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Almost Complex Structures on (n-1)-connected 2n-manifolds

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Abstract

Let M be a closed (n-1)-connected 2n-dimensional smooth manifold with n > 2. In terms of the system of invariants for such manifolds introduced by Wall, we obtain necessary and sufficient conditions for M to admit an almost complex structure.
arXiv:1110.1798v1 [math.AT] 9 Oct 2011
ALMOST COMPLEX STRUCTURES
ON (n1)-CONNECTED 2n-MANIFOLDS
HUIJUN YANG
Abstract. Let Mbe a closed (n1)-connected 2n-dimensional smooth mani-
fold with n3. In terms of the system of invariants for such manifolds intro-
duced by Wall, we obtain necessary and sucient conditions for Mto admit an
almost complex structure.
1. Introduction
First we introduce some notations. For a topological space X, let VectC(X) (resp.
VectR(X)) be the set of isomorphic classes of complex (resp. real) vector bundles
on X, and let r:VectC(X)VectR(X) be the real reduction, which induces the
real reduction homomorphism ˜r:e
K(X)g
KO(X) from the reduced KU-group
to the reduced KO-group of X. For a map f:XYbetween topological spaces
Xand Y, denote by f
u:e
K(Y)e
K(X) and f
o:g
KO(Y)g
KO(X) the induced
homomorphisms.
Let Mbe a 2n-dimensional smooth manifold with tangent bundle T M. We say
that Madmits an almost complex structure (resp. a stable almost complex struc-
ture) if TM Imr(resp. TM Im˜r). Clearly, Madmits an almost complex
structure implies that Madmits a stable almost complex structure. It is a classi-
cal topic in geometry to determine which Madmits an almost complex structure.
See for instance [15, 5, 9, 13]. In this paper we determine those closed (n1)-
connected 2n-dimensional smooth manifolds Mwith n3 that admit an almost
complex structure.
Throughout this paper, Mwill be a closed oriented (n1)-connected 2n- dimen-
sional smooth manifold with n3. In [14], C.T.C. Wall assigned to each Ma
system of invariants as follows.
1) H=Hn(M;Z)Hom(Hn(M;Z); Z)k
j=1Z, the cohomology group of M,
with kthe n-th Betti number of M,
2000 Mathematics Subject Classification. 55N15, 19L64.
Key words and phrases. Almost complex structure, stable almost complex structure, reduced
KU-group, reduced KO-group, real reduction. 1
2 HUIJUN YANG
2) I:H×HZ, the intersection form of Mwhich is unimodular and n- sym-
metric, defined by I(x,y)=<xy,[M]>,
where the homology class [M] is the orientation class of M,
3) A map α:Hn(M;Z)πn1(SOn) that assigns each element xHn(M;Z) to
the characteristic map α(x) for the normal bundle of the embedded n-sphere
Sn
xrepresenting x.
These invariants satisfy the relation ([14, Lemma 2])
(1.1) α(x+y)=α(x)+α(y)+I(x,y)∂ιn,
where is the boundary homomorphism in the exact sequence
(1.2) · · · πn(Sn)
πn1(SOn)S
πn1(SOn+1) · · ·
of the fiber bundle S On֒SOn+1Sn, and ιnπn(Sn) is the class of the
identity map.
Denote by χ=Sα:Hn(M;Z)πn1(SOn+1)g
KO(Sn) the composition
map, then from (1.1) and (1.2)
(1.3) χ=SαHn(M;g
KO(Sn)) =Hom(Hn(M;Z); g
KO(Sn))
can be viewed as an n-dimensional cohomology class of M, with coecient in
g
KO(Sn). It follows from Kervaire [8, Lemma 1.1] and Hirzebruch index Theorem
[7] that the Pontrjagin classes pj(M)H4j(M;Z) of Mcan be expressed in terms
of the cohomology class χand the index τof the intersection form I(when nis
even) as follows (cf. Wall [14, p. 179-180]).
Lemma 1.1. Let M be a closed oriented (n1)-connected 2n-dimensionalsmooth
manifold with n 3. Then
pj(M)=
±an/4(n/21)!χ, n0(mod 4),j=n/4,
a2
n/4
2((n/21)!)2{1(2n/211)2
2n11n
n/2B2
n/4
Bn/2}I(χ, χ)
+n!
