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Cohomology of loop spaces of Hermitian symmetric spaces of classical types
Topology and its Applications 160 (2013) 280–291
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Topology and its Applications www.elsevier.com/locate/topol
Cohomology of loop spaces of Hermitian symmetric spaces of classical types ✩ Younggi Choi ∗ Department of Mathematics Education, Seoul National University, Seoul 151-748, Republic of Korea
a r t i c l e
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a b s t r a c t
Article history: Received 30 July 2012 Received in revised form 1 November 2012 Accepted 2 November 2012
We study the mod p cohomology of loop spaces and free loop spaces of irreducible compact Hermitian symmetric spaces of classical types. We also prove that the integral cohomology of loop spaces of them is torsion free, or has only 2-torsion of order 2. © 2012 Elsevier B.V. All rights reserved.
MSC: 55R20 55T10 55T20 Keywords: Hermitian symmetric spaces of classical types Loop space Free loop space Totally non-homologous to zero
1. Introduction A Hermitian symmetric space is a connected complex manifold with a hermitian metric such that the geodesic symmetries are hermitian isometries. The irreducible compact Hermitian symmetric spaces of classical type are divided into the following four classes:
AIII BDI CI DIII
G m,n (C) = U (m + n)/ U (m) × U (n)
Q n = SO(n + 2)/ SO(2) × SO(n)
m , n 1, n 3,
Sp(n)/U (n)
n 3,
SO(2n)/U (n)
n 4.
The purpose of this paper is to study the cohomology of (based) loop spaces and free loop spaces of irreducible compact Hermitian symmetric spaces of classical types. In this paper, first we determine the mod p cohomology of loop spaces of them by exploiting the Serre spectral sequence and the Eilenberg–Moore spectral sequence converging to the same target space. In [4], Bott and Samelson asked whether ✩ This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012-0001320). Fax: +82 2 889 1747. E-mail address: [email protected].
*
0166-8641/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.topol.2012.11.003
Y. Choi / Topology and its Applications 160 (2013) 280–291
281
the integral cohomology of loop spaces of symmetric spaces has only two torsion. So in Section 4, we study torsion in the integral cohomology of the loop spaces of irreducible compact Hermitian symmetric spaces M of classical types, and prove that the integral cohomology of loop spaces of them is torsion free, or has only 2-torsion of order 2. Finally we determine when Ω M is totally non-homologous to zero in Λ M with respect to F p for the free loop space fibration Ω M → Λ M → M. Throughout the paper E (x) denotes the exterior algebra on x and Γ (x) denotes the divided power algebra on x. The subscript of an element denotes the degree of the element, that is, deg(xi ) = i. 2. The mod 2 cohomology of loop spaces Theorem 2.1. The mod 2 cohomology of loop spaces of complex Grassmannian spaces are as follows:
H ∗ Ω U (m + n)/ U (m) × U (n) ; F2 = H ∗ Ω U (m + n)/U (m) ; F2 ⊗ H ∗ U (n); F2
= Γ (a2m+2i : 0 i n − 1) ⊗ E (x2i +1 : 0 i n − 1) where m n. Proof. First of all, we compute the Eilenberg–Moore spectral sequence converging to H ∗ (Ω(U (n + n)/U (m)); F2 ) with
E 2 = Tor H ∗ (U (m+n)/U (m);F2 ) (F2 , F2 ) = Tor E (x2m+2i+1 : 0i n−1) (F2 , F2 ) = Γ (a2m+2i : 0 i n − 1). Then the spectral sequence collapses at the E 2 -term because E 2 vanishes in all odd total degrees. Moreover since m n, we have |a22m | = 4m > 2m + 2n − 2 = |a2m+2n−2 |, so that there is no algebra extension. Hence H ∗ (Ω(U (m + n)/U (m)); F2 ) = Γ (a2m+2i : 0 i n − 1). Consider the Serre spectral sequence converging to H ∗ (Ω(U (m + n)/(U (m) × U (n))); F2 ) for the fibration
Ω U (m + n)/U (m) −→ Ω U (m + n)/ U (m) × U (n) −→ U (n)
where H ∗ (U (n); F2 ) = E (x2i +1 : 0 i n − 1). Since this spectral sequence is the spectral sequence of a Hopf algebra, the source of the first non-trivial differential is an indecomposable element and its target is a primitive element. Since the degree of indecomposables in H ∗ (Ω(U (m + n)/U (m)); F2 ) is greater than 2m − 1, the degree of target for the first nontrivial differential is greater than 2m. Since m n, there is no primitive element of degree greater than 2m in H ∗ (U (n); F2 ). Hence the Serre spectral sequence collapses at E 2 . Now we consider the algebra extension problems. First, we consider the case of m 2n. Since |a2m | = 2m > 2(2n − 1) = |x22n−1 |, there is no extension problem. For the case of n m 2n, there may be some possibility of extension, such as, x22 j +1 = a2m+2k for some 0 k n and 0 j n − 1. Consider the following morphism of fibration.
Ω(U (m + n)/U (m)) −−−−→ Ω(U (m + n)/(U (m) × U (n))) −−−−→ U (n) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ Ω j Ω i id Ω(U (2m + n)/U (2m)) −−−−→ Ω(U (2m + n)/(U (2m) × U (n))) −−−−→ U (n) Since 2m 2n, there is no extension problem in H ∗ (Ω(U (2m + n)/(U (2m) × U (n))); F2 ). If there were an extension x22 j +1 = a2m+2k in H ∗ (Ω(U (m + n)/(U (m) × U (n))); F2 ), then it would follow that
a2m+2k = x22 j +1 = (Ω j )∗ (x2 j +1 )(Ω j )∗ (x2 j +1 ) = (Ω j )∗ x22 j +1 = (Ω j )∗ (0) = 0. So if there is such an extension, this leads a contradiction. So we get the conclusion.
2
Theorem 2.2. The mod 2 cohomology of loop space of complex quadric spaces Q n for n 3 is as follows:
H ∗ (Ω Q n ; F2 ) = E (a1 ) ⊗ Γ (an−1 ) ⊗ Γ (an ),
Sq1 an−1 =
an
if n is odd,
0
if n is even.
Proof. It is known [10, Chap. 3] that
H ∗ SO(n + 2)/SO(n); F2 = E (xn ) ⊗ E (xn+1 ),
Sq1 xn =
xn+1
if n is odd,
0
if n is even.
Consider the Serre spectral sequence the following fibration:
Ω S n −→ Ω SO(n + 2)/SO(n) −→ Ω S n+1
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with
E 2 = H ∗ Ω S n+1 ; F2 ⊗ H ∗ Ω S n ; F2 = Γ (cn ) ⊗ Γ (cn−1 ). Since it is the spectral sequence of a Hopf algebra, the only possible non-trivial first differential is
d(cn−1 ) = cn .
(2.1)
Now we consider the Eilenberg–Moore spectral sequence of the path loop fibration converging to H ∗ (Ω(SO(n 2)/SO(n)); F2 ) with
+
E 2 = Tor H ∗ (SO(n+2)/SO(n);F2 ) (F2 , F2 ) = Γ (an , an−1 ). Then the differential (2.1) implies the differential from an of bidegree (−1, n + 1) to an−1 of bidegree (−1, n) in the Eilenberg–Moore spectral sequence, however it is impossible because of bidegree reason. Hence the Serre spectral sequence collapses at the E 2 -term, so does the Eilenberg–Moore spectral sequence. So we obtain
H ∗ Ω SO(n + 2)/SO(n) ; F2 = Γ (an−1 ) ⊗ Γ (an ),
Sq1 an−1 =
an
if n is odd,
0
if n is even.
Consider the Serre spectral sequence converging to H ∗ (Ω Q n ; F2 ) for the fibration
Ω SO(n + 2)/SO(n) −→ Ω Q n −→ S 1 . Then by dimensional reason the Serre spectral sequence collapses at E 2 . So we get the conclusion.
2
Theorem 2.3.
H ∗ Ω Sp(n)/U (n) ; F2 = E (x2i −1 : 1 i n) ⊗ Γ (x4i −2 : 1 i n). Proof. Consider the following morphism of fibrations:
Ω Sp(n) −−−−→ Ω(Sp(n)/U (n)) −−−−→ U (n) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ j i Ω Sp
−−−−→
Ω(Sp/U )
−−−−→
(2.2)
U
We recall the following facts from [10, Chap. 4] and [2]:
H ∗ U (n); F2 = E (x2i −1 : 1 i n), H ∗ (U ; F2 ) = E (x2i −1 : i 1),
H ∗ Ω Sp(n); F2 = Γ (x4i −2 : 1 i n), H ∗ (Ω Sp; F2 ) = F2 [x4i −2 : i 1],
H ∗ Ω(Sp/U ); F2 = H ∗ (U / O ; F2 ) = E (xi : i 1). Consider the Serre spectral sequence for the bottom row fibration in (2.2) converging to H ∗ (Ω Sp/U ; F2 ) with
E 2 = H ∗ (U ; F2 ) ⊗ H ∗ (Ω Sp; F2 ). Since the cohomology of the total space has the same size in every total degree as tensor products of those of the base and the fiber space as a graded vector spaces, the spectral sequence collapses at the E 2 . Since i ∗ and j ∗ are epimorphism, the spectral sequence for the top fibration also collapses at the E 2 by the naturality of differentials. Since H ∗ (U ; F2 ) = E (x2i −1 : i 1), there are no multiplicative extensions in the Serre spectral sequence for the bottom row fibration, such as x22i −1 = x4i −2 , i 1. Hence there are also no multiplicative extensions in the Serre spectral sequence for the top row fibration, such as x22i −1 = x4i −2 for 1 i n. Hence we obtain
H ∗ Ω Sp(n)/U (n) ; F2 = E (x2i −1 : 1 i n) ⊗ Γ (x4i −2 : 1 i n).
2
We recall the following result from (72) in [5], and Theorem 6.11 in Chapter 4 of [10].
Y. Choi / Topology and its Applications 160 (2013) 280–291
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Theorem 2.4.
