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Homotopy exponents of real Stiefel manifolds
Topology and its Applications 160 (2013) 589–595
Contents lists available at SciVerse ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Homotopy exponents of real Stiefel manifolds ✩ Hao Zhao a , Ping Zhang b,∗ a b
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China
a r t i c l e
i n f o
Article history: Received 18 May 2012 Received in revised form 23 December 2012 Accepted 26 December 2012
a b s t r a c t Let p be an odd prime. For the real Stiefel manifold V m+k,k = SO(m + k)/SO(m), we obtain an upper bound of its p-primary homotopy exponent by decomposing Ω V m+k,k as the product of some simpler spaces for m + k ( p − 1)2 + 2. © 2012 Elsevier B.V. All rights reserved.
MSC: 55P45 55Q52 57T20 Keywords: Real Stiefel manifold Homotopy exponent Homotopy decomposition
1. Introduction One of the central problems in homotopy theory is to determine the homotopy groups of topological spaces. However, computation of homotopy groups is a very difficult problem. The homotopy groups, even for a one-cell complex sphere, are far beyond the reach of current tools. Considering the calculation complexity and the fact that higher homotopy groups are abelian, we were motivated to study the homotopy exponent problem. We first fix some notions and notation. Let p be an odd prime and q = 2( p − 1). Throughout the paper, all spaces have the homotopy types of p-local CW-complexes and all maps are localized at p. The mod p homology and reduced mod p homology of a space X are denoted by H ∗ ( X ) and H ∗ ( X ), respectively. The notation xi ∈ H i ( X ) denotes a generator of H i ( X ) with degree i. The exterior algebra generated by a vector space V over Z/ p Z is denoted by E ( V ). For a space X , we define the p-primary homotopy exponent of a space X , written exp( X ), as pt if t is the minimal power of p that annihilates the p-torsion of π∗ ( X ). In what follows, the term homotopy exponent is used for the p-primary homotopy exponent. We recall some results on homotopy exponents. We define X n to be the n-connected covering of X . Cohen et al. showed that for the sphere S 2n+1 , the power map pn : Ω 2n+1 ( S 2n+1 2n + 1) → Ω 2n+1 ( S 2n+1 2n + 1) is null-homotopic and then follows exp( S 2n+1 ) pn [1]. Since Gray showed that there exists an element of order pn in π∗ ( S 2n+1 ) [3], it follows that exp( S 2n+1 ) = pn . Neisendorfer showed that the power map p r +1 : Ω 2 P m ( p r ) → Ω 2 P m ( p r ) is null-homotopic for a mod p r Moore space P m ( p r ) whose homotopy group contains an element of order p r +1 [10]. This implies that exp( P m ( p r )) = p r +1 . Theriault gave upper bounds for the homotopy exponents of some classical Lie groups of low-rank and two exceptional ✩ H.Z. is supported by NSFC (No. 11101161) and the Research Fund for the Doctoral Program of Higher Education of China (No. 20114407120011). P.Z. is supported by the Fundamental Research Funds for the Central Universities (No. CDJZR11100003). Corresponding author. E-mail addresses: [email protected] (H. Zhao), [email protected] (P. Zhang).
*
0166-8641/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.topol.2012.12.008
590
H. Zhao, P. Zhang / Topology and its Applications 160 (2013) 589–595
groups such as E 7 and E 8 [12–14]. Using homotopy decomposition and constructing spherical fibrations, Grbic and Zhao studied the homotopy exponents of some homogeneous spaces such as low-rank complex Stiefel manifolds and complex Grassmann manifolds, SU (2n)/Sp(n), E 6 / F 4 and F 4 /G 2 [5]. They also gave the corresponding upper bounds for the homotopy exponents of the above homogeneous spaces. Besides the spaces mentioned, there are still a large number of spaces whose homotopy exponents have not been considered. This motivated us to study the homotopy exponents of the real Stiefel manifolds. Before stating our main results, we define some notations. Let a/b denote the maximal integer that is less than a/b. For some positive integer n we define r ni = (n − i − 1)/( p − 1) + 1 with 1 i p − 1 and define the notation n by
n=
n − 1, n ≡ 1 mod 2, n − 2, n ≡ 0 mod 2.
We fix two positive integers m and k that satisfy m + k ( p − 1)2 + 2. Then, for 1 i p − 1 we define j i = i + r m ( p − 1) i
− i )/( p − 1) + 1. Explicit explanations of the above notations are provided in Section 3. and ki = (m +k−m The main results are stated as follows.
Theorem 1.1. Let m + k ( p − 1)2 + 2 and m > 0. Then for the real Stiefel manifold V m+k,k = SO(m + k)/SO(m), the following results hold: 1. When m + k is odd, let i 0 with 1 i 0 ( p − 1)/2 satisfy 2 j 2i 0 −1 + (k2i 0 −1 − 1)q + 1 = 2(m + k) − 3. Then we have: ¯
(a) If j 2i 0 −1 + (t − 1)( p − 1) ≡ 0 mod p for every 1 t k2i 0 −1 − 1, then exp( V m+k,k ) pm+k+k2i0 −1 −3 ; and m+k+k2i 0 −1 −4
(b) If j 2i 0 −1 + (t − 1)( p − 1) ≡ 0 mod p for some 1 t k2i 0 −1 − 1, then exp( V m+k,k ) p . 2. When m + k is even, let i 0 with 1 i 0 ( p − 1)/2 satisfy 2 j 2i 0 −1 + (k2i 0 −1 − 1)q + 1 = 2(m + k) − 5. Then we have: (a) If j 2i 0 −1 + (t − 1)( p − 1) ≡ 0 mod p for every 1 t k2i 0 −1 − 1, then exp( V m+k,k ) pm+k+k2i0 −1 −4 ; and
(b) If j 2i 0 −1 + (t − 1)( p − 1) ≡ 0 mod p for some 1 t k2i 0 −1 − 1, then exp( V m+k,k ) pm+k+k2i0 −1 −5 .
