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arXiv:1110.1798v1 [math.AT] 9 Oct 2011

ALMOST COMPLEX STRUCTURES

ON (n−1)-CONNECTED 2n-MANIFOLDS

HUIJUN YANG

Abstract. Let Mbe a closed (n−1)-connected 2n-dimensional smooth mani-

fold with n≥3. In terms of the system of invariants for such manifolds intro-

duced by Wall, we obtain necessary and suﬃcient 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:X→Ybetween 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 (n−1)-

connected 2n-dimensional smooth manifolds Mwith n≥3 that admit an almost

complex structure.

Throughout this paper, Mwill be a closed oriented (n−1)-connected 2n- dimen-

sional smooth manifold with n≥3. 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×H→Z, the intersection form of Mwhich is unimodular and n- sym-

metric, deﬁned by I(x,y)=<x∪y,[M]>,

where the homology class [M] is the orientation class of M,

3) A map α:Hn(M;Z)→πn−1(SOn) that assigns each element x∈Hn(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)∂

−→ πn−1(SOn)S

−→ πn−1(SOn+1)→ · · ·

of the ﬁber bundle S On֒→SOn+1→Sn, and ιn∈πn(Sn) is the class of the

identity map.

Denote by χ=S◦α:Hn(M;Z)→πn−1(SOn+1)g

KO(Sn) the composition

map, then from (1.1) and (1.2)

(1.3) χ=S◦α∈Hn(M;g

KO(Sn)) =Hom(Hn(M;Z); g

KO(Sn))

can be viewed as an n-dimensional cohomology class of M, with coeﬃcient 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 (n−1)-connected 2n-dimensionalsmooth

manifold with n ≥3. Then

pj(M)=

±an/4(n/2−1)!χ, n≡0(mod 4),j=n/4,

a2

n/4

2((n/2−1)!)2{1−(2n/2−1−1)2

2n−1−1n

n/2B2

n/4

Bn/2}I(χ, χ)

+n!

2n(2n−1−1)Bn/2τ, n≡0(mod 4),j=n/2,

n!

2n(2n−1−1)Bn/2τ, n≡2(mod 4),j=n/2,

where

an/4=

1,n≡0 (mod 8),

2,n≡4 (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 (n−1)-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 suﬃcient conditions

for M to admit a stable almost complex structure are:

ALMOST COMPLEX STRUCTURES ON (n−1)-CONNECTED 2n-MANIFOLDS 3

1) n ≡2,3,5,6,7 (mod 8), or

2) if n ≡0 (mod 8): χ≡0 (mod 2) and (Bn/2−Bn/4)

Bn/2Bn/4·nτ

2n≡0 (mod 2),

3) if n ≡4 (mod 8): (Bn/2+Bn/4)

Bn/2Bn/4·τ

2n−2≡0 (mod 2),

4) if n ≡1 (mod 8): χ=0.

Theorem 2. Let M be a closed oriented (n−1)-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

x≡χ(mod 2) and I(x,x)=(2(k+2) +pn/2(M))/((n/2−1)!)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/2−1)!)2,

4) if n ≡1 (mod 4): 2((n−1)!) |(2 −k),

5) if n ≡3 (mod 4): (n−1)! |(2 −k).

Remark 1.2.i) Since the rational numbers (Bn/2−Bn/4)

Bn/2Bn/4·nτ

2nand (Bn/2+Bn/4)

Bn/2Bn/4·τ

2n−2in

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 f∈π2n−1(∨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

λ→Mbe the inclusion map of the n-skeleton of Mand p:M→S2n

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(S2n−1)

(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(S2n−1)

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

λ→Sn

jwhich collapses ∨λ,jSn

λto the base

point. Then we have:

Lemma 2.1. Let M be a closed oriented (n−1)-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)

0Zξ⊕Lk

j=1ZηjZγ⊕Lk

j=1Zζj˜r(ξ)=2γ,˜r(ηj)=2ζj

1ZξZ2γ⊕Lk

j=1Z2ζj˜r(ξ)=γ

2Zξ⊕Lk

j=1ZηjZγ⊕Lk

j=1Z2ζj˜r(ξ)=γ,˜r(ηj)=ζj

4Zξ⊕Lk

j=1ZηjZγ⊕Lk

j=1Zζj˜r(ξ)=2γ,˜r(ηj)=ζj

5ZξZ2γ˜r(ξ)=γ

6Zξ⊕Lk

j=1ZηjZγ˜r(ξ)=γ,˜r(ηj)=0

3,7Zξ0 ˜r=0

ALMOST COMPLEX STRUCTURES ON (n−1)-CONNECTED 2n-MANIFOLDS 5

where the generatorsξ,ηj,γ,ζj,1≤j≤k, 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

0→e

K(S2n)p∗

u

−→ e

K(M)i∗

u

−→ e

K(∨k

λ=1Sn

λ)→0(i)

0→g

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 ˜r◦c=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 Σ9f∈π2n+8(∨k

λ=1Sn+9

λ) and Σ8f∈π2n+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

n−1, where πS

n−1is the (n−1)-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 n≡1 (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 n≡1 (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 x∈g

KO(M), we have 2x=˜r◦c(x)=

0, which implies (ii) of assertion b) in this case.