2n(2n11)Bn/2τ, n0(mod 4),j=n/2,
n!
2n(2n11)Bn/2τ, n2(mod 4),j=n/2,
where
an/4=
1,n0 (mod 8),
2,n4 (mod 8),
Bmis the m-th Bernoulli number.
Now we can state the main results as follows.
Theorem 1. Let M be a closed oriented (n1)-connected 2n-dimensional smooth
manifold with n 3,χbe the cohomology class defined in (1.3), τthe index of the
intersection form I (when n is even). Then the necessary and sucient conditions
for M to admit a stable almost complex structure are:
ALMOST COMPLEX STRUCTURES ON (n1)-CONNECTED 2n-MANIFOLDS 3
1) n 2,3,5,6,7 (mod 8), or
2) if n 0 (mod 8): χ0 (mod 2) and (Bn/2Bn/4)
Bn/2Bn/4·nτ
2n0 (mod 2),
3) if n 4 (mod 8): (Bn/2+Bn/4)
Bn/2Bn/4·τ
2n20 (mod 2),
4) if n 1 (mod 8): χ=0.
Theorem 2. Let M be a closed oriented (n1)-connected 2n-dimensional smooth
manifold with n 3, k be the n-th Betti number, I be the intersection form, and
pj(M)be the Pontrjagin class of M as in Lemma 1.1. Then M admits an almost
complex structure if and only if M admits a stable almost complex structure and
one of the following conditions are satisfied:
1) If n 0 (mod 4): 4pn/2(M)I(pn/4(M),pn/4(M)) =8 (k+2),
2) if n 2 (mod 8): there exists an element x Hn(M;Z)such that
xχ(mod 2) and I(x,x)=(2(k+2) +pn/2(M))/((n/21)!)2,
3) if n 6 (mod 8): there exists an element x Hn(M;Z)such that
I(x,x)=(2(k+2) +pn/2(M))/((n/21)!)2,
4) if n 1 (mod 4): 2((n1)!) |(2 k),
5) if n 3 (mod 4): (n1)! |(2 k).
Remark 1.2.i) Since the rational numbers (Bn/2Bn/4)
Bn/2Bn/4·nτ
2nand (Bn/2+Bn/4)
Bn/2Bn/4·τ
2n2in
Theorem 1 can be viewed as 2-adic integers (see the proof of Theorem 1), it
makes sense to take congruent classes modulo 2.
ii) In the cases 2) and 3) of Theorem 2, when the conditions are satisfied, the
almost complex structure on Mdepends on the choice of x.
This paper is arranged as follows. In §2 we obtain presentations for the groups
g
KO(M), e
K(M) and determine the real reduction ˜r:e
K(M)g
KO(M) accord-
ingly. In §3 we determine the expression of T M g
KO(M) with respect to the
presentation of g
KO(M) obtained in §2. With these preliminary results, Theorem
1 and Theorem 2 are established in §4.
I would like to thank my supervisor H. B. Duan, Dr. Su and Dr. Lin for their help
with the preparation of this paper.
2. The real reduction ˜r:e
K(M)g
KO(M)
According to Wall [14], Mis homotopic to a CW complex (k
λ=1Sn
λ)fD2n, where
kis the n-th Betti number of M,k
λ=1Sn
λis the wedge sum of n-spheres which is
the n-skeleton of Mand fπ2n1(k
λ=1Sn
λ) is the attaching map of D2nwhich is
determined by the intersection form Iand the map α(cf. Duan and Wang [4]).
Let i:k
λ=1Sn
λMbe the inclusion map of the n-skeleton of Mand p:MS2n
be the map collapsing the n-skeleton k
λ=1Sn
λto the base point. Then by the
naturality of the Puppe sequence, we have the following exact ladder:
4 HUIJUN YANG
e
K(k
λ=1Sn+1
λ)Σf
u
e
K(S2n)p
u
e
K(M)i
u
e
K(k
λ=1Sn
λ)f
u
e
K(S2n1)
(2.1) ˜r˜r˜r˜r˜r
g
KO(k
λ=1Sn+1
λ)Σf
o
g
KO(S2n)p
o
g
KO(M)i
o
g
KO(k
λ=1Sn
λ)f
o
g
KO(S2n1)
where the horizontal homomorphisms Σf
u,Σf
o,p
u,p
o,i
u,i
oand f
u,f
oare
induced by Σf,p,iand frespectively, and where Σdenotes the suspension.