The remainder of the paper is organized as follows. First we recall some related knowledge on finite H -spaces in Section 2. The homotopy decomposition of Ω V m+k,k under some dimension is given in Section 3. Combining the decomposition in Section 3 with the results obtained in [5], we give the proof of Theorem 1.1. 2. Some preliminaries on finite H -spaces For a given space X , Selick and Wu showed that there exists a natural decomposition [11]
ΩΣ X A min ( X ) × B max ( X ), where A min ( X ) is the natural minimal retract whose mod p homology contains H ∗ ( X ). As retracts of ΩΣ X , A min ( X ) and max B ( X ) are both H -spaces. Let λ be the composite r
E X λ: X − → ΩΣ X −→ A min ( X ),
where E is the adjoint of the identity id : Σ X → Σ X and r is the natural retract. Then λ induces an injection in homology. Let X l denote the p-localization of a space with a CW-complex structure S n1 ∪ en2 ∪ · · · ∪ enl , where n1 n2 · · · nl and each ni is odd. For 1 l < p − 1, Cohen and Neisendorfer showed that A min ( X l ) (they used the notation M ( X l )) is a finite H -space and H ∗ ( A min ( X l )) ∼ H ∗ ( X l )) [2]. When l = p − 1, A min ( X p −1 ) is an infinite H -space and a similar finite = E ( p −1 H -space for 1 l < p − 1 does not exist in general [4]. For X p −1 , we define b = i =1 ni . Wu proved the following theorem on A min ( X p −1 ) [15].
Theorem 2.1. ([15]) For the space X p −1 , there is a fiber sequence
P p −1 E p min p −1 H p min b p −1 Σ X Ω A min Σ b X p −1 − →E X −→ A X −−→ A such that the following properties hold: 1. H ∗ ( A min ( X p −1 )) ∼ = H ∗ ( E ( X p −1 )) ⊗ H ∗ ( A min (Σ b X p −1 )) as coalgebras; 2. H ∗ ( E ( X p −1 )) ∼ H ∗ ( X p −1 )) as coalgebras; and = E ( 3. E p induces an injection and H p induces an epimorphism in homology. In the above theorem, E ( X p −1 ) is finite but might not be an H -space. It has been proved that there exists a map λ : X p −1 → E ( X p −1 ) that induces an injection in homology [5]. For 0 l p − 1, we define X 0 := ∗ and
H. Zhao, P. Zhang / Topology and its Applications 160 (2013) 589–595
l
M X
:=
⎧ ⎨ ∗, ⎩
591
l = 0,
A min ( X l ),
E(X
p −1
1 l < p − 1,
), l = p − 1.
For 1 j l, we denote the cofiber of the inclusion X j −1 → X l by X lj . Then we have the following theorem. Theorem 2.2. ([5]) For l p − 1, the cofibration X j −1 → X l → X lj induces a fibration
M X j −1 → M X l → M X lj
(1)
and the following commutative diagram holds
X j −1 λ
M ( X j −1 )
Xl
X lj
λ
λ
(2)
M ( X lj ).
M( Xl)
3. Homotopy decomposition H ∗ (ΣC P n−1 ) = Z/ p Z{x3 , x5 , . . . , x2n−1 }. There is a Steenrod operation on The space ΣC P n−1 has reduced homology n−1 H ∗ (ΣC P ) given by
P j (x2r +1 ) = r !/ j ! · (r − j )! · x2r + jq+1 (2 r n − 1).
(3)
It has been shown that there exists a wedge decomposition [8]
ΣC P n−1
p −1
A i (n),
(4)
i =1
where A i (n) has a reduced homology
H ∗ A i (n) = Z/ p Z x2i +1 , x2i +q+1 , . . . , x2i +(rn −1)q+1 r ni = (n − i − 1)/( p − 1) + 1 i that inherits a Steenrod operation from H ∗ (ΣC P n−1 ). Suppose that m > 0; then the cofibration m+k−1 ΣC P m−1 → ΣC P m+k−1 → ΣC P m
and the wedge decomposition (4) induce the following wedge decomposition: m+k−1 ΣC P m
p −1
i =1
k¯ ji
A ¯i ,
(5)
k¯ ji
k¯ ji
where A ¯ i is defined by the cofibration A i (m) → A i (m + k) → A ¯ i and its reduced homology is
k¯ ¯ H ∗ A ¯ i = Z/ p Z x2 ¯j +1 , x2 ¯j +q+1 , . . . , x2 ¯j +(k¯ −1)q+1 ¯j i = i + r m i ( p − 1), k i = (k − i )/( p − 1) + 1 . i i i i ji
+k Since for m + k ( p − 1)2 + 1, r m , rm and k¯ i do not exceed p − 1, it follows from Theorem 2.2 that the cofibration i i k¯
A i (m) → A i (m + k) → A ¯ i induces the following fibration: ji
fi
gi
k¯
M A i (m) − → M A i (m + k) − → M A ¯i . ji
(6)
In what follows we apply the construction M ( A i (n)) to give a homotopy decomposition of Ω V m+k,k under some dimensional restriction. We know that [9]
H ∗ SO(2n + 1) = E (x3 , x7 , . . . , x4n−1 )
(7)
and
H ∗ SO(2n + 2) = E (x3 , x7 , . . . , x4n−1 , x¯ 2n−1 ).
(8)
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H. Zhao, P. Zhang / Topology and its Applications 160 (2013) 589–595
Moreover, there exist the homotopy decompositions [6,9]
SO(2n + 2) S 2n+1 × SO(2n + 1)
(9)
and
SU (2n) SO(2n + 1) × SU (2n)/Sp(n).
