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CONNECTED SUMS OF ALMOST COMPLEX MANIFOLDS

HUIJUN YANG

Abstract. In this paper, ﬁrstly, for any 4n-dimensional almost complex man-

ifolds Mi,1≤i≤α, we prove that ]α

i=1Mi](α−1)CP2nmust admits an

almost complex structure, where αis a positive integer. Secondly, for a 2n-

dimensional almost complex manifold M, we get that M]CPnalso admits an al-

most complex structure. At last, as an application, we obtain that αCP2n]βCP2n

admits an almost complex structure if and only if αis odd.

1. Introduction

Through out this paper, all the manifolds are closed, connected, oriented

and smooth.

Let Mbe a manifold. Denote by T M the tangent bundle of Mand kthe

k-dimensional trivial real vector bundle over M. We say that Mis an almost

(resp. a stable almost) complex manifolds, i.e., admits an almost (resp. a stable

almost) complex structure, if there exists a endomorphism J:T M →T M (resp.

J:T M ⊕k→T M ⊕kfor some k) such that J2=−1. If Madmits an almost

complex structure, it follows from the deﬁnition that Mmust be orientable and

with even dimension.

Suppose that Madmits an almost complex structure with dim M=4n. De-

note by χ(M) and τ(M) the Euler characteristic and signature of Mrespectively.

Then Hirzebruch [5, p. 777] tell us that we must have

(1.1) χ(M)≡(−1)nτ(M) mod 4.

Remark 1.1.For n>1, the congruence (1.1) can also be deduced from Tang and

Zhang [11, Corollary 3.8].

Example 1.2. Denote by CPnthe n-dimensional complex projective space with

the natural orientation induced by the complex structure, and CPnthe same man-

ifold with the opposite orientation. We know that χ(CP2n)=χ(CP2n)=2n+1,

τ(CP2n)=1 and τ(CP2n)=−1. Therefore, it follows from the congruence (1.1)

that CP2ndoes not admit an almost complex structure for n≥1 (see also for

instance Tang and Zhang [11, Corollary 3.9]).

2010 Mathematics Subject Classiﬁcation. 53C15, 55S35.

Key words and phrases. Almost complex manifolds, Connected sums, Obstructions.

1

2 HUIJUN YANG

Let αbe a positive integer, and Mi,1≤i≤α, be 4n-dimensional almost

complex manifolds (may be diﬀerent). Denote by αM1the connected sum of α

copies of M1, and ]α

i=1Mithe connected sum of Mi,1≤i≤α. Then we have

χ(]α

i=1Mi)= Σα

i=1χ(Mi)−2(α−1),(1.2)

τ(]α

i=1Mi)= Σα

i=1τ(Mi).(1.3)

Hence it follows immediately from the congruence (1.1) that

Proposition 1.3. Let Mi,1≤i≤α, be 4n-dimensional almost complex mani-

folds. If αis even, then ]α

i=1Midoes not admit an almost complex structure.

Remark 1.4.In Proposition 1.3, the conditions Mi,1≤i≤αare almost complex

manifolds and dim Mi=4nare necessary because of the following facts.

(1) Denote by Snthe n-dimensional standard sphere. It can be deduced easily

from M¨

uller and Geiges [8, Theorem 4] that S4×S4]CP4do admits

an almost complex structure. However, it follows from Sutherland [10,

Theorem 3.1] (or Yang [12, Theorems 1 and 2] or Datta and Subramanian

[2, Theorem 1]) and Example 1.2 that both S4×S4and CP4do not admit

an almost complex structure.

(2) We know that αCP2n+1is diﬀeomorphism to CP2n+1](α−1)CP2n+1which

is a blow up of CP2n+1in (α−1) point. Therefore, αCP2n+1must be

K¨

ahler, hence admits almost complex structure for any positive integer

α.

Now, based on the facts of Proposition 1.3 and Remark 1.4, it is natural to

ask the following naive question:

Question 1. For any odd positive integer αand any 4n-dimensional almost com-

plex manifolds Mi,1≤i≤α, does ]α

i=1Miadmits an almost complex structure?