H ∗ SO(2n)/U (n); F2 = V (c 2i : 1 i n − 1), where V (c 2i : 1 i n − 1) is the commutative associative algebra over F2 satisfying the following conditions: 1. {(c 2 )1 · · · (c 2i )i · · · (c 2n−2 )n−1 : i = 0, 1 for 1 i n − 1} is a basis, 2 2 2. c 2i = c 4i for 4i 2n − 2, and c 2i = 0 if 4i > 2n − 2. 2 Hence c 4i +2 = 0 for n − 1 < 4i + 2 2n − 2. For c 4i +2 with 2 4i + 2 n − 1, let (2n − 1) < (4i + 2)2νi 4n − 4. Then we can rewrite it as follows:
νi
νi be the positive integer such that
2 H ∗ SO(2n)/U (n); F2 = F2 [c 4i +2 : 0 4i n − 3]/ c 4i +2 ⊗ E (c 4i +2 : n − 3 < 4i 2n − 4).
(2.3)
H ∗ (Ω(SO(2n)/U (n)); F
To determine 2 ), we will exploit two spectral sequences converging to the same space H ∗ (Ω(SO(2n)/U (n)); F2 ). One is the Serre spectral sequence and the other is the Eilenberg–Moore spectral sequence for the path loop fibration. Lemma 2.5. In the Eilenberg–Moore spectral sequence of the path loop fibration converging to H ∗ (Ω(SO(2n)/U (n)); F2 ),
E 2 = E (u 4i +1 : 0 4i n − 3) ⊗ Γ ( w 8i +6 : n − 4 4i 2n − 6) ⊗ Γ (u 4i +1 : n − 3 < 4i 2n − 4). Proof.
E 2 = TorH∗ (Ω(SO(2n)/U(n));F2 ) (F2 , F2 )
= TorF
2 [c4i+2 :
νi (F2 , F2 ) 04in−3]/(c24i+2 )⊗E(c4i+2 : n−3<4i2n−4)
= E (u 4i +1 : 0 4i n − 3) ⊗ Γ ( w (4i +2)2νi −2 : 0 4i n − 3) ⊗ Γ (u 4i +1 : n − 3 < 4i 2n − 4) = E (u 4i +1 : 0 4i n − 3) ⊗ Γ ( w 8i +6 : n − 4 4i 2n − 6) ⊗ Γ (u 4i +1 : n − 3 < 4i 2n − 4) where νi be the positive integer such that (2n − 1) < (4i + 2)2νi 4n − 4. Note that {(4i + 2)2νi − 2: 0 4i n − 3} = {8i + 6: n − 4 4i 2n − 6}. 2 Theorem 2.6. The mod 2 cohomology of loop space of SO(2n)/U (n) is as follows:
H ∗ Ω SO(2n)/U (n) ; F2 = E (u 4i +1 : 0 4i n − 3) ⊗ Γ ( w 8i +6 : n − 4 4i 2n − 6)
⊗ Γ (u 4i +1 : n − 3 < 4i 2n − 4). Proof. Consider the Serre spectral sequence for the following fibration for i = 1, 2, 3, 4:
Ω0 SO(8n + 2i ) −→ Ω SO(8n + 2i )/U (4n + i ) −→ U (2n + i ). Note that the structures of H ∗ (Ω0 SO(n); F2 ) = H ∗ (Ω Spin(n); F2 ) depend the congruence of n mod 8 [6]. Now we will prove the case of i = 1. The other cases are similar to the case of i = 1. We have the following morphism of fibrations:
Ω0 SO(8n + 2) −−−−→ Ω(SO(8n + 2)/U (4n + 1)) −−−−→ U (4n + 1) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ Ω0 SO
−−−−→
−−−−→
Ω(SO/U )
(2.4)
U
We recall in [6] that
ν
H ∗ Ω0 SO(8n + 2); F2 = F2 [a4i −2 : 1 i n]/ a4ii −2
⊗ Γ a4n+2+4i : 0 i (n − 1) ⊗ Γ c 8n+2+2i : 0 i (4n − 2), i ≡ 1 mod 4 4 ⊗ F2 γ2i (α8n ): i 0 / γ2i (α8n )
where
νi is the power of 2 such that 8n + 8 νi (4i − 2) 16n. We also recall in [10, Chap. 4] and [2] that
(2.5)
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H ∗ U (4n + 1); F2 = E (x2i +1 : 0 i 4n), H ∗ (U ; F2 ) = E (x2i +1 : i 0), H ∗ (Ω0 SO; F2 ) = F2 [a4i +2 : i 0],
H ∗ Ω(SO/U ); F2 = H ∗ (U /Sp; F2 ) = E (x4i +1 : i 0).
(2.6)
Then we have the following differentials in the Serre spectral sequence for the bottom fibration in (2.4):
d(a4i +2 ) = x4i +3 ,
i 0.
Then by naturality, we have the following differentials in the Serre spectral sequence for the top fibration in (2.4):
d(a4i +2 ) = x4i +3 ,
0 i n − 1,
d(a(4i +2)2 j ) = x(4i +2)2 j +1 , d(a4n+4i +2 ) = x4n+4i +3 ,
0 i n − 1, 1 j ν i − 1, 0 i n − 1.
By Lemma 2.5, in the Eilenberg–Moore spectral sequence of the path loop fibration converging to H ∗ (Ω(SO(8n + 2)/U (4n + 1)); F2 )
E 2 = E (u 4i +1 : 0 i n − 1) ⊗ Γ ( w 8i +6 : n i 2n − 1) ⊗ Γ (u 4i +1 : n i 2n − 1).
(2.7)
Then generators u 4i +1 of bidegree (−1, 4i + 2), 0 i 2n − 1, survive by bidegree reason, moreover, there are no generators of odd degree greater than 8n. This implies the following differentials in the Serre spectral sequence for the top fibration in (2.4):
d(α8n ) = x4n+1 a22n , d d d
γ2k+1 (α8n ) = x4n+1a22n γ2k (α8n ),
2 = x8n+1 a42n , α8n
2 2 = x8n+1 a42n γ2k α8n , γ2k+1 α8n
d(c 8n+8i ) = x4n+4i +1 a(4l+2)2 j ,
1 i n − 1, d γ2k+1 (c 8n+8i ) = x4n+4i +1 a(4l+2)2 j γ2k (c 8n+8i ),
k 0, 1 i n − 1
where j, l are some natural numbers with 0 l n − 1 satisfying |(4l + 2)2 j | = 4n + 4i for 1 i n − 1. In fact, j, l are uniquely determined according to i. Then γ2k (c 8n+8i +2 ), k 0, 0 i n − 1, and γ2k (c 8n+8i +6 ), k 0, 0 i n − 1, survive permanently. Moreover x2i +1 a(4l+2)2 j , 1 i 2n − 1, also survive permanently where j , l are some natural numbers with 0 l n − 1 satisfy-
ing |(4l + 2)2 j | = 2i for 1 i 2n − 1. Hence
E ∞ = E (x1 ) ⊗ E (x2i +1 a(4l+2)2 j : 1 i 2n − 1) ⊗ Γ (c 8n+8i +2 : 0 i n − 1) ⊗ Γ (c 8n+8i +6 : 0 i n − 1). Then simple calculation shows that this E ∞ -term has the same size in every total degree as the E 2 -term (2.7) of above Eilenberg–Moore spectral sequence as a graded vector space. This implies that the Eilenberg–Moore spectral sequence collapses at the E 2 -term. So we obtain
H ∗ Ω SO(8n + 2)/U (4n + 1) ; F2 = E (u 4i +1 : 0 i n − 1) ⊗ Γ ( w 8i +6 : n i 2n − 1)
⊗ Γ (u 4i +1 : n i 2n − 1). Hence we get the conclusion.
2
3. The mod p cohomology of loop spaces for odd primes p From now on, p always denotes odd primes. Theorem 3.1. The mod p cohomology of loop spaces of complex Grassmannian spaces is as follows:
H ∗ Ω U (m + n)/ U (m) × U (n) ; F p = H ∗ Ω U (m + n)/U (m) ; F p ⊗ H ∗ U (n); F p
= Γ (a2m+2i : 0 i n − 1) ⊗ E (x2i +1 : 0 i n − 1) where m n.
Y. Choi / Topology and its Applications 160 (2013) 280–291
Proof. By the same method as the mod 2 case, we get the conclusion.
285
2
Theorem 3.2. The mod p cohomology of loop space of complex quadric spaces Q n for n 3 is as follows:
∗
H (Ω Q n ; F p ) =
E (x1 ) ⊗ Γ ( y 2n )
if n is odd,
E (x1 , xn−1 ) ⊗ Γ ( y 2n−2 , yn )
if n is even.
Proof. For an odd prime p, we have the following equivalence [8]:
SO(2n + 1) p Sp(n). Hence we get
H ∗ Ω SO(2n + 1)/SO(2n − 1) ; F p = H ∗ Ω Sp(n)/Sp(n − 1) ; F p
= H ∗ Ω S 4n−1 ; F p
= Γ ( y 4n−2 ). As is known [10, Chap. 3],
H ∗ SO(2n + 2)/SO(2n); F p = E (x2n , x2n+1 ). Consider the Serre spectral sequence the following fibration:
Ω S 2n −→ Ω SO(2n + 2)/SO(2n) −→ Ω S 2n+1 . Then by similar method as the mod 2 case, it collapses at the E 2 -term. Hence we get
H ∗ Ω SO(2n + 2)/SO(2n) ; F p = H ∗ Ω S 2n+1 ; F p ⊗ H ∗ Ω S 2n ; F p
= Γ ( y 2n ) ⊗ E (x2n−1 ) ⊗ Γ ( y 4n−2 ). Consider the Serre spectral sequence converging to H ∗ (Ω Q n ; F p ), n 3, for the fibration
Ω SO(n + 2)/SO(n) −→ Ω Q n −→ S 1 . Then by dimensional reason the Serre spectral sequence collapses at E 2 . Hence we get the conclusion.
2
Theorem 3.3. ([7, Theorem 3.1])
H ∗ Ω Sp(n)/U (n); F p = E (x4i +1 : 0 2i n − 1) ⊗ Γ (x4i +2 : n − 1 2i 2n − 2). Theorem 3.4. The mod p cohomology of loop space of SO(2n)/U (n) is as follows:
H ∗ Ω SO(4n)/U (2n) ; F p = E (u 4i +1 : 0 i n − 1) ⊗ Γ ( w 4i +2 : n − 1 i 2n − 2),
H ∗ Ω SO(4n + 2)/U (2n + 1) ; F p = E (u 4i +1 : 0 i n − 1) ⊗ Γ ( w 4i +2 : n i 2n − 1). Proof. First we consider the stable elements [10, Chap. 4] and [2]:
H ∗ Ω(SO/U ); F p = H ∗ (U /Sp; F p ) = E (u 4i +1 : i 0).
(3.1)
We will prove the theorem by induction. Now we have
H ∗ Ω SO(4)/U (2) ; F p = H ∗ Ω S 2 ; F p = E (u 1 ) ⊗ Γ ( w 2 ). Consider the Serre spectral sequence for the following fibration:
Ω SO(4)/U (2) −→ Ω SO(6)/U (3) −→ Ω S 4 , where H ∗ (Ω S 4 ; F p ) = E (u 3 ) ⊗ Γ ( w 6 ). Then from the information of (3.1), we get the following transgression, d( w 2 ) = u 3 . Hence we obtain
H ∗ Ω SO(6)/U (3) ; F p = E (u 1 ) ⊗ Γ ( w 6 ).