(10)
Then, for m + k ( p − 1)2 + 2 we have m + k ( p − 1)2 + 1. This implies a fibration similar to (6) as
M A i ( m) − → M A i (m + k) − → M A ji , fi
gi
k
(11)
i
− i )/( p − 1) + 1. where j i = i + r m ( p − 1) and ki = (m +k−m i
Theorem 3.1. Let m + k ( p − 1)2 + 2. Then for the real Stiefel manifold V m+k,k = SO(m + k)/SO(m), there exists a homotopy decomposition
⎧ ( p −1)/2 k ⎪ Ω M ( A j 2i2i−−11 ), ⎪ S m−1 × Ω S m+k−1 × i =1 ⎪ ⎪ ⎪ ⎪ 1)/2 k ⎪ ⎨ S m−1 × (i =p − Ω M ( A j 2i2i−−11 ), 1 Ω V m+k,k ( p −1)/2 k ⎪ ⎪ Ω S m+k−1 × i =1 Ω M ( A j 2i2i−−11 ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ( p −1)/2 Ω M ( Ak2i−1 ), i =1
j 2i −1
m ≡ 0, k ≡ 0 mod 2, m ≡ 0, k ≡ 1 mod 2, m ≡ 1, k ≡ 1 mod 2, m ≡ 1, k ≡ 0 mod 2.
Proof. We know that there exists a canonical map ΣC P 2n−1 → SU (2n) that induces an injection in homology [9]. By (2n) decompositions (4) and (10), we define a map j 2i −1 as the composite
j 2i −1 : A 2i −1 (2n) → ΣC P 2n−1 → SU (2n) → SO(2n + 1), (2n)
(2n)
where the middle map is the canonical map and the rightmost map is a projection. We use the same notation j 2i −1 as above (2n) j 2i −1
to denote the composite A 2i −1 (2n) −−−→ SO(2n + 1) → SO(2n + 2) (the context makes the difference). By decompositions (2n) (4), (9) and (10), the map j 2i −1 induces an injection in homology. Since SO(2n + 1) and SO(2n + 2) are loop spaces on their classifying spaces, by the universal property of James construction [7] there is an H -map
¯j (2n) : ΩΣ A 2i −1 (2n) → SO(2n + 1) or ¯j (2n) : ΩΣ A 2i −1 (2n) → SO(2n + 2) 2i −1 2i −1 (2n) (2n) such that j 2i −1 ¯j 2i −1 ◦ E. Then we obtain the following commutative diagram: f 2i −1
M ( A 2i −1 ( m))
M ( A 2i −1 (m + k))
ΩΣ A 2i −1 (m + k)
ΩΣ A 2i −1 ( m)
(12)
m+k) ¯j (
m) ¯j (
2i −1
2i −1
SO(m + k)
SO(m)
where the commutativity of the upper square is due to the naturality of the constructions M ( A 2i −1 ( m)) and M ( A 2i −1 (m +k)), and the commutativity of the lower square is due to the naturality of the James construction [7]. In the above diagram, we ( m+k)
m)) → SO(m) and use ˜j 2i −1 : M ( A 2i −1 (m + k)) → SO(m + k) to denote the composite in the left column by ˜j 2i −1 : M ( A 2i −1 ( denote the composite in the right column. Thus, the following commutative diagram holds ( m)
( p −1)/2 i =1
m)) M ( A 2i −1 (
Π f 2i −1
( p −1)/2 i =1
M ( A 2i −1 (m + k)) ( m+k)
Π ˜j 2i −1 ( m)
( p −1)/2 i =1
SO(m)
μ
SO(m)
( p −1)/2 i =1
Π ˜j 2i −1
SO(m + k) μ
SO(m + k)
(13)
H. Zhao, P. Zhang / Topology and its Applications 160 (2013) 589–595
593
where μ is the multiplication on SO(n). We write the maps in the left and right columns in the above diagram as φ and ψ , respectively. Then we obtain the following simplified commutative diagram:
( p −1)/2
M ( A 2i −1 ( m))
i =1
Π f 2i −1
( p −1)/2 i =1
M ( A 2i −1 (m + k)) (14)
ψ
φ
SO(m + k).
SO(m) Let hn be the composite ∂ ◦ E : S
n−1
→ Ω S → SO(n), where ∂ is the connecting map of the fibration sequence n
∂ Ω Sn − → SO(n) → SO(n + 1) → S n .
(15)
We define X (n) as
X (n) :=
n ≡ 1 mod 2,
∗, S
n −1
, n ≡ 0 mod 2.
When n is even we define the map ϕn : X (n) → SO(n) as hn and let sequence (15) follows the commutative diagram
X (m)
θ
ϕn be trivial when n is odd. Thus, the exactness of fiber
X (m + k)
ϕm
ϕm+k
SO(m)
SO(m + k)
(16)
where θ is a trivial map. Let X be the homotopy fiber of θ . Then we obtain the fibration θ X → X (m) − → X (m + k)
(17)
from which it follows that X = X (m) × Ω X (m + k). Multiplying (14) by (16) gives the following commutative diagram:
X×
p −1 i =1
k
Ω M ( A j 2i2i−−11 )
X (m) ×
( p −1)/2 i =1
M ( A 2i −1 ( m))
ϕm ×φ h
SO(m) × SO(m) μ
Ω V m+k,k
SO(m)
θ ×Π f 2i −1
X (m + k) ×
( p −1)/2 i =1
M ( A 2i −1 (m + k))
ϕm+k ×ψ
SO(m + k) × SO(m + k) μ
SO(m + k) (18)
where the top horizontal sequence is the multiplication of fibrations (11) and (17) and the bottom horizontal sequence is the principal fibration induced by
SO(m) → SO(m + k) → V m+k,k . The map h is the induced map between the fibers. By the homology of SO(m) and SO(m + k) given in (7) and (8) and the definitions of X (m) and X (m + k), we see that the right two column maps induce coalgebra homomorphisms in homology that are isomorphisms on the sets of generators of odd degree. Dualizing to cohomology algebra, the dualities of homology generators become the cohomology algebra generators of the same odd degree. Thus, the corresponding two column maps induce algebra homomorphisms that are isomorphisms on the sets of generators. In addition, the coalgebra homomorphisms in homology are dualized to algebra homomorphisms in cohomology. It follows that these two homomorphisms in cohomology are isomorphisms. Thus, the right two-column maps are homotopy equivalences. From the Steenrod 5-lemma we see that h is a homotopy equivalence. The definitions of X (m) and X (m + k) give the stated homotopy decomposition in Theorem 3.1. 2 4. Proof of Theorem 1.1 We prove Theorem 1.1 by virtue of the homotopy decomposition given in Theorem 3.1. Recall that ¯j i = i + r m i ( p − 1) and
k¯ i = (k − i )/( p − 1) + 1 for 1 i p − 1. We first introduce a result on the upper bound for the homotopy exponent of k¯ ji
M ( A ¯ i ) given in [5]. By the cofibration
594
H. Zhao, P. Zhang / Topology and its Applications 160 (2013) 589–595
m+k−1 ΣC P m−1 → ΣC P m+k−1 → ΣC P m , m+k−1 the reduced cohomology of ΣC P m ,
m+k−1 = Z/ p {x2m+1 , x2m+3 , . . . , x2(m+k)−1 }, H ∗ ΣC P m
inherits a Steenrod operation from H ∗ (ΣC P m+k−1 ), which is dual to (3). In particular, we have P j (x2(m+k)−1 ) = 0 for m+k−1 j > 0 and P 1 (x2r +1 ) = r · x2r +q+1 for m r m + k − 1. By the wedge decomposition ΣC P m k¯ ji
p −1 i =1
k¯ ji
A ¯ i , the reduced
cohomology of A ¯ i ,
k¯ H ∗ A ¯ i = Z/ p {x2 ¯j +1 , x2 ¯j +q+1 , . . . , x2 ¯j +(k¯ −1)q+1 }, i i i i ji
m+k−1 also inherits a corresponding Steenrod operation from H ∗ (ΣC P m ). In particular, for 1 t k¯ i − 1 we have
P 1 (x2 ¯j i +(t −1)q+1 ) = ¯j i + (t − 1)( p − 1) · x2 ¯j i +tq+1 .