Let Zβ(resp. Z2β) be the infinite cyclic group (resp. finite cyclic group of or-
der 2) generated by β. Then the generators ωm
C(resp. ωm
R) of the cyclic group
e
K(Sm) (resp. g
KO(Sm)) with m>0 can be so chosen such that the real reduction
˜r:e
K(Sm)g
KO(Sm) can be summarized as in Table 1 (cf. Mimura and Toda [12,
Theorem 6.1, p. 211]).
Table 1. Real reduction ˜r:e
K(Sm)g
KO(Sm)
m(mod 8) e
K(Sm)g
KO(Sm)˜r:e
K(Sm)g
KO(Sm)
0Zωm
CZωm
R˜r(ωm
C)=2ωm
R
1 0 Z2ωm
R˜r=0
2Zωm
CZ2ωm
R˜r(ωm
C)=ωm
R
4Zωm
CZωm
R˜r(ωm
C)=ωm
R
6Zωm
C0 ˜r=0
3, 5, 7 0 0 ˜r=0
Denoted by t
ju :e
K(Sn
j)e
K(k
λ=1Sn
λ) and t
jo :g
KO(Sn
j)g
KO(k
λ=1Sn
λ) the ho-
momorphisms induced by tj:k
λ=1Sn
λSn
jwhich collapses λ,jSn
λto the base
point. Then we have:
Lemma 2.1. Let M be a closed oriented (n1)-connected 2n-dimensionalsmooth
manifold with n 3. Then the presentations of the groups e
K(M)and g
KO(M)as
well as the real reduction ˜r:e
K(M)g
KO(M)can be given as in Table 2.
Table 2. Real reduction ˜r:e
K(M)g
KO(M)
n(mod 8) e
K(M)g
KO(M)˜r:e
K(M)g
KO(M)
0ZξLk
j=1ZηjZγLk
j=1Zζj˜r(ξ)=2γ,˜r(ηj)=2ζj
1ZξZ2γLk
j=1Z2ζj˜r(ξ)=γ
2ZξLk
j=1ZηjZγLk
j=1Z2ζj˜r(ξ)=γ,˜r(ηj)=ζj
4ZξLk
j=1ZηjZγLk
j=1Zζj˜r(ξ)=2γ,˜r(ηj)=ζj
5ZξZ2γ˜r(ξ)=γ
6ZξLk
j=1ZηjZγ˜r(ξ)=γ,˜r(ηj)=0
3,7Zξ0 ˜r=0
ALMOST COMPLEX STRUCTURES ON (n1)-CONNECTED 2n-MANIFOLDS 5
where the generatorsξ,ηj,γ,ζj,1jk, satisfy:
ξ=p
u(ω2n
C),i
u(ηj)=t
ju(ωn
C);
γ=p
o(ω2n
R),i
o(ζj)=t
jo(ωn
R).
Proof. We assert that
a) the induced homomorphisms f
u,f
o,Σf
uand Σf
oin (2.1) are trivial, moreover,
b) the short exact sequences
0e
K(S2n)p
u
e
K(M)i
u
e
K(k
λ=1Sn
λ)0(i)
0g
KO(S2n)p
o
g
KO(M)i
o
g
KO(k
λ=1Sn
λ)0(ii)
split.
Denote by c:g
KO(X)e
K(X) the complexification. Then by (2.1), combining
these assertions with the fact that ˜rc=2, all the results in Table 2 are easily
verified.
Now we prove assertions a) and b).