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Suppose that
H ∗ Ω SO(4n)/U (2n) ; F p = E (u 4i +1 : 0 i n − 1) ⊗ Γ ( w 4i +2 : n − 1 i 2n − 2),
H ∗ Ω SO(4n + 2)/U (2n + 1) ; F p = E (u 4i +1 : 0 i n − 1) ⊗ Γ ( w 4i +2 : n i 2n − 1). Consider the Serre spectral sequence for the following fibrations:
Ω SO(4n)/U (2n) −→ Ω SO(4n + 2)/U (2n + 1) −→ Ω S 4n , Ω SO(4n + 2)/U (2n + 1) −→ Ω SO(4n + 4)/U (2n + 2) −→ Ω S 4n+2 .
(3.2) (3.3)
Then from the information of (3.1), we get the following transgression in the Serre spectral sequence for (3.2),
d( w 4n−2 ) = u 4n−1 where H ∗ (Ω S 4n ; F p ) = E (u 4n−1 ) ⊗ Γ ( w 8n−2 ). So we get
H ∗ Ω SO(4n + 2)/U (2n + 1) ; F p = E (u 4i +1 : 0 i n − 1) ⊗ Γ ( w 4i +2 : n i 2n − 1). On the other hand, H ∗ (Ω S 4n+2 ; F p ) = E (u 4n+1 ) ⊗ Γ ( w 8n+2 ) and there are no primitive elements of degree 4n or 8n + 1 in H ∗ (Ω(SO(4n + 2)/U (2n + 1)); F p ). So there are no non-trivial differentials in the Serre spectral sequence for (3.3) because of the degree reason. So we get
H ∗ Ω SO(4n + 4)/U (2n + 2) ; F p = E (u 4i +1 : 0 i n) ⊗ Γ ( w 4i +2 : n − 1 i 2n). Hence we get the conclusion.
2
Remark 3.5. We can rewrite H ∗ (Ω(SO(2n)/U (n)); F p ) as follows:
H ∗ Ω SO(2n)/U (n) ; F p =
E (u 4i +1 : 0 2i n − 2) ⊗ Γ ( w 4i +2 : n − 2 2i 2n − 4) for even n,
E (u 4i +1 : 0 2i n − 3) ⊗ Γ ( w 4i +2 : n − 1 2i 2n − 4) for odd n = E (u 4i +1 : 0 2i n − 2) ⊗ Γ ( w 4i +2 : n − 2 2i 2n − 4).
Note that {i: 2i n − 2} = {i: 2i n − 3}, {i: n − 1 2i } = {i: n − 2 2i } for odd n. Moreover if we check the degrees of generators in the mod 2 cohomology in Theorem 2.6, we can obtain
dimF2 H ∗ Ω SO(2n)/U (n) ; F2
= dimF p H ∗ Ω SO(2n)/U (n) ; F p .
(3.4)
4. Torsion in the integral cohomology of loop spaces Consider the exact couple i
H ∗ ( X ; Z ) −−−−→ H ∗ ( X ; Z ) k
ρ H ∗( X ; Fp )
where i is induced by multiplication by p in Z and ρ is induced by the reduction homomorphism and k is the Bockstein homomorphism in the long exact sequence. From this exact couple we get the Bockstein spectral sequence { E r , dr } with E 1 = H ∗ ( X ; F p ) converging to ( H ∗ ( X ; Z )/torsion) ⊗ F p . For more details, we refer [11]. Note that
dimF p H ∗ ( X ; F p ) dimF p H ∗ ( X ; Z )/torsion ⊗ F p
= dimQ H ∗ ( X ; Q) .
Hence if dimF p ( H ∗ ( X ; F p )) = dimQ ( H ∗ ( X ; Q)), then the Bockstein spectral sequence collapses at E 1 and this implies that H ∗ ( X ; Z ) is p-torsion free. Lemma 4.1. The integral cohomology of loop spaces of complex Grassmannian spaces is torsion free. Proof. Consider the Serre spectral sequence converging to H ∗ (Ω(U (m + n)/(U (m) × U (n))); Q) for the fibration
Ω U (m + n)/U (m) −→ Ω U (m + n)/ U (m) × U (n) −→ U (n)
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where H ∗ (Ω(U (m + n)/U (m)); Q) = Q[a2m+2i : 0 i n] and H ∗ (U (n); Q) = E (x2i +1 : 0 i n − 1). By the same method as the mod 2 case, we can show that it collapses at E 2 and obtain
H ∗ Ω U (m + n)/ U (m) × U (n) ; Q = Q[a2m+2i : 0 i n − 1] ⊗ E (x2i +1 : 0 i n − 1). Since
dimF2 H ∗ Ω U (m + n)/U (m) ; F2
= dimF p H ∗ Ω U (m + n)/U (m) ; F p = dimQ H ∗ Ω U (m + n)/U (m) ; Q ,
the integral cohomology of loop spaces of complex Grassmannian spaces is torsion free.
2
Lemma 4.2. The integral cohomology of loop space of Q n for n 3 is torsion free for even n, and has only 2-torsion of order 2 for odd n. Proof. We have the following results from [1, Proposition 10.1],
H ∗ SO(2n + 1)/SO(2n − 1); Q = E (x4n−1 ),
H ∗ SO(2n + 2)/SO(2n); Q = E (x2n , x2n+1 ). By the same method as the mod p case in Theorem 3.2, we get
H ∗ Ω SO(2n + 1)/SO(2n − 1) ; Q = Q[ y 4n−2 ],
H ∗ Ω SO(2n + 2)/SO(2n) ; Q = E (x2n−1 ) ⊗ Q[ y 2n , y 4n−2 ]. So we obtain ∗
H (Ω Q n ; Q) =
E (x1 ) ⊗ Q[ y 2n ]
if n is odd,
E (x1 , xn−1 ) ⊗ Q[ yn , y 2n−2 ]
if n is even.
Then by Theorem 3.2,
dimF p H ∗ (Ω Q n ; F p ) = dimQ H ∗ (Ω Q n ; Q) . This implies that H ∗ (Ω Q n ; Z ) is odd torsion free for each n. Now we turn to the 2-torsion case. Consider the Bockstein spectral sequence with E 1 = H ∗ (Ω Q n ; F2 ) converging to ( H ∗ (Ω Q n ; Z )/torsion) ⊗ F2 . By Theorem 2.2,
d1 (an−1 ) = Sq1 (an−1 ) = Then
E2 =
an
if n is odd,
0
if n is even.
E (a1 ) ⊗ Γ (a2n )
if n is odd,
E (a1 ) ⊗ Γ (an−1 ) ⊗ Γ (an )
if n is even
γ2 (an ) = a2n . Now we have dimF2 ( E 2 ) = dimQ H ∗ (Ω Q n ; Q) .
where we let
It follows that E 2 = E ∞ . Therefore H ∗ (Ω Q n ; Z ) is 2-torsion free for even n, and has only 2-torsion of order 2 for odd n. Lemma 4.3. The integral cohomology of loop space of Sp(n)/U (n) has only 2-torsion of order 2, and is odd torsion free. Proof. First we compute H ∗ (Ω(Sp(2n)/U (2n)); Q) using the following map of fibrations:
Ω Sp(n + 1) −−−−→ Ω Sp(n + 1)/U (n + 1) −−−−→ U (n + 1) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ Ω Sp
−−−−→
Ω Sp/U
−−−−→
U
We can obtain the following basic data from [10, Chap. 4] and [2]:
2
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Y. Choi / Topology and its Applications 160 (2013) 280–291
H ∗ U (n + 1); Q = E (x2i +1 : 0 i n),
H ∗ Ω Sp(n + 1); Q = Q[x4i +2 : 0 i n], H ∗ (Ω Sp; Q) = Q[x4i +2 : i 0],
H ∗ Ω(Sp/U ); Q = H ∗ (U / O ; Q) = E (x4i +1 : i 0). From above data we can get the following transgressions for the Serre spectral sequence for the bottom row fibration:
d4i +3 (x4i +2 ) = x4i +3 ,
i 0.
By the naturality of differentials, we also have the following transgressions for the top row fibration:
d4i +3 (x4i +2 ) = x4i +3 ,
0 2i n − 2.
So we have
H ∗ Ω Sp(n)/U (n); Q = E (x4i +1 : 0 2i n − 1) ⊗ Q[x4i +2 : n − 1 2i 2n − 2]. From Theorem 3.3, we obtain
dimF p H ∗ Ω Sp(n)/U (n) ; F p
= dimQ H ∗ Ω Sp(n)/U (n) ; Q ,
so that H ∗ (Ω(Sp(n)/U (n)); Z ) is odd torsion free. Consider the following fibration: i
U −→ U / O −→ B O . Then i ∗ ( w i ) = xi , i 1 where H ∗ ( B O ; F2 ) = F2 [ w i : i 1]. For more details, we refer [10, Chap. 3, Theorem 6.7]. Since
Sq1 w i = we have
Sq1 xi =
w i +1
if i is even,
0
if i is odd,
x i +1
if i is even,
0
if i is odd H ∗ (Ω(Sp/U ); F
∗ in H ∗ (U / O ; F2 ) = 2 ), and also in H (Ω(Sp(n)/U (n)); F2 ). Consider the Bockstein spectral sequence with E 1 = H ∗ (Ω(Sp(n)/U (n)); F2 ) converging to ( H ∗ (Ω(Sp(n)/U (n)); Z )/ torsion) ⊗ F2 . Then
d1 (xi ) = Sq1 xi =
x i +1
if i is even,
0
if i is odd,
xi +1 γ2k (xi ) if i is even, 1 d1 γ2k+1 (xi ) = Sq γ2k+1 (xi ) =
0
if i is odd
for all k 1. Then
E2 =
E (x4i +1 : 0 2i n − 2) ⊗ Γ (x4i +2 : n 2i 2n − 2)
if n is even,
E (x4i +1 : 0 2i n − 1) ⊗ Γ (x4i +2 : n − 1 2i 2n − 2) if n is odd = E (x4i +1 : 0 2i n − 1) ⊗ Γ (x4i +2 : n − 1 2i 2n − 2).
Note that for even n, {i: 2i n − 2} = {i: 2i n − 1}, {i: n 2i } = {i: n − 1 2i }. Now we have
dimF2 ( E 2 ) = dimQ H ∗ Ω Sp(n)/U (n) ; Q . Hence E 2 = E ∞ , and this implies that H ∗ (Ω(Sp(n)/U (n)); Z ) has 2-torsion of order 2 only.