When m + k ( p − 1)2 + 2, we have k¯ i p − 1, from which it follows that 1 t p − 2. Thus, there exists at most one t ¯
k H ∗( A ¯i ) such that ¯j i + (t − 1)( p − 1) ≡ 0 mod p. It follows that on the set of generators of ji
{x2 ¯j i +1 , x2 ¯j i +q+1 , . . . , x2 ¯j i +(k¯ i −2)q+1 }, P 1 acts trivially on at most one generator. Lemma 4.1. For m + k ( p − 1)2 + 1, let 1 i p − 1 and 1 t k¯ i − 1. Then: ¯ ¯ k¯ 1. If for every t there is ¯j i + (t − 1)( p − 1) ≡ 0 mod p, then exp( M ( A ¯ i )) p j i +(ki −1) p . ji
¯ ¯ k¯ 2. If there exists some t such that ¯j i + (t − 1)( p − 1) ≡ 0 mod p, then exp( M ( A ¯ i )) p j i +(ki −1) p −1 . ji
In the following we give the proof of Theorem 1.1 by Theorem 3.1 and Lemma 4.1. Proof of Theorem 1.1. We only give the proof of (1). The proof of (2) is similar. From Theorem 3.1 we have the homotopy decomposition
Ω V m+k,k X×
( p− 1)/2 i =1
k Ω M A j 2i2i−−11 ,
where the definition of X is given by (16). Thus, by Lemma 4.1 it follows that
1 i ( p − 1)/2 k max p (m+k)/2 , exp M A j 2i2i−−11 1 i ( p − 1)/2
max p (m+k)/2 , p j 2i−1 +(k2i−1 −1) p 1 i ( p − 1)/2
= max p j 2i−1 +(k2i−1 −1) p 1 i ( p − 1)/2 .
exp( V m+k,k ) = max exp( X ), exp M A j 2i−1 k
2i −1
(1)(a) If for every 1 t k2i 0 −1 − 1 there is j 2i 0 −1 + (t − 1)( p − 1) ≡ 0 mod p, then by Theorem 3.1 we obtain k2i 0 −1
exp( M ( A j
2i 0 −1
)) p j 2i0 −1 +(k2i0 −1 −1) p . For 1 i ( p − 1)/2, if i = i 0 , then from the definitions of j i and ki it follows that
j 2i 0 −1 + (k2i 0 −1 − 1) p − j 2i −1 + (k2i −1 − 1) p
= ( j 2i 0 −1 − j 2i −1 ) + (k2i 0 −1 − k2i −1 ) p = ( j 2i 0 −1 − j 2i −1 ) + (k2i 0 −1 − k2i −1 ) p m m = 2(i 0 − i ) + r2i − r2i −1 ( p − 1) + (k2i 0 −1 − k2i −1 ) p 0 −1 > 0. From the above it follows that p j 2i−1 +(k2i−1 −1) p < p j 2i0 −1 +(k2i0 −1 −1) p . Thus, we have
exp( V m+k,k ) p j 2i0 −1 +(k2i0 −1 −1) p
H. Zhao, P. Zhang / Topology and its Applications 160 (2013) 589–595
595
or, equivalently,
exp( V m+k,k ) pm+k+k2i0 −1 −3 by 2 j 2i 0 −1 + (k2i 0 −1 − 1)q + 1 = 2(m + k) − 3. (1)(b) If there exists some 1 t k2i 0 −1 − 1 such that j 2i 0 −1 + (t − 1)( p − 1) ≡ 0 mod p, then by Theorem 3.1 we obtain k2i 0 −1
exp( M ( A j
2i 0 −1
)) p j 2i0 −1 +(k2i0 −1 −1) p −1 . According to the proof of (1)(a), for 1 i ( p − 1)/2, if i = i 0 , then
k
exp M A j 2i−1 2i −1
p j 2i−1 +(k2i−1 −1) p p j 2i0 −1 +(k2i0 −1 −1) p −1 .
Thus,
exp( V m+k,k ) p j 2i0 −1 +(k2i0 −1 −1) p −1 or, equivalently,
exp( V m+k,k ) pm+k+k2i0 −1 −4 , by 2 j 2i 0 −1 + (k2i 0 −1 − 1)q + 1 = 2(m + k) − 5.