Firstly, by the Bott periodicity Theorem [3], we may assume that the horizon-
tal homomorphisms Σf
u,Σf
o,p
u,p
o,i
u,i
oand f
u,f
oin (2.1) are induced by
Σ9f,Σ8p,Σ8iand Σ8frespectively, where Σjdenotes the j-th iterated suspen-
sion. Note that Σ9fπ2n+8(k
λ=1Sn+9
λ) and Σ8fπ2n+7(k
λ=1Sn+8
λ), and the
groups π2n+8(k
λ=1Sn+9
λ) and π2n+7(k
λ=1Sn+8
λ) are all in their stable range, that is
π2n+8(k
λ=1Sn+9
λ)π2n+7(k
λ=1Sn+8
λ)k
λ=1πs
n1, where πS
n1is the (n1)-th stable
homotopy group of spheres. Thus the fact that Σf
uand f
uare trivial can be de-
duced easily from Table 1 and Adams [1, proposition 7.1]; the fact that Σf
oand
f
oare trivial when n.1 (mod 8) follows from Table 1 while the fact that Σf
o
and f
oare trivial when n1 (mod 8) follows from Adams [1, proposition 7.1].
This proves assertion a).
Secondly, (i) of assertion b) is true since the abelian group e
K(k
λ=1Sn
λ) is free.
Finally we prove (ii) of assertion b). For the cases n.1,2 (mod 8) the proof is
similar to (i).
Case n1 (mod 8). From (2.1), Table 1 and (i) we get that e
K(M)Zand
g
KO(M) is a finite group. Therefore, for each xg
KO(M), we have 2x=˜rc(x)=
0, which implies (ii) of assertion b) in this case.
Case n2 (mod 8). By (i), we may write e
K(M) as
e
K(M)=Zξ
k
M
j=1
Zηj,
6 HUIJUN YANG
where the generators ξ,ηj, 1 jk, satisfy ξ=p
u(ω2n
C), i
u(ηj)=t
ju(ωn
C). By
Hilton-Milnor theorem [16, p. 511] we know that the group π2n1(k
j=1Sn
j) can be
decomposed as:
π2n1(k
j=1Sn
j)k
j=1π2n1(Sn
j)1i<jkπ2n1(S2n1
ij ),
where S2n1
ij =S2n1, the group π2n1(Sn
j) is embedded in π2n1(k
j=1Sn
j) by the
natural inclusion, and the group π2n1(S2n1
ij ) is embedded by composition with
the Whitehead product of certain elements in πn(k
j=1Sn
j). Hence by Duan and
Wang [4, Lemma 3], the attaching map fcan be decomposed accordingly as:
f= Σk
j=1fj+g,
where fjImJ π2n1(Sn)
Jbeing the J-homomorphism and g∈ ⊕1i<jkπ2n1(S2n1
ij ). Moreover, since the
suspension of the Whitehead product is trivial, it follows that the homotopy group
π2n+7(k
j=1Sn+8
j) can be decomposed as:
π2n+7(k
j=1Sn+8
j)k
j=1π2n+7(Sn+8
j),
and accordingly Σ8fcan be decomposed as:
Σ8f=k
j=1Σ8fj∈ ⊕k
j=1π2n+7(Sn+8
j)
with
Σ8fjImJ π2n+7(Sn+8
j)πs
n1.
Denote by eC(Σ8fj) the eCinvariant of Σ8fjdefined in Adams [1], Ψ1
C:e
K(M)
e
K(M) and Ψ1
R=id:g
KO(M)g
KO(M) the Adams operations, where id is the
identity map. Then it follows from Adams [1, Proposition 7.19] that
eC(Σ8fj)=0,
for each 1 jk. Hence, by considering the map
˜
tj: (k
λ=1Sn+8
λ)Σ8fD2n+8Sn+8
jΣ8fjD2n+8
which collapses λ,jSn+8
λto a point, it’s easy to see from [1, proposition 7.5,
Proposition 7.8] and the naturality of Adams operation that
Ψ1
C(ηj)=(1)n/2ηj+l·((1)n(1)n/2)ξe
K(M)
for each ηj, and for some lZ. Therefore from
˜rΨ1
C= Ψ1
R˜r,
we have
Ψ1
R˜r(ηj)=˜r(ηj)+2l˜r(ξ).
That is r(ηjlξ)=0.
But from (2.1) and Table 1, we get
i
o˜r(ηjlξ)=t
oj(ωn
R).
ALMOST COMPLEX STRUCTURES ON (n1)-CONNECTED 2n-MANIFOLDS 7
That is ˜r(ηjlξ),0g
KO(M).
Thus (ii) of assertion b) in this case is established and the proof is finished.