2
Lemma 4.4. The integral cohomology of loop space of SO(2n)/U (n) is torsion free. Proof. First we will compute the rational cohomology of loop space of SO(2n)/U (n). Then by the same way as the mod 2 case, we consider the Serre spectral sequence converging to H ∗ (Ω(SO(8n + 2i )/U (4n + i )); Q) for the following fibration for i = 1, 2, 3, 4:
Ω0 SO(8n + 2i ) −→ Ω(SO(8n + 2i )/U (4n + i )) −→ U (2n + i ).
Y. Choi / Topology and its Applications 160 (2013) 280–291
289
Now we will prove the case of i = 1. Since H ∗ (Ω0 SO(8n + 2); F2 ) is even dimensional, so that it is torsion free. So by (2.5) we obtain
ν
H ∗ Ω0 SO(8n + 2); Q = Q[a4i −2 : 1 i n]/ a4ii −2
⊗ Q a4n+2+4k : 0 k (n − 1) ⊗ Q c 8n+2+2k : 0 k (4n − 2), k ≡ 1 mod 4 4 ⊗ Q γ2i (α8n ): i 0 / γ2i (α8n ) .
Consider
Ω0 SO(8n + 2) −−−−→ Ω(SO(8n + 2)/U (4n + 1)) −−−−→ U (4n + 1) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ Ω0 SO
−−−−→
Ω(SO/U )
−−−−→
U
From (2.6), we can get
H ∗ U (4n + 1); Q = E (x2i +1 : 0 i 4n), H ∗ (U ; Q) = E (x2i +1 : i 0), H ∗ (Ω0 SO; Q) = Q[a4i +2 : i 0],
H ∗ Ω(SO/U ); Q = E (x4i +1 : i 0). Then we can determine differentials in the Serre spectral sequence by the exact same way as the mod 2 case and get
H ∗ Ω SO(8n + 2)/U (4n + 1) ; Q = E (u 4i +1 : 0 i n − 1) ⊗ Q[ w 8n+8i +6 : 0 i n − 1]
⊗ Q[u 4i +1 : n i 2n − 1]. The other cases are similar to the case of i = 1. Then we get
H ∗ Ω SO(2n)/U (n) ; Q = E (u 4i +1 : 0 4i n − 3) ⊗ Q[ w 8i +6 : n − 4 4i 2n − 6]
⊗ Q[u 4i +1 : n − 3 < 4i 2n − 4]. Now from Theorem 2.6 and (3.4) we obtain
dimQ H ∗ Ω SO(2n)/U (n) ; Q
= dimF2 H ∗ Ω SO(2n)/U (n) ; F2 = dimF p H ∗ Ω SO(2n)/U (n) ; F p .
Therefore the integral cohomology of loop space of SO(2n)/U (n) is torsion free.
2
Summarizing above lemmas, we get Theorem 4.5. The integral cohomology of loop spaces of irreducible compact Hermitian symmetric spaces of classical types is torsion free, or has only 2-torsion of order 2. 5. The mod p cohomology of free loop space i
Given a fibration F → E → B, we say that F is totally non-homologous to zero in E with respect to F p if the homomorphism i ∗ : H ∗ ( E ; F p ) → H ∗ ( F ; F p ) is onto. Let Λ M be a space of free loops on M. Even though the free loop space fibration Ω M → Λ M → M has a cross section given by sending each x in M into the constant loop at x, it is well known the fibre Ω M need not be totally non-homologous to zero in Λ M. Hence it is the general question when Ω M is totally non-homologous to zero in Λ M. Following the notation by Kuribayashi, let T ( M ) be a set of prime numbers p such that Ω M is totally non-homologous to zero in Λ M with respect to F p . Theorem 5.1. ([9])
/ T (U (m + n)/U (m) × U (n)) for any prime p. (a) If m, n 2, then p ∈ (b) p ∈ T (Sp(n)/U (n)) iff p = 2.
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Hence the remaining cases are Q n = SO(n + 2)/(SO(2) × SO(n)) and SO(2n)/U (n).
/ T ( Q n ) for n 3. For the case of p = 2, p ∈ T ( Q n ) iff (n + 1)/2 + 1 is a power of 2 when Theorem 5.2. If p is any odd prime, then p ∈ n is odd, and n/2 + 1 is a power of 2 when n is even. Proof. We can extract the following results from [3, §9, 10]:
H ∗ ( Q 2n−1 ; F2 ) = F2 [c 2 ]/ cn2 ⊗ E (a2n ),
H ∗ ( Q 2n ; F2 ) = F2 [c 2 ]/ cn2+1 ⊗ E (a2n ),
H ∗ ( Q 2n−1 ; F p ) = F p [c 2 ]/ c 22n ,
H ∗ ( Q 2n ; F p ) = F p [c 2 ]/ cn2+1 ⊗ E (a2n ). For the case of p = 2. Consider the Serre spectral sequence converging to H ∗ (Λ Q 2n−1 ; F2 ) for Ω Q 2n−1 → Λ Q 2n−1 → Q 2n−1 and H ∗ (Λ Q 2n ; F2 ) for Ω Q 2n → Λ Q 2n → Q 2n . Then by Proposition 3.5 in [12], we get the following non-trivial differentials:
d(a2n−2 ) = ncn2−1 a1 d(a2n ) = (n
for Q 2n−1 if n is not a power of 2,
+ 1)cn2 a1
for Q 2n if n + 1 is not a power of 2.
Consider the remaining cases: Q 2n−1 where n is a power of 2, and Q 2n where n + 1 is a power of 2. Then they satisfy the condition of Theorem 2 in [13], so that Ω Q 2n−1 , Ω Q 2n are totally non-homologous to zero in Λ Q 2n−1 , Λ Q 2n with respect to F2 by Theorem 2 in [13] and Proposition 1.7 in [9]. Now we turn to the case of odd primes. By Proposition 3.5 in [12] we obtain the following differentials in the Serre spectral sequence converging to H ∗ (Λ Q n ; F p ) for Ω Q n → Λ Q n → Q n :
d( y 2n ) = (n + 1)cn2 x1
if n is odd,
d( y 2n−2 ) = 2an xn−1
if n is even.
So p ∈ / T ( Q n ) if p is an odd prime.
2
Theorem 5.3. ([10, Theorem 6.11]) For odd primes p, H ∗ (SO(2n)/U (n); F p ) is
Z [c 2 , . . . , c 2n−2 ]/
(−1)i c 2i c 2 j , k 1 .
i + j =2k
Theorem 5.4. p ∈ T (SO(2n)/U (n)) iff p = 2. Proof. Since H ∗ (SO(2n)/U (n); F2 ) is even dimensional by Theorem 2.4, Sq1 x = 0 for all x ∈ H ∗ (SO(2n)/U (n); F2 ). And by (2.3), H ∗ (SO(2n)/U (n); F2 ) satisfies the condition of Theorem 2 in [13]. Then by Theorem 2 in [13] and Proposition 1.7 in [9], 2 ∈ T (SO(2n)/U (n)). For the case of odd primes, in order to use Proposition 3.5 in [12] we should express H ∗ (SO(2n)/U (n); F p ) as
E ( yl1 , . . . , yli ) ⊗ F p [xn1 , . . . , xn j ]/(ρm1 , . . . , ρmk ) where deg( yl1 ), . . . , deg( yli ) are odd, and deg(xn1 ), . . . , deg(xn j ) are even, and able. Then we can express
ρm1 , . . . , ρmk ∈ F p [x1 , . . . , xn ] are decompos-
H ∗ SO(4n)/U (2n); F p = F p [c 4i +2 : 0 i n − 1]/(ρ4n , . . . , ρ8n−4 ),
H ∗ Ω SO(4n + 2)/U (2n + 1) ; F p = F p [c 4i +4 : 0 i n − 1]/(ρ4n+2 , . . . , ρ8n−4 ), where
ρ4n+4 = c23 − 2c6 c4n−2 +
n −1
g 4n−4k+6 (c 2 , . . . , c 4k−6 )c 4k−2
for SO(4n + 2)/U (2n + 1),
k =1
ρ4n = (2c1 )c4n−2 +
n −1
g 4n−4k+2 (c 2 , . . . , c 4k−6 )c 4k−2
for SO(4n)/U (2n).
k =1
Here g 4n−4k+6 (c 2 , . . . , c 4k−6 ), g 4n−4k+2 (c 2 , . . . , c 4k−6 ) ∈ F p [c 2 , . . . , c 4k−6 ] are homogeneous polynomials of degrees 4n − 4k + 6, 4n − 4k + 2.
Y. Choi / Topology and its Applications 160 (2013) 280–291
291
From the information of Theorem 2.6, we can easily deduce that the Serre spectral sequence for Ω(SO(2n)/U (n)) → ∗ → SO(2n)/U (n) collapses at E 2 . So by Proposition 3.5 in [12] we obtain the following differentials in the Serre spectral sequence converging to H ∗ (Λ(SO(2n)/U (n)); F p ) for Ω(SO(2n)/U (n)) → Λ(SO(2n)/U (n)) → SO(2n)/U (n):
d( w 4n+2 ) = c 23 − 2c 6 u 4n−3 +
n −1
∂ g 4n−4k+6 (c 2 , . . . , c 4k−6 )c 4k−2 /∂ c 4k−2 u 4k−3 k =1
= c 23 − 2c 6 u 4n−3 +
n −1
g 4n−4k+6 (c 2 , . . . , c 4k−6 )u 4k−3
k =1
= 0 for SO(4n + 2)/U (2n + 1), d( w 4n−2 ) = 2c 1 u 4n−3 +
n −1
∂ g 4n−4k+2 (c 2 , . . . , c 4k−6 )c 4k−2 /∂ c 4k−2 u 4k−3 k =1
= 2c 1 u 4n−3 +
n −1
g 4n−4k+2 (c 2 , . . . , c 4k−6 )u 4k−3
k =1
= 0 for SO(4n)/U (2n). Hence p ∈ / T (SO(2n)/U (n)) if p is any odd prime.