2
Acknowledgement We are grateful to the anonymous referee for many helpful suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
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Contents lists available at SciVerse ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Homotopy exponents of real Stiefel manifolds ✩ Hao Zhao a , Ping Zhang b,∗ a b
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China
a r t i c l e
i n f o
Article history: Received 18 May 2012 Received in revised form 23 December 2012 Accepted 26 December 2012
a b s t r a c t Let p be an odd prime. For the real Stiefel manifold V m+k,k = SO(m + k)/SO(m), we obtain an upper bound of its p-primary homotopy exponent by decomposing Ω V m+k,k as the product of some simpler spaces for m + k ( p − 1)2 + 2. © 2012 Elsevier B.V. All rights reserved.
MSC: 55P45 55Q52 57T20 Keywords: Real Stiefel manifold Homotopy exponent Homotopy decomposition
1. Introduction One of the central problems in homotopy theory is to determine the homotopy groups of topological spaces. However, computation of homotopy groups is a very difficult problem. The homotopy groups, even for a one-cell complex sphere, are far beyond the reach of current tools. Considering the calculation complexity and the fact that higher homotopy groups are abelian, we were motivated to study the homotopy exponent problem. We first fix some notions and notation. Let p be an odd prime and q = 2( p − 1). Throughout the paper, all spaces have the homotopy types of p-local CW-complexes and all maps are localized at p. The mod p homology and reduced mod p homology of a space X are denoted by H ∗ ( X ) and H ∗ ( X ), respectively. The notation xi ∈ H i ( X ) denotes a generator of H i ( X ) with degree i. The exterior algebra generated by a vector space V over Z/ p Z is denoted by E ( V ). For a space X , we define the p-primary homotopy exponent of a space X , written exp( X ), as pt if t is the minimal power of p that annihilates the p-torsion of π∗ ( X ). In what follows, the term homotopy exponent is used for the p-primary homotopy exponent. We recall some results on homotopy exponents. We define X n to be the n-connected covering of X . Cohen et al. showed that for the sphere S 2n+1 , the power map pn : Ω 2n+1 ( S 2n+1 2n + 1) → Ω 2n+1 ( S 2n+1 2n + 1) is null-homotopic and then follows exp( S 2n+1 ) pn [1]. Since Gray showed that there exists an element of order pn in π∗ ( S 2n+1 ) [3], it follows that exp( S 2n+1 ) = pn . Neisendorfer showed that the power map p r +1 : Ω 2 P m ( p r ) → Ω 2 P m ( p r ) is null-homotopic for a mod p r Moore space P m ( p r ) whose homotopy group contains an element of order p r +1 [10]. This implies that exp( P m ( p r )) = p r +1 . Theriault gave upper bounds for the homotopy exponents of some classical Lie groups of low-rank and two exceptional ✩ H.Z. is supported by NSFC (No. 11101161) and the Research Fund for the Doctoral Program of Higher Education of China (No. 20114407120011). P.Z. is supported by the Fundamental Research Funds for the Central Universities (No. CDJZR11100003). Corresponding author. E-mail addresses: [email protected] (H. Zhao), [email protected] (P. Zhang).
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0166-8641/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.topol.2012.12.008
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H. Zhao, P. Zhang / Topology and its Applications 160 (2013) 589–595
groups such as E 7 and E 8 [12–14]. Using homotopy decomposition and constructing spherical fibrations, Grbic and Zhao studied the homotopy exponents of some homogeneous spaces such as low-rank complex Stiefel manifolds and complex Grassmann manifolds, SU (2n)/Sp(n), E 6 / F 4 and F 4 /G 2 [5]. They also gave the corresponding upper bounds for the homotopy exponents of the above homogeneous spaces. Besides the spaces mentioned, there are still a large number of spaces whose homotopy exponents have not been considered. This motivated us to study the homotopy exponents of the real Stiefel manifolds. Before stating our main results, we define some notations. Let a/b denote the maximal integer that is less than a/b. For some positive integer n we define r ni = (n − i − 1)/( p − 1) + 1 with 1 i p − 1 and define the notation n by
n=
n − 1, n ≡ 1 mod 2, n − 2, n ≡ 0 mod 2.
We fix two positive integers m and k that satisfy m + k ( p − 1)2 + 2. Then, for 1 i p − 1 we define j i = i + r m ( p − 1) i
− i )/( p − 1) + 1. Explicit explanations of the above notations are provided in Section 3. and ki = (m +k−m The main results are stated as follows.
Theorem 1.1. Let m + k ( p − 1)2 + 2 and m > 0. Then for the real Stiefel manifold V m+k,k = SO(m + k)/SO(m), the following results hold: 1. When m + k is odd, let i 0 with 1 i 0 ( p − 1)/2 satisfy 2 j 2i 0 −1 + (k2i 0 −1 − 1)q + 1 = 2(m + k) − 3. Then we have: ¯
(a) If j 2i 0 −1 + (t − 1)( p − 1) ≡ 0 mod p for every 1 t k2i 0 −1 − 1, then exp( V m+k,k ) pm+k+k2i0 −1 −3 ; and m+k+k2i 0 −1 −4
(b) If j 2i 0 −1 + (t − 1)( p − 1) ≡ 0 mod p for some 1 t k2i 0 −1 − 1, then exp( V m+k,k ) p . 2. When m + k is even, let i 0 with 1 i 0 ( p − 1)/2 satisfy 2 j 2i 0 −1 + (k2i 0 −1 − 1)q + 1 = 2(m + k) − 5. Then we have: (a) If j 2i 0 −1 + (t − 1)( p − 1) ≡ 0 mod p for every 1 t k2i 0 −1 − 1, then exp( V m+k,k ) pm+k+k2i0 −1 −4 ; and
(b) If j 2i 0 −1 + (t − 1)( p − 1) ≡ 0 mod p for some 1 t k2i 0 −1 − 1, then exp( V m+k,k ) pm+k+k2i0 −1 −5 .