Remark 2.2.Since the induced homomorphisms i:Hn(M;Z)Hn(k
λ=1Sn
λ;Z)
and p:H2n(S2n;Z)H2n(M;Z) are both isomorphisms, and the generator
ω2n
Ce
K(S2n) can be chosen such that its n-th chern class cn(ω2n
C)=(n1)!(cf.
Hatcher [6, p. 101]), from the naturality of the chern class, we get
ci(ξ)=
(n1)!,i=n,
0,others.
Similarly, when nis even, ηj, 1 jk, can be chosen such that
cn/2(Σk
j=1xjηj)=(n/21)!(x1,x2, ..., xk)Hn(M;Z),
where xjZfor all 1 jk(since Hn(M;Z)k
j=1Z, we can write an element
xHn(M;Z), under the isomorphism i, as the form (x1,x2, ..., xk) ).
Remark 2.3.As in Remark 2.2, if we write χas (χ1, ..., χk)Hn(M;g
KO(Sn)),
where
χjg
KO(Sn)
Z,n0 (mod 4),
Z2,n1,2 (mod 8),
0,others,
then since the tangent bundle of sphere is stably trivial, it follows that
i
o(TM)= Σk
j=1χjt
jo(ωn
R).
3. The tangent bundle of M
Denote by dimcαthe dimension of αVectC(M). When n0 (mod 4), we set
ˆ
A(M)=<ˆ
A(M),[M]>,
ˆ
AC(M)=<ch(TM C)·ˆ
A(M),[M]>,
ˆ
Aχ(M)=<ch(Σk
j=1χjηj)·ˆ
A(M),[M]>,
where ch denotes the chern character, and ˆ
A(M) is the A-class of M(cf. Atiyah
and Hirzebruch [2]). It follows from the dierentiable Riemann-Roch theorem
(cf. Atiyah and Hirzebruch [2]) that ˆ
A(M), ˆ
AC(M) and ˆ
Aχ(M) are all integers. In
particular, ˆ
Aχ(M) is even when χ0 (mod 2).
Using the notation above, we get
Lemma 3.1. Let M be a closed oriented (n1)-connected 2n-dimensionalsmooth
manifold with n 3. Then TM can be expressed by the generators γ,ζj,1j
8 HUIJUN YANG
k of g
KO(M)as follows:
TM =
lγ+ Σk
j=1χjζj,n0,2,4 (mod 8),
lγ, n6 (mod 8),
Σk
j=1χjζj,n1 (mod 8),
0,n3,5,7 (mod 8),
where
l=
ˆ
AC(M)+(Σk
j=1an/4χjdimcηj2n)ˆ
A(M)an/4ˆ
Aχ(M),n0 (mod 4),
pn/2(M)
2((n1)!),n2 (mod 4).
Proof. Case n0 (mod 8). By Remark 2.3, we may suppose that
TM =lγ+ Σk
j=1χjζjg
KO(M),
where lZ. Hence from ˜rc=2 and Table 2, we have
c(TM)=TM C(3.1)
=lξ+ Σk
j=1χjηje
K(M).
Now if we regard ξand χjas complex vector bundles, then from (3.1) we have
TM Cεslξ
k
M
j=1
χjηjεt,
for some s,tZsatisfying
st=l·dimcξ+ Σk
j=1χjdimcηj2n,
where εjis the trivial complex vector bundle of dimension j. Thus we have
ˆ
AC(M)=(l·dimcξ+ Σk
j=1χjdimcηj2n)ˆ
A(M)
+<ch(lξ+ Σk
j=1χjηj)·ˆ
A(M),[M]>,
that is l=ˆ
AC(M)+(Σk
j=1χjdimcηj2n)ˆ
A(M)ˆ
Aχ(M).
Cases n2,4,6 (mod 8) can be proved by the same way as above. Note that in
the case n2 (mod 4) the calculation of ˆ
AC(M) is replaced by the calculation of
the n-th chern class of TM C.