2
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
A. Borel, Sur la cohomologie des espaces fibres principaux et des espaces homogenes de groupes de Lie compacts, Ann. of Math. 57 (1953) 115–207. R. Bott, The stable homotopy of the classical groups, Proc. Nat. Acad. Sci. 43 (1957) 933–935. R. Bott, The space of loops on a Lie group, Michigan Math. J. 5 (1958) 35–61. R. Bott, H. Samelson, Applications of the theory of Morse to symmetric spaces, Amer. J. Math. 80 (1958) 964–1029. H. Cartan, Démonstration homologique des théorèmes de périodicitè de Bott II, III, in: Séminaire Cartan et Moore 1959/1960, Exposés 17, Ecole Norm. Sup. Paris. Y. Choi, Homology of the double and triple loop space of SO(n), Math. Z. 222 (1996) 59–80. Y. Choi, Homology of the double loop space of the homogeneous space Sp(n)/U (n), Bull. Korean Math. Soc. 36 (1999) 747–754. B. Harris, On the homotopy groups of the classical groups, Ann. Math. 74 (1961) 407–413. K. Kuribayashi, On the mod p cohomology of the spaces of free loops on the Grassmann and Stiefel manifolds, J. Math. Soc. Japan 43 (1991) 331–346. M. Mimura, H. Toda, Topology of Lie Groups I, Trans. Math. Monographs, vol. 91, Amer. Math. Soc., 1991. R.E. Mosher, M.C. Tangora, Cohomology Operations and Applications in Homotopy Theory, Harper & Row Publisher, 1968. L. Smith, On the characteristic zero cohomology of the free loop space, Amer. J. Math. 103 (1981) 887–910. L. Smith, The Eilenberg–Moore spectral sequence and the mod 2 cohomology of certain free loop spaces, Illinois J. Math. 28 (1984) 516–522.
Contents lists available at SciVerse ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Cohomology of loop spaces of Hermitian symmetric spaces of classical types ✩ Younggi Choi ∗ Department of Mathematics Education, Seoul National University, Seoul 151-748, Republic of Korea
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 30 July 2012 Received in revised form 1 November 2012 Accepted 2 November 2012
We study the mod p cohomology of loop spaces and free loop spaces of irreducible compact Hermitian symmetric spaces of classical types. We also prove that the integral cohomology of loop spaces of them is torsion free, or has only 2-torsion of order 2. © 2012 Elsevier B.V. All rights reserved.
MSC: 55R20 55T10 55T20 Keywords: Hermitian symmetric spaces of classical types Loop space Free loop space Totally non-homologous to zero
1. Introduction A Hermitian symmetric space is a connected complex manifold with a hermitian metric such that the geodesic symmetries are hermitian isometries. The irreducible compact Hermitian symmetric spaces of classical type are divided into the following four classes:
AIII BDI CI DIII
G m,n (C) = U (m + n)/ U (m) × U (n)
Q n = SO(n + 2)/ SO(2) × SO(n)
m , n 1, n 3,
Sp(n)/U (n)
n 3,
SO(2n)/U (n)
n 4.
The purpose of this paper is to study the cohomology of (based) loop spaces and free loop spaces of irreducible compact Hermitian symmetric spaces of classical types. In this paper, first we determine the mod p cohomology of loop spaces of them by exploiting the Serre spectral sequence and the Eilenberg–Moore spectral sequence converging to the same target space. In [4], Bott and Samelson asked whether ✩ This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012-0001320). Fax: +82 2 889 1747. E-mail address: [email protected].
*
0166-8641/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.topol.2012.11.003
Y. Choi / Topology and its Applications 160 (2013) 280–291
281
the integral cohomology of loop spaces of symmetric spaces has only two torsion. So in Section 4, we study torsion in the integral cohomology of the loop spaces of irreducible compact Hermitian symmetric spaces M of classical types, and prove that the integral cohomology of loop spaces of them is torsion free, or has only 2-torsion of order 2. Finally we determine when Ω M is totally non-homologous to zero in Λ M with respect to F p for the free loop space fibration Ω M → Λ M → M. Throughout the paper E (x) denotes the exterior algebra on x and Γ (x) denotes the divided power algebra on x. The subscript of an element denotes the degree of the element, that is, deg(xi ) = i. 2. The mod 2 cohomology of loop spaces Theorem 2.1. The mod 2 cohomology of loop spaces of complex Grassmannian spaces are as follows:
H ∗ Ω U (m + n)/ U (m) × U (n) ; F2 = H ∗ Ω U (m + n)/U (m) ; F2 ⊗ H ∗ U (n); F2
= Γ (a2m+2i : 0 i n − 1) ⊗ E (x2i +1 : 0 i n − 1) where m n. Proof. First of all, we compute the Eilenberg–Moore spectral sequence converging to H ∗ (Ω(U (n + n)/U (m)); F2 ) with
E 2 = Tor H ∗ (U (m+n)/U (m);F2 ) (F2 , F2 ) = Tor E (x2m+2i+1 : 0i n−1) (F2 , F2 ) = Γ (a2m+2i : 0 i n − 1). Then the spectral sequence collapses at the E 2 -term because E 2 vanishes in all odd total degrees. Moreover since m n, we have |a22m | = 4m > 2m + 2n − 2 = |a2m+2n−2 |, so that there is no algebra extension. Hence H ∗ (Ω(U (m + n)/U (m)); F2 ) = Γ (a2m+2i : 0 i n − 1). Consider the Serre spectral sequence converging to H ∗ (Ω(U (m + n)/(U (m) × U (n))); F2 ) for the fibration
Ω U (m + n)/U (m) −→ Ω U (m + n)/ U (m) × U (n) −→ U (n)
where H ∗ (U (n); F2 ) = E (x2i +1 : 0 i n − 1). Since this spectral sequence is the spectral sequence of a Hopf algebra, the source of the first non-trivial differential is an indecomposable element and its target is a primitive element. Since the degree of indecomposables in H ∗ (Ω(U (m + n)/U (m)); F2 ) is greater than 2m − 1, the degree of target for the first nontrivial differential is greater than 2m. Since m n, there is no primitive element of degree greater than 2m in H ∗ (U (n); F2 ). Hence the Serre spectral sequence collapses at E 2 . Now we consider the algebra extension problems. First, we consider the case of m 2n. Since |a2m | = 2m > 2(2n − 1) = |x22n−1 |, there is no extension problem. For the case of n m 2n, there may be some possibility of extension, such as, x22 j +1 = a2m+2k for some 0 k n and 0 j n − 1. Consider the following morphism of fibration.
Ω(U (m + n)/U (m)) −−−−→ Ω(U (m + n)/(U (m) × U (n))) −−−−→ U (n) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ Ω j Ω i id Ω(U (2m + n)/U (2m)) −−−−→ Ω(U (2m + n)/(U (2m) × U (n))) −−−−→ U (n) Since 2m 2n, there is no extension problem in H ∗ (Ω(U (2m + n)/(U (2m) × U (n))); F2 ). If there were an extension x22 j +1 = a2m+2k in H ∗ (Ω(U (m + n)/(U (m) × U (n))); F2 ), then it would follow that
a2m+2k = x22 j +1 = (Ω j )∗ (x2 j +1 )(Ω j )∗ (x2 j +1 ) = (Ω j )∗ x22 j +1 = (Ω j )∗ (0) = 0. So if there is such an extension, this leads a contradiction. So we get the conclusion.
2
Theorem 2.2. The mod 2 cohomology of loop space of complex quadric spaces Q n for n 3 is as follows:
H ∗ (Ω Q n ; F2 ) = E (a1 ) ⊗ Γ (an−1 ) ⊗ Γ (an ),
Sq1 an−1 =
an
if n is odd,
0
if n is even.
Proof. It is known [10, Chap. 3] that
H ∗ SO(n + 2)/SO(n); F2 = E (xn ) ⊗ E (xn+1 ),
Sq1 xn =
xn+1
if n is odd,
0
if n is even.
Consider the Serre spectral sequence the following fibration:
Ω S n −→ Ω SO(n + 2)/SO(n) −→ Ω S n+1
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Y. Choi / Topology and its Applications 160 (2013) 280–291
with
E 2 = H ∗ Ω S n+1 ; F2 ⊗ H ∗ Ω S n ; F2 = Γ (cn ) ⊗ Γ (cn−1 ). Since it is the spectral sequence of a Hopf algebra, the only possible non-trivial first differential is
d(cn−1 ) = cn .
(2.1)
Now we consider the Eilenberg–Moore spectral sequence of the path loop fibration converging to H ∗ (Ω(SO(n 2)/SO(n)); F2 ) with
+
E 2 = Tor H ∗ (SO(n+2)/SO(n);F2 ) (F2 , F2 ) = Γ (an , an−1 ). Then the differential (2.1) implies the differential from an of bidegree (−1, n + 1) to an−1 of bidegree (−1, n) in the Eilenberg–Moore spectral sequence, however it is impossible because of bidegree reason. Hence the Serre spectral sequence collapses at the E 2 -term, so does the Eilenberg–Moore spectral sequence. So we obtain
H ∗ Ω SO(n + 2)/SO(n) ; F2 = Γ (an−1 ) ⊗ Γ (an ),
Sq1 an−1 =
an
if n is odd,
0
if n is even.
Consider the Serre spectral sequence converging to H ∗ (Ω Q n ; F2 ) for the fibration
Ω SO(n + 2)/SO(n) −→ Ω Q n −→ S 1 . Then by dimensional reason the Serre spectral sequence collapses at E 2 . So we get the conclusion.
2
Theorem 2.3.
H ∗ Ω Sp(n)/U (n) ; F2 = E (x2i −1 : 1 i n) ⊗ Γ (x4i −2 : 1 i n). Proof. Consider the following morphism of fibrations:
Ω Sp(n) −−−−→ Ω(Sp(n)/U (n)) −−−−→ U (n) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ j i Ω Sp
−−−−→
Ω(Sp/U )
−−−−→
(2.2)
U
We recall the following facts from [10, Chap. 4] and [2]:
H ∗ U (n); F2 = E (x2i −1 : 1 i n), H ∗ (U ; F2 ) = E (x2i −1 : i 1),
H ∗ Ω Sp(n); F2 = Γ (x4i −2 : 1 i n), H ∗ (Ω Sp; F2 ) = F2 [x4i −2 : i 1],
H ∗ Ω(Sp/U ); F2 = H ∗ (U / O ; F2 ) = E (xi : i 1). Consider the Serre spectral sequence for the bottom row fibration in (2.2) converging to H ∗ (Ω Sp/U ; F2 ) with
E 2 = H ∗ (U ; F2 ) ⊗ H ∗ (Ω Sp; F2 ). Since the cohomology of the total space has the same size in every total degree as tensor products of those of the base and the fiber space as a graded vector spaces, the spectral sequence collapses at the E 2 . Since i ∗ and j ∗ are epimorphism, the spectral sequence for the top fibration also collapses at the E 2 by the naturality of differentials. Since H ∗ (U ; F2 ) = E (x2i −1 : i 1), there are no multiplicative extensions in the Serre spectral sequence for the bottom row fibration, such as x22i −1 = x4i −2 , i 1. Hence there are also no multiplicative extensions in the Serre spectral sequence for the top row fibration, such as x22i −1 = x4i −2 for 1 i n. Hence we obtain
H ∗ Ω Sp(n)/U (n) ; F2 = E (x2i −1 : 1 i n) ⊗ Γ (x4i −2 : 1 i n).