The remainder of the paper is organized as follows. First we recall some related knowledge on finite H -spaces in Section 2. The homotopy decomposition of Ω V m+k,k under some dimension is given in Section 3. Combining the decomposition in Section 3 with the results obtained in [5], we give the proof of Theorem 1.1. 2. Some preliminaries on finite H -spaces For a given space X , Selick and Wu showed that there exists a natural decomposition [11]
ΩΣ X A min ( X ) × B max ( X ), where A min ( X ) is the natural minimal retract whose mod p homology contains H ∗ ( X ). As retracts of ΩΣ X , A min ( X ) and max B ( X ) are both H -spaces. Let λ be the composite r
E X λ: X − → ΩΣ X −→ A min ( X ),
where E is the adjoint of the identity id : Σ X → Σ X and r is the natural retract. Then λ induces an injection in homology. Let X l denote the p-localization of a space with a CW-complex structure S n1 ∪ en2 ∪ · · · ∪ enl , where n1 n2 · · · nl and each ni is odd. For 1 l < p − 1, Cohen and Neisendorfer showed that A min ( X l ) (they used the notation M ( X l )) is a finite H -space and H ∗ ( A min ( X l )) ∼ H ∗ ( X l )) [2]. When l = p − 1, A min ( X p −1 ) is an infinite H -space and a similar finite = E ( p −1 H -space for 1 l < p − 1 does not exist in general [4]. For X p −1 , we define b = i =1 ni . Wu proved the following theorem on A min ( X p −1 ) [15].
Theorem 2.1. ([15]) For the space X p −1 , there is a fiber sequence
P p −1 E p min p −1 H p min b p −1 Σ X Ω A min Σ b X p −1 − →E X −→ A X −−→ A such that the following properties hold: 1. H ∗ ( A min ( X p −1 )) ∼ = H ∗ ( E ( X p −1 )) ⊗ H ∗ ( A min (Σ b X p −1 )) as coalgebras; 2. H ∗ ( E ( X p −1 )) ∼ H ∗ ( X p −1 )) as coalgebras; and = E ( 3. E p induces an injection and H p induces an epimorphism in homology. In the above theorem, E ( X p −1 ) is finite but might not be an H -space. It has been proved that there exists a map λ : X p −1 → E ( X p −1 ) that induces an injection in homology [5]. For 0 l p − 1, we define X 0 := ∗ and
H. Zhao, P. Zhang / Topology and its Applications 160 (2013) 589–595
l
M X
:=
⎧ ⎨ ∗, ⎩
591
l = 0,
A min ( X l ),
E(X
p −1
1 l < p − 1,
), l = p − 1.
For 1 j l, we denote the cofiber of the inclusion X j −1 → X l by X lj . Then we have the following theorem. Theorem 2.2. ([5]) For l p − 1, the cofibration X j −1 → X l → X lj induces a fibration
M X j −1 → M X l → M X lj
(1)
and the following commutative diagram holds
X j −1 λ
M ( X j −1 )
Xl
X lj
λ
λ
(2)
M ( X lj ).
M( Xl)
3. Homotopy decomposition H ∗ (ΣC P n−1 ) = Z/ p Z{x3 , x5 , . . . , x2n−1 }. There is a Steenrod operation on The space ΣC P n−1 has reduced homology n−1 H ∗ (ΣC P ) given by
P j (x2r +1 ) = r !/ j ! · (r − j )! · x2r + jq+1 (2 r n − 1).
(3)
It has been shown that there exists a wedge decomposition [8]
ΣC P n−1
p −1
A i (n),
(4)
i =1
where A i (n) has a reduced homology
H ∗ A i (n) = Z/ p Z x2i +1 , x2i +q+1 , . . . , x2i +(rn −1)q+1 r ni = (n − i − 1)/( p − 1) + 1 i that inherits a Steenrod operation from H ∗ (ΣC P n−1 ). Suppose that m > 0; then the cofibration m+k−1 ΣC P m−1 → ΣC P m+k−1 → ΣC P m
and the wedge decomposition (4) induce the following wedge decomposition: m+k−1 ΣC P m
p −1
i =1
k¯ ji
A ¯i ,
(5)
k¯ ji
k¯ ji
where A ¯ i is defined by the cofibration A i (m) → A i (m + k) → A ¯ i and its reduced homology is
k¯ ¯ H ∗ A ¯ i = Z/ p Z x2 ¯j +1 , x2 ¯j +q+1 , . . . , x2 ¯j +(k¯ −1)q+1 ¯j i = i + r m i ( p − 1), k i = (k − i )/( p − 1) + 1 . i i i i ji
+k Since for m + k ( p − 1)2 + 1, r m , rm and k¯ i do not exceed p − 1, it follows from Theorem 2.2 that the cofibration i i k¯
A i (m) → A i (m + k) → A ¯ i induces the following fibration: ji
fi
gi
k¯
M A i (m) − → M A i (m + k) − → M A ¯i . ji
(6)
In what follows we apply the construction M ( A i (n)) to give a homotopy decomposition of Ω V m+k,k under some dimensional restriction. We know that [9]
H ∗ SO(2n + 1) = E (x3 , x7 , . . . , x4n−1 )
(7)
and
H ∗ SO(2n + 2) = E (x3 , x7 , . . . , x4n−1 , x¯ 2n−1 ).
(8)
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H. Zhao, P. Zhang / Topology and its Applications 160 (2013) 589–595
Moreover, there exist the homotopy decompositions [6,9]
SO(2n + 2) S 2n+1 × SO(2n + 1)
(9)
and
SU (2n) SO(2n + 1) × SU (2n)/Sp(n).