Case n1 (mod 4). From Milnor and Kervaire [10, Lemma 1] and Adams [1,
Theorem 1.3], we get that χ=0 implies TM =0g
KO(M). Then
i) case n5 (mod 8). T M =0g
KO(M) because χ=0 in this case.
ii) case n1 (mod 8). By Remark 2.3, we may suppose that
TM =lγ+ Σk
j=1χjζj,
ALMOST COMPLEX STRUCTURES ON (n1)-CONNECTED 2n-MANIFOLDS 9
where lZ2. Then if χ=0, we have l=0 because TM =0. If χ,0 and l,0,
suppose that χλ,0 for some 1 λk, set
ζ
j=
ζjif j,λ,
ζj+γif j=λ.
Hence γ,ζ
j, 1 jk, which satisfy the conditions in Lemma 2.1, are also
the generators of g
KO(M), and we have TM = Σk
j=1χjζ
j. This implies that the
generators γ,ζj, 1 jk, of g
KO(M) in Lemma 2.1 can always be chosen such
that TM = Σk
j=1χjζj.
Case n3 (mod 4). TM =0 because g
KO(M)=0 in this case.
4. Almost complex structure on M
We are now ready to prove Theorem 1 and Theorem 2.
Proof of Theorem 1. Cases 1) and 2) n0 (mod 4). In these cases, we get that
(cf. Wall [14, p. 179-180])
ˆ
A(M)=Bn/2
2(n!)pn/2(M)+1
2{B2
n/4
4((n/2)!)2+Bn/2
2(n!)}I(pn/4(M),pn/4(M)),
ˆ
A(M)=1Bn/4
2((n/2)!)pn/4(M)+ˆ
A(M),
ch(TM C)=2n+(1)n/4+1pn/4(M)
(n/21)! +I(pn/4(M),pn/4(M)) 2pn/2(M)
2((n1)!) .
Hence by Lemma 1.1 we have
ˆ
AC(M)=2n{1+1
Bn/2+(2n11)
(2n/21)2·(1)n/4Bn/2Bn/4
Bn/2Bn/4}ˆ
A(M)(4.1)
+1
(2n/21)2·(1)n/4Bn/2Bn/4
Bn/2Bn/4·nτ
2n.
Moreover since the denominator of Bm, when written as the most simple fraction,
is always square free and divisible by 2 (cf. Milnor [11, p. 284]), we may set
Bm=bm/(2cm), where cmand bmare odd integers. Then multiply each side of
(4.1) by (2n/21)2·bn/2·bn/4, we get that
(2n/21)2bn/2bn/4ˆ
AC(M)=2n{(2n/21)2·bn/2·bn/4+2(2n/21)2bn/4cn/2
+2(2n11)((1)n/4bn/2cn/4bn/4cn/2)}ˆ
A(M)
+2((1)n/4bn/2cn/4bn/4cn/2)nτ
2n.
10 HUIJUN YANG
Since ˆ
AC(M) and ˆ
A(M) are integers and (2n/21)2·bn/2·bn/4is an odd integer,
it follows that (1)n/4Bn/2Bn/4
Bn/2Bn/4·nτ
2n
is a 2-adic integer, and hence
ˆ
AC(M)0 (mod 2) (1)n/4Bn/2Bn/4
Bn/2Bn/4·nτ
2n0 (mod 2).
Then by combining these facts with Lemma 2.1 and Lemma 3.1, one verifies the
results in these cases.
Cases 3) and 4) n.0 (mod 4) can be deduced easily from Lemma 2.1 and
Lemma 3.1.
To prove Theorem 2, we need the following lemma (see Sutherland [13] for the
proof).
Lemma 4.1. Let N be a closed smooth 2n-manifold. Then N admits an almost
complex structure if and only if it admits a stable almost complex structure α
satisfying cn(α)=e(N), where e(N)is the Euler class of N.
Proof of Theorem 2. Firstly, it follows from Lemma 4.1 that Madmits an almost
complex structure if and only if there exists an element αe
K(M) such that
(4.2)
˜r(α)=T M g
KO(M),
cn(α)=e(M).
Secondly, if there exists an element αe
K(M) such that ˜r(α)=TM g
KO(M), then
we have the following identity (cf. Milnor [11, p. 177]):
(4.3) (X
j
(1)jcj(α)) ·(X
j
cj(α)) =X
j
(1)jpj(M).
Now we prove Theorem 2 case by case.