2
We recall the following result from (72) in [5], and Theorem 6.11 in Chapter 4 of [10].
Y. Choi / Topology and its Applications 160 (2013) 280–291
283
Theorem 2.4.
H ∗ SO(2n)/U (n); F2 = V (c 2i : 1 i n − 1), where V (c 2i : 1 i n − 1) is the commutative associative algebra over F2 satisfying the following conditions: 1. {(c 2 )1 · · · (c 2i )i · · · (c 2n−2 )n−1 : i = 0, 1 for 1 i n − 1} is a basis, 2 2 2. c 2i = c 4i for 4i 2n − 2, and c 2i = 0 if 4i > 2n − 2. 2 Hence c 4i +2 = 0 for n − 1 < 4i + 2 2n − 2. For c 4i +2 with 2 4i + 2 n − 1, let (2n − 1) < (4i + 2)2νi 4n − 4. Then we can rewrite it as follows:
νi
νi be the positive integer such that
2 H ∗ SO(2n)/U (n); F2 = F2 [c 4i +2 : 0 4i n − 3]/ c 4i +2 ⊗ E (c 4i +2 : n − 3 < 4i 2n − 4).
(2.3)
H ∗ (Ω(SO(2n)/U (n)); F
To determine 2 ), we will exploit two spectral sequences converging to the same space H ∗ (Ω(SO(2n)/U (n)); F2 ). One is the Serre spectral sequence and the other is the Eilenberg–Moore spectral sequence for the path loop fibration. Lemma 2.5. In the Eilenberg–Moore spectral sequence of the path loop fibration converging to H ∗ (Ω(SO(2n)/U (n)); F2 ),
E 2 = E (u 4i +1 : 0 4i n − 3) ⊗ Γ ( w 8i +6 : n − 4 4i 2n − 6) ⊗ Γ (u 4i +1 : n − 3 < 4i 2n − 4). Proof.
E 2 = TorH∗ (Ω(SO(2n)/U(n));F2 ) (F2 , F2 )
= TorF
2 [c4i+2 :
νi (F2 , F2 ) 04in−3]/(c24i+2 )⊗E(c4i+2 : n−3<4i2n−4)
= E (u 4i +1 : 0 4i n − 3) ⊗ Γ ( w (4i +2)2νi −2 : 0 4i n − 3) ⊗ Γ (u 4i +1 : n − 3 < 4i 2n − 4) = E (u 4i +1 : 0 4i n − 3) ⊗ Γ ( w 8i +6 : n − 4 4i 2n − 6) ⊗ Γ (u 4i +1 : n − 3 < 4i 2n − 4) where νi be the positive integer such that (2n − 1) < (4i + 2)2νi 4n − 4. Note that {(4i + 2)2νi − 2: 0 4i n − 3} = {8i + 6: n − 4 4i 2n − 6}. 2 Theorem 2.6. The mod 2 cohomology of loop space of SO(2n)/U (n) is as follows:
H ∗ Ω SO(2n)/U (n) ; F2 = E (u 4i +1 : 0 4i n − 3) ⊗ Γ ( w 8i +6 : n − 4 4i 2n − 6)
⊗ Γ (u 4i +1 : n − 3 < 4i 2n − 4). Proof. Consider the Serre spectral sequence for the following fibration for i = 1, 2, 3, 4:
Ω0 SO(8n + 2i ) −→ Ω SO(8n + 2i )/U (4n + i ) −→ U (2n + i ). Note that the structures of H ∗ (Ω0 SO(n); F2 ) = H ∗ (Ω Spin(n); F2 ) depend the congruence of n mod 8 [6]. Now we will prove the case of i = 1. The other cases are similar to the case of i = 1. We have the following morphism of fibrations:
Ω0 SO(8n + 2) −−−−→ Ω(SO(8n + 2)/U (4n + 1)) −−−−→ U (4n + 1) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ Ω0 SO
−−−−→
−−−−→
Ω(SO/U )
(2.4)
U
We recall in [6] that
ν
H ∗ Ω0 SO(8n + 2); F2 = F2 [a4i −2 : 1 i n]/ a4ii −2
⊗ Γ a4n+2+4i : 0 i (n − 1) ⊗ Γ c 8n+2+2i : 0 i (4n − 2), i ≡ 1 mod 4 4 ⊗ F2 γ2i (α8n ): i 0 / γ2i (α8n )
where
νi is the power of 2 such that 8n + 8 νi (4i − 2) 16n. We also recall in [10, Chap. 4] and [2] that
(2.5)
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H ∗ U (4n + 1); F2 = E (x2i +1 : 0 i 4n), H ∗ (U ; F2 ) = E (x2i +1 : i 0), H ∗ (Ω0 SO; F2 ) = F2 [a4i +2 : i 0],
H ∗ Ω(SO/U ); F2 = H ∗ (U /Sp; F2 ) = E (x4i +1 : i 0).
(2.6)
Then we have the following differentials in the Serre spectral sequence for the bottom fibration in (2.4):
d(a4i +2 ) = x4i +3 ,
i 0.
Then by naturality, we have the following differentials in the Serre spectral sequence for the top fibration in (2.4):
d(a4i +2 ) = x4i +3 ,
0 i n − 1,
d(a(4i +2)2 j ) = x(4i +2)2 j +1 , d(a4n+4i +2 ) = x4n+4i +3 ,
0 i n − 1, 1 j ν i − 1, 0 i n − 1.
By Lemma 2.5, in the Eilenberg–Moore spectral sequence of the path loop fibration converging to H ∗ (Ω(SO(8n + 2)/U (4n + 1)); F2 )
E 2 = E (u 4i +1 : 0 i n − 1) ⊗ Γ ( w 8i +6 : n i 2n − 1) ⊗ Γ (u 4i +1 : n i 2n − 1).
(2.7)
Then generators u 4i +1 of bidegree (−1, 4i + 2), 0 i 2n − 1, survive by bidegree reason, moreover, there are no generators of odd degree greater than 8n. This implies the following differentials in the Serre spectral sequence for the top fibration in (2.4):
d(α8n ) = x4n+1 a22n , d d d
γ2k+1 (α8n ) = x4n+1a22n γ2k (α8n ),
2 = x8n+1 a42n , α8n
2 2 = x8n+1 a42n γ2k α8n , γ2k+1 α8n
d(c 8n+8i ) = x4n+4i +1 a(4l+2)2 j ,
1 i n − 1, d γ2k+1 (c 8n+8i ) = x4n+4i +1 a(4l+2)2 j γ2k (c 8n+8i ),
k 0, 1 i n − 1
where j, l are some natural numbers with 0 l n − 1 satisfying |(4l + 2)2 j | = 4n + 4i for 1 i n − 1. In fact, j, l are uniquely determined according to i. Then γ2k (c 8n+8i +2 ), k 0, 0 i n − 1, and γ2k (c 8n+8i +6 ), k 0, 0 i n − 1, survive permanently. Moreover x2i +1 a(4l+2)2 j , 1 i 2n − 1, also survive permanently where j , l are some natural numbers with 0 l n − 1 satisfy-
ing |(4l + 2)2 j | = 2i for 1 i 2n − 1. Hence
E ∞ = E (x1 ) ⊗ E (x2i +1 a(4l+2)2 j : 1 i 2n − 1) ⊗ Γ (c 8n+8i +2 : 0 i n − 1) ⊗ Γ (c 8n+8i +6 : 0 i n − 1). Then simple calculation shows that this E ∞ -term has the same size in every total degree as the E 2 -term (2.7) of above Eilenberg–Moore spectral sequence as a graded vector space. This implies that the Eilenberg–Moore spectral sequence collapses at the E 2 -term. So we obtain
H ∗ Ω SO(8n + 2)/U (4n + 1) ; F2 = E (u 4i +1 : 0 i n − 1) ⊗ Γ ( w 8i +6 : n i 2n − 1)
⊗ Γ (u 4i +1 : n i 2n − 1). Hence we get the conclusion.
2
3. The mod p cohomology of loop spaces for odd primes p From now on, p always denotes odd primes. Theorem 3.1. The mod p cohomology of loop spaces of complex Grassmannian spaces is as follows:
H ∗ Ω U (m + n)/ U (m) × U (n) ; F p = H ∗ Ω U (m + n)/U (m) ; F p ⊗ H ∗ U (n); F p
= Γ (a2m+2i : 0 i n − 1) ⊗ E (x2i +1 : 0 i n − 1) where m n.
Y. Choi / Topology and its Applications 160 (2013) 280–291
Proof. By the same method as the mod 2 case, we get the conclusion.
285
2
Theorem 3.2. The mod p cohomology of loop space of complex quadric spaces Q n for n 3 is as follows:
∗
H (Ω Q n ; F p ) =
E (x1 ) ⊗ Γ ( y 2n )
if n is odd,
E (x1 , xn−1 ) ⊗ Γ ( y 2n−2 , yn )
if n is even.
Proof. For an odd prime p, we have the following equivalence [8]:
SO(2n + 1) p Sp(n). Hence we get
H ∗ Ω SO(2n + 1)/SO(2n − 1) ; F p = H ∗ Ω Sp(n)/Sp(n − 1) ; F p
= H ∗ Ω S 4n−1 ; F p
= Γ ( y 4n−2 ). As is known [10, Chap. 3],
H ∗ SO(2n + 2)/SO(2n); F p = E (x2n , x2n+1 ). Consider the Serre spectral sequence the following fibration:
Ω S 2n −→ Ω SO(2n + 2)/SO(2n) −→ Ω S 2n+1 . Then by similar method as the mod 2 case, it collapses at the E 2 -term. Hence we get
H ∗ Ω SO(2n + 2)/SO(2n) ; F p = H ∗ Ω S 2n+1 ; F p ⊗ H ∗ Ω S 2n ; F p
= Γ ( y 2n ) ⊗ E (x2n−1 ) ⊗ Γ ( y 4n−2 ). Consider the Serre spectral sequence converging to H ∗ (Ω Q n ; F p ), n 3, for the fibration
Ω SO(n + 2)/SO(n) −→ Ω Q n −→ S 1 . Then by dimensional reason the Serre spectral sequence collapses at E 2 . Hence we get the conclusion.
2
Theorem 3.3. ([7, Theorem 3.1])
H ∗ Ω Sp(n)/U (n); F p = E (x4i +1 : 0 2i n − 1) ⊗ Γ (x4i +2 : n − 1 2i 2n − 2). Theorem 3.4. The mod p cohomology of loop space of SO(2n)/U (n) is as follows:
H ∗ Ω SO(4n)/U (2n) ; F p = E (u 4i +1 : 0 i n − 1) ⊗ Γ ( w 4i +2 : n − 1 i 2n − 2),
H ∗ Ω SO(4n + 2)/U (2n + 1) ; F p = E (u 4i +1 : 0 i n − 1) ⊗ Γ ( w 4i +2 : n i 2n − 1). Proof. First we consider the stable elements [10, Chap. 4] and [2]:
H ∗ Ω(SO/U ); F p = H ∗ (U /Sp; F p ) = E (u 4i +1 : i 0).