(10)
Then, for m + k ( p − 1)2 + 2 we have m + k ( p − 1)2 + 1. This implies a fibration similar to (6) as
M A i ( m) − → M A i (m + k) − → M A ji , fi
gi
k
(11)
i
− i )/( p − 1) + 1. where j i = i + r m ( p − 1) and ki = (m +k−m i
Theorem 3.1. Let m + k ( p − 1)2 + 2. Then for the real Stiefel manifold V m+k,k = SO(m + k)/SO(m), there exists a homotopy decomposition
⎧ ( p −1)/2 k ⎪ Ω M ( A j 2i2i−−11 ), ⎪ S m−1 × Ω S m+k−1 × i =1 ⎪ ⎪ ⎪ ⎪ 1)/2 k ⎪ ⎨ S m−1 × (i =p − Ω M ( A j 2i2i−−11 ), 1 Ω V m+k,k ( p −1)/2 k ⎪ ⎪ Ω S m+k−1 × i =1 Ω M ( A j 2i2i−−11 ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ( p −1)/2 Ω M ( Ak2i−1 ), i =1
j 2i −1
m ≡ 0, k ≡ 0 mod 2, m ≡ 0, k ≡ 1 mod 2, m ≡ 1, k ≡ 1 mod 2, m ≡ 1, k ≡ 0 mod 2.
Proof. We know that there exists a canonical map ΣC P 2n−1 → SU (2n) that induces an injection in homology [9]. By (2n) decompositions (4) and (10), we define a map j 2i −1 as the composite
j 2i −1 : A 2i −1 (2n) → ΣC P 2n−1 → SU (2n) → SO(2n + 1), (2n)
(2n)
where the middle map is the canonical map and the rightmost map is a projection. We use the same notation j 2i −1 as above (2n) j 2i −1
to denote the composite A 2i −1 (2n) −−−→ SO(2n + 1) → SO(2n + 2) (the context makes the difference). By decompositions (2n) (4), (9) and (10), the map j 2i −1 induces an injection in homology. Since SO(2n + 1) and SO(2n + 2) are loop spaces on their classifying spaces, by the universal property of James construction [7] there is an H -map
¯j (2n) : ΩΣ A 2i −1 (2n) → SO(2n + 1) or ¯j (2n) : ΩΣ A 2i −1 (2n) → SO(2n + 2) 2i −1 2i −1 (2n) (2n) such that j 2i −1 ¯j 2i −1 ◦ E. Then we obtain the following commutative diagram: f 2i −1
M ( A 2i −1 ( m))
M ( A 2i −1 (m + k))
ΩΣ A 2i −1 (m + k)
ΩΣ A 2i −1 ( m)
(12)
m+k) ¯j (
m) ¯j (
2i −1
2i −1
SO(m + k)
SO(m)
where the commutativity of the upper square is due to the naturality of the constructions M ( A 2i −1 ( m)) and M ( A 2i −1 (m +k)), and the commutativity of the lower square is due to the naturality of the James construction [7]. In the above diagram, we ( m+k)
m)) → SO(m) and use ˜j 2i −1 : M ( A 2i −1 (m + k)) → SO(m + k) to denote the composite in the left column by ˜j 2i −1 : M ( A 2i −1 ( denote the composite in the right column. Thus, the following commutative diagram holds ( m)
( p −1)/2 i =1
m)) M ( A 2i −1 (
Π f 2i −1
( p −1)/2 i =1
M ( A 2i −1 (m + k)) ( m+k)
Π ˜j 2i −1 ( m)
( p −1)/2 i =1
SO(m)
μ
SO(m)
( p −1)/2 i =1
Π ˜j 2i −1
SO(m + k) μ
SO(m + k)
(13)
H. Zhao, P. Zhang / Topology and its Applications 160 (2013) 589–595
593
where μ is the multiplication on SO(n). We write the maps in the left and right columns in the above diagram as φ and ψ , respectively. Then we obtain the following simplified commutative diagram:
( p −1)/2
M ( A 2i −1 ( m))
i =1
Π f 2i −1
( p −1)/2 i =1
M ( A 2i −1 (m + k)) (14)
ψ
φ
SO(m + k).
SO(m) Let hn be the composite ∂ ◦ E : S
n−1
→ Ω S → SO(n), where ∂ is the connecting map of the fibration sequence n
∂ Ω Sn − → SO(n) → SO(n + 1) → S n .
(15)
We define X (n) as
X (n) :=
n ≡ 1 mod 2,
∗, S
n −1
, n ≡ 0 mod 2.
When n is even we define the map ϕn : X (n) → SO(n) as hn and let sequence (15) follows the commutative diagram
X (m)
θ
ϕn be trivial when n is odd. Thus, the exactness of fiber
X (m + k)
ϕm
ϕm+k
SO(m)
SO(m + k)
(16)
where θ is a trivial map. Let X be the homotopy fiber of θ . Then we obtain the fibration θ X → X (m) − → X (m + k)
(17)
from which it follows that X = X (m) × Ω X (m + k). Multiplying (14) by (16) gives the following commutative diagram:
X×
p −1 i =1
k
Ω M ( A j 2i2i−−11 )
X (m) ×
( p −1)/2 i =1
M ( A 2i −1 ( m))
ϕm ×φ h
SO(m) × SO(m) μ
Ω V m+k,k
SO(m)
θ ×Π f 2i −1
X (m + k) ×
( p −1)/2 i =1
M ( A 2i −1 (m + k))
ϕm+k ×ψ
SO(m + k) × SO(m + k) μ
SO(m + k) (18)
where the top horizontal sequence is the multiplication of fibrations (11) and (17) and the bottom horizontal sequence is the principal fibration induced by
SO(m) → SO(m + k) → V m+k,k . The map h is the induced map between the fibers. By the homology of SO(m) and SO(m + k) given in (7) and (8) and the definitions of X (m) and X (m + k), we see that the right two column maps induce coalgebra homomorphisms in homology that are isomorphisms on the sets of generators of odd degree. Dualizing to cohomology algebra, the dualities of homology generators become the cohomology algebra generators of the same odd degree. Thus, the corresponding two column maps induce algebra homomorphisms that are isomorphisms on the sets of generators. In addition, the coalgebra homomorphisms in homology are dualized to algebra homomorphisms in cohomology. It follows that these two homomorphisms in cohomology are isomorphisms. Thus, the right two-column maps are homotopy equivalences. From the Steenrod 5-lemma we see that h is a homotopy equivalence. The definitions of X (m) and X (m + k) give the stated homotopy decomposition in Theorem 3.1. 2 4. Proof of Theorem 1.1 We prove Theorem 1.1 by virtue of the homotopy decomposition given in Theorem 3.1. Recall that ¯j i = i + r m i ( p − 1) and
k¯ i = (k − i )/( p − 1) + 1 for 1 i p − 1. We first introduce a result on the upper bound for the homotopy exponent of k¯ ji
M ( A ¯ i ) given in [5]. By the cofibration
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H. Zhao, P. Zhang / Topology and its Applications 160 (2013) 589–595
m+k−1 ΣC P m−1 → ΣC P m+k−1 → ΣC P m , m+k−1 the reduced cohomology of ΣC P m ,
m+k−1 = Z/ p {x2m+1 , x2m+3 , . . . , x2(m+k)−1 }, H ∗ ΣC P m
inherits a Steenrod operation from H ∗ (ΣC P m+k−1 ), which is dual to (3). In particular, we have P j (x2(m+k)−1 ) = 0 for m+k−1 j > 0 and P 1 (x2r +1 ) = r · x2r +q+1 for m r m + k − 1. By the wedge decomposition ΣC P m k¯ ji
p −1 i =1
k¯ ji
A ¯ i , the reduced
cohomology of A ¯ i ,
k¯ H ∗ A ¯ i = Z/ p {x2 ¯j +1 , x2 ¯j +q+1 , . . . , x2 ¯j +(k¯ −1)q+1 }, i i i i ji
m+k−1 also inherits a corresponding Steenrod operation from H ∗ (ΣC P m ). In particular, for 1 t k¯ i − 1 we have
P 1 (x2 ¯j i +(t −1)q+1 ) = ¯j i + (t − 1)( p − 1) · x2 ¯j i +tq+1 .