Case n≡2 (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 ≤j≤k, 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 π2n−1(∨k

j=1Sn

j) can be

decomposed as:

π2n−1(∨k

j=1Sn

j)⊕k

j=1π2n−1(Sn

j)⊕1≤i<j≤kπ2n−1(S2n−1

ij ),

where S2n−1

ij =S2n−1, the group π2n−1(Sn

j) is embedded in π2n−1(∨k

j=1Sn

j) by the

natural inclusion, and the group π2n−1(S2n−1

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 fj∈ImJ ⊂π2n−1(Sn)

Jbeing the J-homomorphism and g∈ ⊕1≤i<j≤kπ2n−1(S2n−1

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

Σ8fj∈ImJ ⊂π2n+7(Sn+8

j)πs

n−1.

Denote by eC(Σ8fj) the eCinvariant of Σ8fjdeﬁned in Adams [1], Ψ−1

C:e

K(M)→

e

K(M) and Ψ−1

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 ≤j≤k. Hence, by considering the map

˜

tj: (∨k

λ=1Sn+8

λ)∪Σ8fD2n+8→Sn+8

j∪Σ8fjD2n+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

Ψ−1

C(ηj)=(−1)n/2ηj+l·((−1)n−(−1)n/2)ξ∈e

K(M)

for each ηj, and for some l∈Z. Therefore from

˜r◦Ψ−1

C= Ψ−1

R◦˜r,

we have

Ψ−1

R˜r(ηj)=−˜r(ηj)+2l˜r(ξ).

That is 2˜r(ηj−lξ)=0.

But from (2.1) and Table 1, we get

i∗

o˜r(ηj−lξ)=t∗

oj(ωn

R).

ALMOST COMPLEX STRUCTURES ON (n−1)-CONNECTED 2n-MANIFOLDS 7

That is ˜r(ηj−lξ),0∈g

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

C∈e

K(S2n) can be chosen such that its n-th chern class cn(ω2n

C)=(n−1)!(cf.

Hatcher [6, p. 101]), from the naturality of the chern class, we get

ci(ξ)=

(n−1)!,i=n,

0,others.

Similarly, when nis even, ηj, 1 ≤j≤k, can be chosen such that

cn/2(Σk

j=1xjηj)=(n/2−1)!(x1,x2, ..., xk)∈Hn(M;Z),

where xj∈Zfor all 1 ≤j≤k(since Hn(M;Z)⊕k

j=1Z, we can write an element

x∈Hn(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

χj∈g

KO(Sn)

Z,n≡0 (mod 4),

Z2,n≡1,2 (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 α∈VectC(M). When n≡0 (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 diﬀerentiable 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 χ≡0 (mod 2).

Using the notation above, we get

Lemma 3.1. Let M be a closed oriented (n−1)-connected 2n-dimensionalsmooth

manifold with n ≥3. Then TM can be expressed by the generators γ,ζj,1≤j≤

8 HUIJUN YANG

k of g

KO(M)as follows:

TM =

lγ+ Σk

j=1χjζj,n≡0,2,4 (mod 8),

lγ, n≡6 (mod 8),

Σk

j=1χjζj,n≡1 (mod 8),

0,n≡3,5,7 (mod 8),

where

l=

ˆ

AC(M)+(Σk

j=1an/4χjdimcηj−2n)ˆ

A(M)−an/4ˆ

Aχ(M),n≡0 (mod 4),

−pn/2(M)

2((n−1)!),n≡2 (mod 4).

Proof. Case n≡0 (mod 8). By Remark 2.3, we may suppose that

TM =lγ+ Σk

j=1χjζj∈g

KO(M),

where l∈Z. Hence from ˜r◦c=2 and Table 2, we have

c(TM)=TM ⊗C(3.1)

=lξ+ Σk

j=1χjηj∈e

K(M).

