topology 7 1968 4753.
[5] Gitler, S.; Handel, D., The projective Stiefel manifolds. I. Topology 7 1968 3946.
[6] Lam, Kee Yuen, Sectioning vector bundles over real projective spaces. Quart. J. Math. Oxford Ser.
(2) 23 (1972), 97106.
[7] Milnor, John W.and Moore, John C. On the structure of Hopf algebras. Ann. of Math. (2) 81 1965
211264.
[8] N. E. Steenrod, Cohomology operations. Lectures by N. E. STeenrod written and revised by D. B.
A. Epstein. Annals of Mathematics Studies, No. 50, Princeton University Press, Princeton.