When m + k ( p − 1)2 + 2, we have k¯ i p − 1, from which it follows that 1 t p − 2. Thus, there exists at most one t ¯
k H ∗( A ¯i ) such that ¯j i + (t − 1)( p − 1) ≡ 0 mod p. It follows that on the set of generators of ji
{x2 ¯j i +1 , x2 ¯j i +q+1 , . . . , x2 ¯j i +(k¯ i −2)q+1 }, P 1 acts trivially on at most one generator. Lemma 4.1. For m + k ( p − 1)2 + 1, let 1 i p − 1 and 1 t k¯ i − 1. Then: ¯ ¯ k¯ 1. If for every t there is ¯j i + (t − 1)( p − 1) ≡ 0 mod p, then exp( M ( A ¯ i )) p j i +(ki −1) p . ji
¯ ¯ k¯ 2. If there exists some t such that ¯j i + (t − 1)( p − 1) ≡ 0 mod p, then exp( M ( A ¯ i )) p j i +(ki −1) p −1 . ji
In the following we give the proof of Theorem 1.1 by Theorem 3.1 and Lemma 4.1. Proof of Theorem 1.1. We only give the proof of (1). The proof of (2) is similar. From Theorem 3.1 we have the homotopy decomposition
Ω V m+k,k X×
( p− 1)/2 i =1
k Ω M A j 2i2i−−11 ,
where the definition of X is given by (16). Thus, by Lemma 4.1 it follows that
1 i ( p − 1)/2 k max p (m+k)/2 , exp M A j 2i2i−−11 1 i ( p − 1)/2
max p (m+k)/2 , p j 2i−1 +(k2i−1 −1) p 1 i ( p − 1)/2
= max p j 2i−1 +(k2i−1 −1) p 1 i ( p − 1)/2 .
exp( V m+k,k ) = max exp( X ), exp M A j 2i−1 k
2i −1
(1)(a) If for every 1 t k2i 0 −1 − 1 there is j 2i 0 −1 + (t − 1)( p − 1) ≡ 0 mod p, then by Theorem 3.1 we obtain k2i 0 −1
exp( M ( A j
2i 0 −1
)) p j 2i0 −1 +(k2i0 −1 −1) p . For 1 i ( p − 1)/2, if i = i 0 , then from the definitions of j i and ki it follows that
j 2i 0 −1 + (k2i 0 −1 − 1) p − j 2i −1 + (k2i −1 − 1) p
= ( j 2i 0 −1 − j 2i −1 ) + (k2i 0 −1 − k2i −1 ) p = ( j 2i 0 −1 − j 2i −1 ) + (k2i 0 −1 − k2i −1 ) p m m = 2(i 0 − i ) + r2i − r2i −1 ( p − 1) + (k2i 0 −1 − k2i −1 ) p 0 −1 > 0. From the above it follows that p j 2i−1 +(k2i−1 −1) p < p j 2i0 −1 +(k2i0 −1 −1) p . Thus, we have
exp( V m+k,k ) p j 2i0 −1 +(k2i0 −1 −1) p
H. Zhao, P. Zhang / Topology and its Applications 160 (2013) 589–595
595
or, equivalently,
exp( V m+k,k ) pm+k+k2i0 −1 −3 by 2 j 2i 0 −1 + (k2i 0 −1 − 1)q + 1 = 2(m + k) − 3. (1)(b) If there exists some 1 t k2i 0 −1 − 1 such that j 2i 0 −1 + (t − 1)( p − 1) ≡ 0 mod p, then by Theorem 3.1 we obtain k2i 0 −1
exp( M ( A j
2i 0 −1
)) p j 2i0 −1 +(k2i0 −1 −1) p −1 . According to the proof of (1)(a), for 1 i ( p − 1)/2, if i = i 0 , then
k
exp M A j 2i−1 2i −1
p j 2i−1 +(k2i−1 −1) p p j 2i0 −1 +(k2i0 −1 −1) p −1 .
Thus,
exp( V m+k,k ) p j 2i0 −1 +(k2i0 −1 −1) p −1 or, equivalently,
exp( V m+k,k ) pm+k+k2i0 −1 −4 , by 2 j 2i 0 −1 + (k2i 0 −1 − 1)q + 1 = 2(m + k) − 5.
2
Acknowledgement We are grateful to the anonymous referee for many helpful suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
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