(3.1)
We will prove the theorem by induction. Now we have
H ∗ Ω SO(4)/U (2) ; F p = H ∗ Ω S 2 ; F p = E (u 1 ) ⊗ Γ ( w 2 ). Consider the Serre spectral sequence for the following fibration:
Ω SO(4)/U (2) −→ Ω SO(6)/U (3) −→ Ω S 4 , where H ∗ (Ω S 4 ; F p ) = E (u 3 ) ⊗ Γ ( w 6 ). Then from the information of (3.1), we get the following transgression, d( w 2 ) = u 3 . Hence we obtain
H ∗ Ω SO(6)/U (3) ; F p = E (u 1 ) ⊗ Γ ( w 6 ).
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Suppose that
H ∗ Ω SO(4n)/U (2n) ; F p = E (u 4i +1 : 0 i n − 1) ⊗ Γ ( w 4i +2 : n − 1 i 2n − 2),
H ∗ Ω SO(4n + 2)/U (2n + 1) ; F p = E (u 4i +1 : 0 i n − 1) ⊗ Γ ( w 4i +2 : n i 2n − 1). Consider the Serre spectral sequence for the following fibrations:
Ω SO(4n)/U (2n) −→ Ω SO(4n + 2)/U (2n + 1) −→ Ω S 4n , Ω SO(4n + 2)/U (2n + 1) −→ Ω SO(4n + 4)/U (2n + 2) −→ Ω S 4n+2 .
(3.2) (3.3)
Then from the information of (3.1), we get the following transgression in the Serre spectral sequence for (3.2),
d( w 4n−2 ) = u 4n−1 where H ∗ (Ω S 4n ; F p ) = E (u 4n−1 ) ⊗ Γ ( w 8n−2 ). So we get
H ∗ Ω SO(4n + 2)/U (2n + 1) ; F p = E (u 4i +1 : 0 i n − 1) ⊗ Γ ( w 4i +2 : n i 2n − 1). On the other hand, H ∗ (Ω S 4n+2 ; F p ) = E (u 4n+1 ) ⊗ Γ ( w 8n+2 ) and there are no primitive elements of degree 4n or 8n + 1 in H ∗ (Ω(SO(4n + 2)/U (2n + 1)); F p ). So there are no non-trivial differentials in the Serre spectral sequence for (3.3) because of the degree reason. So we get
H ∗ Ω SO(4n + 4)/U (2n + 2) ; F p = E (u 4i +1 : 0 i n) ⊗ Γ ( w 4i +2 : n − 1 i 2n). Hence we get the conclusion.
2
Remark 3.5. We can rewrite H ∗ (Ω(SO(2n)/U (n)); F p ) as follows:
H ∗ Ω SO(2n)/U (n) ; F p =
E (u 4i +1 : 0 2i n − 2) ⊗ Γ ( w 4i +2 : n − 2 2i 2n − 4) for even n,
E (u 4i +1 : 0 2i n − 3) ⊗ Γ ( w 4i +2 : n − 1 2i 2n − 4) for odd n = E (u 4i +1 : 0 2i n − 2) ⊗ Γ ( w 4i +2 : n − 2 2i 2n − 4).
Note that {i: 2i n − 2} = {i: 2i n − 3}, {i: n − 1 2i } = {i: n − 2 2i } for odd n. Moreover if we check the degrees of generators in the mod 2 cohomology in Theorem 2.6, we can obtain
dimF2 H ∗ Ω SO(2n)/U (n) ; F2
= dimF p H ∗ Ω SO(2n)/U (n) ; F p .
(3.4)
4. Torsion in the integral cohomology of loop spaces Consider the exact couple i
H ∗ ( X ; Z ) −−−−→ H ∗ ( X ; Z ) k
ρ H ∗( X ; Fp )
where i is induced by multiplication by p in Z and ρ is induced by the reduction homomorphism and k is the Bockstein homomorphism in the long exact sequence. From this exact couple we get the Bockstein spectral sequence { E r , dr } with E 1 = H ∗ ( X ; F p ) converging to ( H ∗ ( X ; Z )/torsion) ⊗ F p . For more details, we refer [11]. Note that
dimF p H ∗ ( X ; F p ) dimF p H ∗ ( X ; Z )/torsion ⊗ F p
= dimQ H ∗ ( X ; Q) .
Hence if dimF p ( H ∗ ( X ; F p )) = dimQ ( H ∗ ( X ; Q)), then the Bockstein spectral sequence collapses at E 1 and this implies that H ∗ ( X ; Z ) is p-torsion free. Lemma 4.1. The integral cohomology of loop spaces of complex Grassmannian spaces is torsion free. Proof. Consider the Serre spectral sequence converging to H ∗ (Ω(U (m + n)/(U (m) × U (n))); Q) for the fibration
Ω U (m + n)/U (m) −→ Ω U (m + n)/ U (m) × U (n) −→ U (n)
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287
where H ∗ (Ω(U (m + n)/U (m)); Q) = Q[a2m+2i : 0 i n] and H ∗ (U (n); Q) = E (x2i +1 : 0 i n − 1). By the same method as the mod 2 case, we can show that it collapses at E 2 and obtain
H ∗ Ω U (m + n)/ U (m) × U (n) ; Q = Q[a2m+2i : 0 i n − 1] ⊗ E (x2i +1 : 0 i n − 1). Since
dimF2 H ∗ Ω U (m + n)/U (m) ; F2
= dimF p H ∗ Ω U (m + n)/U (m) ; F p = dimQ H ∗ Ω U (m + n)/U (m) ; Q ,
the integral cohomology of loop spaces of complex Grassmannian spaces is torsion free.
2
Lemma 4.2. The integral cohomology of loop space of Q n for n 3 is torsion free for even n, and has only 2-torsion of order 2 for odd n. Proof. We have the following results from [1, Proposition 10.1],
H ∗ SO(2n + 1)/SO(2n − 1); Q = E (x4n−1 ),
H ∗ SO(2n + 2)/SO(2n); Q = E (x2n , x2n+1 ). By the same method as the mod p case in Theorem 3.2, we get
H ∗ Ω SO(2n + 1)/SO(2n − 1) ; Q = Q[ y 4n−2 ],
H ∗ Ω SO(2n + 2)/SO(2n) ; Q = E (x2n−1 ) ⊗ Q[ y 2n , y 4n−2 ]. So we obtain ∗
H (Ω Q n ; Q) =
E (x1 ) ⊗ Q[ y 2n ]
if n is odd,
E (x1 , xn−1 ) ⊗ Q[ yn , y 2n−2 ]
if n is even.
Then by Theorem 3.2,
dimF p H ∗ (Ω Q n ; F p ) = dimQ H ∗ (Ω Q n ; Q) . This implies that H ∗ (Ω Q n ; Z ) is odd torsion free for each n. Now we turn to the 2-torsion case. Consider the Bockstein spectral sequence with E 1 = H ∗ (Ω Q n ; F2 ) converging to ( H ∗ (Ω Q n ; Z )/torsion) ⊗ F2 . By Theorem 2.2,
d1 (an−1 ) = Sq1 (an−1 ) = Then
E2 =
an
if n is odd,
0
if n is even.
E (a1 ) ⊗ Γ (a2n )
if n is odd,
E (a1 ) ⊗ Γ (an−1 ) ⊗ Γ (an )
if n is even
γ2 (an ) = a2n . Now we have dimF2 ( E 2 ) = dimQ H ∗ (Ω Q n ; Q) .
where we let
It follows that E 2 = E ∞ . Therefore H ∗ (Ω Q n ; Z ) is 2-torsion free for even n, and has only 2-torsion of order 2 for odd n. Lemma 4.3. The integral cohomology of loop space of Sp(n)/U (n) has only 2-torsion of order 2, and is odd torsion free. Proof. First we compute H ∗ (Ω(Sp(2n)/U (2n)); Q) using the following map of fibrations:
Ω Sp(n + 1) −−−−→ Ω Sp(n + 1)/U (n + 1) −−−−→ U (n + 1) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ Ω Sp
−−−−→
Ω Sp/U
−−−−→
U
We can obtain the following basic data from [10, Chap. 4] and [2]:
2
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H ∗ U (n + 1); Q = E (x2i +1 : 0 i n),
H ∗ Ω Sp(n + 1); Q = Q[x4i +2 : 0 i n], H ∗ (Ω Sp; Q) = Q[x4i +2 : i 0],
H ∗ Ω(Sp/U ); Q = H ∗ (U / O ; Q) = E (x4i +1 : i 0). From above data we can get the following transgressions for the Serre spectral sequence for the bottom row fibration:
d4i +3 (x4i +2 ) = x4i +3 ,
i 0.
By the naturality of differentials, we also have the following transgressions for the top row fibration:
d4i +3 (x4i +2 ) = x4i +3 ,
0 2i n − 2.
So we have
H ∗ Ω Sp(n)/U (n); Q = E (x4i +1 : 0 2i n − 1) ⊗ Q[x4i +2 : n − 1 2i 2n − 2]. From Theorem 3.3, we obtain
dimF p H ∗ Ω Sp(n)/U (n) ; F p
= dimQ H ∗ Ω Sp(n)/U (n) ; Q ,
so that H ∗ (Ω(Sp(n)/U (n)); Z ) is odd torsion free. Consider the following fibration: i
U −→ U / O −→ B O . Then i ∗ ( w i ) = xi , i 1 where H ∗ ( B O ; F2 ) = F2 [ w i : i 1]. For more details, we refer [10, Chap. 3, Theorem 6.7]. Since
Sq1 w i = we have
Sq1 xi =
w i +1
if i is even,
0
if i is odd,
x i +1
if i is even,
0
if i is odd H ∗ (Ω(Sp/U ); F
∗ in H ∗ (U / O ; F2 ) = 2 ), and also in H (Ω(Sp(n)/U (n)); F2 ). Consider the Bockstein spectral sequence with E 1 = H ∗ (Ω(Sp(n)/U (n)); F2 ) converging to ( H ∗ (Ω(Sp(n)/U (n)); Z )/ torsion) ⊗ F2 . Then
d1 (xi ) = Sq1 xi =
x i +1
if i is even,
0
if i is odd,
xi +1 γ2k (xi ) if i is even, 1 d1 γ2k+1 (xi ) = Sq γ2k+1 (xi ) =
0
if i is odd
for all k 1. Then
E2 =
E (x4i +1 : 0 2i n − 2) ⊗ Γ (x4i +2 : n 2i 2n − 2)
if n is even,
E (x4i +1 : 0 2i n − 1) ⊗ Γ (x4i +2 : n − 1 2i 2n − 2) if n is odd = E (x4i +1 : 0 2i n − 1) ⊗ Γ (x4i +2 : n − 1 2i 2n − 2).