Now if we regard ξand χjas complex vector bundles, then from (3.1) we have

TM ⊗C⊕εslξ⊕

k

M

j=1

χjηj⊕εt,

for some s,t∈Zsatisfying

s−t=l·dimcξ+ Σk

j=1χjdimcηj−2n,

where εjis the trivial complex vector bundle of dimension j. Thus we have

ˆ

AC(M)=−(l·dimcξ+ Σk

j=1χjdimcηj−2n)ˆ

A(M)

+<ch(lξ+ Σk

j=1χjηj)·ˆ

A(M),[M]>,

that is l=ˆ

AC(M)+(Σk

j=1χjdimcηj−2n)ˆ

A(M)−ˆ

Aχ(M).

Cases n≡2,4,6 (mod 8) can be proved by the same way as above. Note that in

the case n≡2 (mod 4) the calculation of ˆ

AC(M) is replaced by the calculation of

the n-th chern class of TM ⊗C.

Case n≡1 (mod 4). From Milnor and Kervaire [10, Lemma 1] and Adams [1,

Theorem 1.3], we get that χ=0 implies TM =0∈g

KO(M). Then

i) case n≡5 (mod 8). T M =0∈g

KO(M) because χ=0 in this case.

ii) case n≡1 (mod 8). By Remark 2.3, we may suppose that

TM =lγ+ Σk

j=1χjζj,

ALMOST COMPLEX STRUCTURES ON (n−1)-CONNECTED 2n-MANIFOLDS 9

where l∈Z2. Then if χ=0, we have l=0 because TM =0. If χ,0 and l,0,

suppose that χλ,0 for some 1 ≤λ≤k, set

ζ′

j=

ζjif j,λ,

ζj+γif j=λ.

Hence γ,ζ′

j, 1 ≤j≤k, 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 ≤j≤k, of g

KO(M) in Lemma 2.1 can always be chosen such

that TM = Σk

j=1χjζj.

Case n≡3 (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) n≡0 (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)=1−Bn/4

2((n/2)!)pn/4(M)+ˆ

A(M),

ch(TM ⊗C)=2n+(−1)n/4+1pn/4(M)

(n/2−1)! +I(pn/4(M),pn/4(M)) −2pn/2(M)

2((n−1)!) .

Hence by Lemma 1.1 we have

ˆ

AC(M)=2n{1+1

Bn/2+(2n−1−1)

(2n/2−1)2·(−1)n/4Bn/2−Bn/4

Bn/2Bn/4}ˆ

A(M)(4.1)

+1

(2n/2−1)2·(−1)n/4Bn/2−Bn/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/2−1)2·bn/2·bn/4, we get that

(2n/2−1)2bn/2bn/4ˆ

AC(M)=2n{(2n/2−1)2·bn/2·bn/4+2(2n/2−1)2bn/4cn/2

+2(2n−1−1)((−1)n/4bn/2cn/4−bn/4cn/2)}ˆ

A(M)

+2((−1)n/4bn/2cn/4−bn/4cn/2)nτ

2n.

10 HUIJUN YANG

Since ˆ

AC(M) and ˆ

A(M) are integers and (2n/2−1)2·bn/2·bn/4is an odd integer,

it follows that (−1)n/4Bn/2−Bn/4

Bn/2Bn/4·nτ

2n

is a 2-adic integer, and hence

ˆ

AC(M)≡0 (mod 2) ⇐⇒ (−1)n/4Bn/2−Bn/4

Bn/2Bn/4·nτ

2n≡0 (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 α∈e

K(M) such that

(4.2)

˜r(α)=T M ∈g

KO(M),

cn(α)=e(M).

Secondly, if there exists an element α∈e

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) n≡0 (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 (n−1)-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) n≡2 (mod 8). In this case e(M)=k+2. Set α=lξ+ Σk

j=1xjηj∈e

K(M)

where l∈Zis the integer as in Lemma 3.1 and xj∈Z, such that xj≡χj(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=1xjηj∈e

K(M),

cn(α)=e(M).

Let x=(x1,x2, ..., xk)∈Hn(M;Z). Then by Remark 2.2

cn/2(α)=(n/2−1)!x.

Now (4.3) is

(1 −cn/2(α)+cn(α)) ·(1 +cn/2(α)+cn(α)) =1−pn/2(M),

therefore

cn(α)=1

2(I(cn/2(α),cn/2(α)) −pn/2(M))

=1

2{((n/2−1)!)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 x∈Hn(M;Z) such that

x≡χ(mod 2),

I(x,x)=(2(k+2) +pn/2(M))/((n/2−1)!)2.

Case 3) n≡6 (mod 8). The proof is similar to the proof of case 2).

Case 4) n≡1 (mod 4). Nowe(M)=2−k. 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(n−1)! =2−k,

where a∈Z. 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(n−1)! |(2 −k).

Case 5) n≡3 (mod 4). The proof is similar to the proof of case 4).

12 HUIJUN YANG

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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