Case 1) n0 (mod 4). In this case e(M)=k+2. From Lemma 4.1 we know that
Madmits an almost complex structure if and only if there exists an element α
e
K(M) such that (4.2) is satisfied. Now (4.3) becomes
(1 +cn/2(α)+cn(α)) ·(1 +cn/2(α)+cn(α)) =1+(1)n/4pn/4(M)+pn/2(M),
it follows that
cn/2(α)=(1)n/41
2pn/4(M),
hence
cn(α)=1
2pn/2(M)1
8I(pn/4(M),pn/4(M)).
ALMOST COMPLEX STRUCTURES ON (n1)-CONNECTED 2n-MANIFOLDS 11
Therefore from (4.2) we get that, Madmits an almost complex structure if and
only if Madmits a stable almost complex structure and satisfies
4pn/2(M)I(pn/4(M),pn/4(M)) =8(k+2).
Case 2) n2 (mod 8). In this case e(M)=k+2. Set α=lξ+ Σk
j=1xjηje
K(M)
where lZis the integer as in Lemma 3.1 and xjZ, such that xjχj(mod 2).
Then from Lemma 2.1 and Lemma 3.1, we know that ˜r(α)=TM g
KO(M).
Hence by (4.2), we see that Madmits an almost complex structure if and only if
α=lξ+ Σk
j=1xjηje
K(M),
cn(α)=e(M).
Let x=(x1,x2, ..., xk)Hn(M;Z). Then by Remark 2.2
cn/2(α)=(n/21)!x.
Now (4.3) is
(1 cn/2(α)+cn(α)) ·(1 +cn/2(α)+cn(α)) =1pn/2(M),
therefore
cn(α)=1
2(I(cn/2(α),cn/2(α)) pn/2(M))
=1
2{((n/21)!)2I(x,x)pn/2(M)}.
Thus it follows from (4.2) that Madmits an almost complex structure if and only
if there exists an element xHn(M;Z) such that
xχ(mod 2),
I(x,x)=(2(k+2) +pn/2(M))/((n/21)!)2.
Case 3) n6 (mod 8). The proof is similar to the proof of case 2).
Case 4) n1 (mod 4). Nowe(M)=2k. From (4.2), Lemma 2.1, Lemma 3.1 and
Remark 2.2, we see that Madmits an almost complex structure if and only if
χ=0,
α=2aξ,
2a(n1)! =2k,
where aZ. Hence by Lemma 3.1 and Lemma 2.1, Madmits an almost complex
structure if and only if Madmits a stable almost complex structure and
2(n1)! |(2 k).
Case 5) n3 (mod 4). The proof is similar to the proof of case 4).
12 HUIJUN YANG
References
[1] J. F. Adams,
On the groups J(X)-IV
, Topology, 5(1966) 21-71.
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Riemann-Roch theorems for dierentiable manifolds
, Bull.
Amer. Math. Soc. 65(1959) 276-281.
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, Ann. Math. 70(1959) 313-337.
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The degrees of maps between manifolds
, Math. Z. 244(2003) 67-89.
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Sur les vari´et´es presque complexes
, Pro. Int. Cong. Math. Vol. II(1950) 412-
419.
[6] A. Hatcher,
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, Version 2.1, 2009, aviliable
http://www.math.cornell.edu/hatcher/VBKT/VBpage.html
[7] F. Hirzebruch,
Topological methods in algebraic geometry
, Classics in Mathematics,
Springer-Verlag, Berlin Heidelberg New York, 1978.
[8] M. A. Kervaire,
A note on obstructions and characteristic classes
, Ameri. J. Math. 81(1959)
773-784.
[9] S. M¨uller and H. Geiges,
Almost complex structures on 8-manifolds
, L’Enseignement
Math´ematique, t. 46(2000) 95-107.
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Bernoulli numbers, homotopy groups, and a theorem of Rohlin
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Princeton University Press, Princeton, 1974.
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Topology of Lie groups, I and II
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Sur les espaces fibres
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Elements of homotopy theory
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Springer-Verlag, Berlin Heidelberg New York, 1978.
Hua Loo-Keng Key laboratory of Mathematics, Academy of Mathematics and Systems Science,
Chinese Academy of Sciences, Beijing 100190, China
E-mail address:yhj@amss.ac.cn
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