Note that for even n, {i: 2i n − 2} = {i: 2i n − 1}, {i: n 2i } = {i: n − 1 2i }. Now we have
dimF2 ( E 2 ) = dimQ H ∗ Ω Sp(n)/U (n) ; Q . Hence E 2 = E ∞ , and this implies that H ∗ (Ω(Sp(n)/U (n)); Z ) has 2-torsion of order 2 only.
2
Lemma 4.4. The integral cohomology of loop space of SO(2n)/U (n) is torsion free. Proof. First we will compute the rational cohomology of loop space of SO(2n)/U (n). Then by the same way as the mod 2 case, we consider the Serre spectral sequence converging to H ∗ (Ω(SO(8n + 2i )/U (4n + i )); Q) for the following fibration for i = 1, 2, 3, 4:
Ω0 SO(8n + 2i ) −→ Ω(SO(8n + 2i )/U (4n + i )) −→ U (2n + i ).
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289
Now we will prove the case of i = 1. Since H ∗ (Ω0 SO(8n + 2); F2 ) is even dimensional, so that it is torsion free. So by (2.5) we obtain
ν
H ∗ Ω0 SO(8n + 2); Q = Q[a4i −2 : 1 i n]/ a4ii −2
⊗ Q a4n+2+4k : 0 k (n − 1) ⊗ Q c 8n+2+2k : 0 k (4n − 2), k ≡ 1 mod 4 4 ⊗ Q γ2i (α8n ): i 0 / γ2i (α8n ) .
Consider
Ω0 SO(8n + 2) −−−−→ Ω(SO(8n + 2)/U (4n + 1)) −−−−→ U (4n + 1) ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ Ω0 SO
−−−−→
Ω(SO/U )
−−−−→
U
From (2.6), we can get
H ∗ U (4n + 1); Q = E (x2i +1 : 0 i 4n), H ∗ (U ; Q) = E (x2i +1 : i 0), H ∗ (Ω0 SO; Q) = Q[a4i +2 : i 0],
H ∗ Ω(SO/U ); Q = E (x4i +1 : i 0). Then we can determine differentials in the Serre spectral sequence by the exact same way as the mod 2 case and get
H ∗ Ω SO(8n + 2)/U (4n + 1) ; Q = E (u 4i +1 : 0 i n − 1) ⊗ Q[ w 8n+8i +6 : 0 i n − 1]
⊗ Q[u 4i +1 : n i 2n − 1]. The other cases are similar to the case of i = 1. Then we get
H ∗ Ω SO(2n)/U (n) ; Q = E (u 4i +1 : 0 4i n − 3) ⊗ Q[ w 8i +6 : n − 4 4i 2n − 6]
⊗ Q[u 4i +1 : n − 3 < 4i 2n − 4]. Now from Theorem 2.6 and (3.4) we obtain
dimQ H ∗ Ω SO(2n)/U (n) ; Q
= dimF2 H ∗ Ω SO(2n)/U (n) ; F2 = dimF p H ∗ Ω SO(2n)/U (n) ; F p .
Therefore the integral cohomology of loop space of SO(2n)/U (n) is torsion free.
2
Summarizing above lemmas, we get Theorem 4.5. The integral cohomology of loop spaces of irreducible compact Hermitian symmetric spaces of classical types is torsion free, or has only 2-torsion of order 2. 5. The mod p cohomology of free loop space i
Given a fibration F → E → B, we say that F is totally non-homologous to zero in E with respect to F p if the homomorphism i ∗ : H ∗ ( E ; F p ) → H ∗ ( F ; F p ) is onto. Let Λ M be a space of free loops on M. Even though the free loop space fibration Ω M → Λ M → M has a cross section given by sending each x in M into the constant loop at x, it is well known the fibre Ω M need not be totally non-homologous to zero in Λ M. Hence it is the general question when Ω M is totally non-homologous to zero in Λ M. Following the notation by Kuribayashi, let T ( M ) be a set of prime numbers p such that Ω M is totally non-homologous to zero in Λ M with respect to F p . Theorem 5.1. ([9])
/ T (U (m + n)/U (m) × U (n)) for any prime p. (a) If m, n 2, then p ∈ (b) p ∈ T (Sp(n)/U (n)) iff p = 2.
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Hence the remaining cases are Q n = SO(n + 2)/(SO(2) × SO(n)) and SO(2n)/U (n).
/ T ( Q n ) for n 3. For the case of p = 2, p ∈ T ( Q n ) iff (n + 1)/2 + 1 is a power of 2 when Theorem 5.2. If p is any odd prime, then p ∈ n is odd, and n/2 + 1 is a power of 2 when n is even. Proof. We can extract the following results from [3, §9, 10]:
H ∗ ( Q 2n−1 ; F2 ) = F2 [c 2 ]/ cn2 ⊗ E (a2n ),
H ∗ ( Q 2n ; F2 ) = F2 [c 2 ]/ cn2+1 ⊗ E (a2n ),
H ∗ ( Q 2n−1 ; F p ) = F p [c 2 ]/ c 22n ,
H ∗ ( Q 2n ; F p ) = F p [c 2 ]/ cn2+1 ⊗ E (a2n ). For the case of p = 2. Consider the Serre spectral sequence converging to H ∗ (Λ Q 2n−1 ; F2 ) for Ω Q 2n−1 → Λ Q 2n−1 → Q 2n−1 and H ∗ (Λ Q 2n ; F2 ) for Ω Q 2n → Λ Q 2n → Q 2n . Then by Proposition 3.5 in [12], we get the following non-trivial differentials:
d(a2n−2 ) = ncn2−1 a1 d(a2n ) = (n
for Q 2n−1 if n is not a power of 2,
+ 1)cn2 a1
for Q 2n if n + 1 is not a power of 2.
Consider the remaining cases: Q 2n−1 where n is a power of 2, and Q 2n where n + 1 is a power of 2. Then they satisfy the condition of Theorem 2 in [13], so that Ω Q 2n−1 , Ω Q 2n are totally non-homologous to zero in Λ Q 2n−1 , Λ Q 2n with respect to F2 by Theorem 2 in [13] and Proposition 1.7 in [9]. Now we turn to the case of odd primes. By Proposition 3.5 in [12] we obtain the following differentials in the Serre spectral sequence converging to H ∗ (Λ Q n ; F p ) for Ω Q n → Λ Q n → Q n :
d( y 2n ) = (n + 1)cn2 x1
if n is odd,
d( y 2n−2 ) = 2an xn−1
if n is even.
So p ∈ / T ( Q n ) if p is an odd prime.
2
Theorem 5.3. ([10, Theorem 6.11]) For odd primes p, H ∗ (SO(2n)/U (n); F p ) is
Z [c 2 , . . . , c 2n−2 ]/
(−1)i c 2i c 2 j , k 1 .
i + j =2k
Theorem 5.4. p ∈ T (SO(2n)/U (n)) iff p = 2. Proof. Since H ∗ (SO(2n)/U (n); F2 ) is even dimensional by Theorem 2.4, Sq1 x = 0 for all x ∈ H ∗ (SO(2n)/U (n); F2 ). And by (2.3), H ∗ (SO(2n)/U (n); F2 ) satisfies the condition of Theorem 2 in [13]. Then by Theorem 2 in [13] and Proposition 1.7 in [9], 2 ∈ T (SO(2n)/U (n)). For the case of odd primes, in order to use Proposition 3.5 in [12] we should express H ∗ (SO(2n)/U (n); F p ) as
E ( yl1 , . . . , yli ) ⊗ F p [xn1 , . . . , xn j ]/(ρm1 , . . . , ρmk ) where deg( yl1 ), . . . , deg( yli ) are odd, and deg(xn1 ), . . . , deg(xn j ) are even, and able. Then we can express
ρm1 , . . . , ρmk ∈ F p [x1 , . . . , xn ] are decompos-
H ∗ SO(4n)/U (2n); F p = F p [c 4i +2 : 0 i n − 1]/(ρ4n , . . . , ρ8n−4 ),
H ∗ Ω SO(4n + 2)/U (2n + 1) ; F p = F p [c 4i +4 : 0 i n − 1]/(ρ4n+2 , . . . , ρ8n−4 ), where
ρ4n+4 = c23 − 2c6 c4n−2 +
n −1
g 4n−4k+6 (c 2 , . . . , c 4k−6 )c 4k−2
for SO(4n + 2)/U (2n + 1),
k =1
ρ4n = (2c1 )c4n−2 +
n −1
g 4n−4k+2 (c 2 , . . . , c 4k−6 )c 4k−2
for SO(4n)/U (2n).
k =1
Here g 4n−4k+6 (c 2 , . . . , c 4k−6 ), g 4n−4k+2 (c 2 , . . . , c 4k−6 ) ∈ F p [c 2 , . . . , c 4k−6 ] are homogeneous polynomials of degrees 4n − 4k + 6, 4n − 4k + 2.
Y. Choi / Topology and its Applications 160 (2013) 280–291
291
From the information of Theorem 2.6, we can easily deduce that the Serre spectral sequence for Ω(SO(2n)/U (n)) → ∗ → SO(2n)/U (n) collapses at E 2 . So by Proposition 3.5 in [12] we obtain the following differentials in the Serre spectral sequence converging to H ∗ (Λ(SO(2n)/U (n)); F p ) for Ω(SO(2n)/U (n)) → Λ(SO(2n)/U (n)) → SO(2n)/U (n):
d( w 4n+2 ) = c 23 − 2c 6 u 4n−3 +
n −1
∂ g 4n−4k+6 (c 2 , . . . , c 4k−6 )c 4k−2 /∂ c 4k−2 u 4k−3 k =1
= c 23 − 2c 6 u 4n−3 +
n −1
g 4n−4k+6 (c 2 , . . . , c 4k−6 )u 4k−3
k =1
= 0 for SO(4n + 2)/U (2n + 1), d( w 4n−2 ) = 2c 1 u 4n−3 +
n −1
∂ g 4n−4k+2 (c 2 , . . . , c 4k−6 )c 4k−2 /∂ c 4k−2 u 4k−3 k =1
= 2c 1 u 4n−3 +
n −1
g 4n−4k+2 (c 2 , . . . , c 4k−6 )u 4k−3
k =1
= 0 for SO(4n)/U (2n). Hence p ∈ / T (SO(2n)/U (n)) if p is any odd prime.
2
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