For example, it is known that S6admits an almost complex structure, hence

S6×S6admits an almost complex structure. Therefore, It can be deduced from

Yang [12, Theorems 1 and 2] that αS6×S6admits an almost complex structure

if and only if αis odd (the proof is left to the reader).

However, the answer to Question 1 is negative in generally.

Example 1.5. We can deduced from Yang [12, Theorem 2] that αS10 ×S10 admits

an almost complex structure if and only if α≡ −1 mod 1152. Therefore let

M=1151S10 ×S10, it must admits an almost complex structure. Moreover, αM

admits an almost complex structure if and only if α≡1 mod 1152.

In fact, if we set M4mbe the set of (4m−1)-connected 8m-dimensional

smooth manifolds for which admit almost complex structure. Then it can be

deduced from Yang [12, Lemma 1, Theorem 2] that

Proposition 1.6. For any Mi∈M4m,1≤i≤α,]α

i=1Miadmits an almost complex

structure if and only if α=1.

CONNECTED SUMS OF ALMOST COMPLEX MANIFOLDS 3

Proof. For a (4m−1)-connected 8m-manifold M, a necessary condition for Mto

admits an almost complex structure is (cf. Yang [12, Theorem 2])

(1.4) 4p2m(M)−p2

m(M)=8χ(M),

where pi(M) is the i-th Pontrjagin class of M. Note that for these manifolds, we

must have (cf. Yang [12, Lemma 1])

p2m(]α

i=1Mi)= Σα

i=1p2m(Mi),(1.5)

p2

m(]α

i=1Mi)= Σα

i=1p2

m(Mi).(1.6)

Then the facts of this proposition follows easily from the necessary condition

(1.4) and the equations (1.2), (1.5) and (1.6).

Remark 1.7.It follows from M¨

uller and Geiges [8, Proposition 6] that

HP2]HP2]S4×S4∈M4,

where HP2is the quaternionic projective plane. Hence M4,∅.

Even though the answer to Question 1 is negative in generally, it may pos-

itive if some Miin Question 1 are ﬁxed into some particular almost complex

manifolds. In this paper, our main results are stated as:

Theorem 1.8. For any positive integer αand 4n-dimensional almost complex

manifolds Mi,1≤i≤α, the connected sum

]α

i=1Mi](α−1)CP2n

must admits an almost complex structure.

Remark 1.9.For n≥2 and 2n-dimensional almost complex manifolds Mi,1≤

i≤α, Geiges has proved in [3, Lemma 2] that ]α

i=1Mi](α−1)S2×S2n−2must

admits an almost complex structure. However, S2×S2n−2does not admits an

almost complex structure whence n≥4 (cf. Sutherland [10, Theorem 3.1] or

Datta and Subramanian [2, Theorem 1]).

Therefore, it follows immediately from Proposition 1.3 and Theorem 1.8

that

Corollary 1.10. αCP2nadmits an almost complex structure if and only if αis

odd.

Remark 1.11.This fact has been got by Goertsches and Konstantis [4].

The facts of theorem 1.8 lead us to consider more about the connected sum

of almost complex manifolds with complex projective spaces.

Theorem 1.12. Let M be a 2n-dimensional almost complex manifolds. Then

M]CPnmust admits an almost complex structure.

4 HUIJUN YANG

Remark 1.13.Even if M]CPnadmits an almost complex structure, Mmay not

admits an almost complex structure. For example:

(1) we known that S2ndoes not admit an almost complex structure for k≥4.

However, for odd n,S2n]CPn=CPnis diﬀeomorphic to CPnwhich admits an

almost complex structure.

(2) although S4×S4does not admit an almost complex structure, Remark 1.4

tell us that S4×S4]CP4admits an almost complex structure.

As an application, combing the congruence (1.1) with Theorem 1.12 and

Corollary 1.10, we can get that

Corollary 1.14. αCP2n]βCP2nadmits an almost complex structure if and only if

αis odd.

The proof of this corollary is left to the reader.

Remark 1.15.For n=1 and 2, the facts of this corollary have been obtained by

Audin [1] and M¨

uller and Geiges [8] respectively.

Remark 1.16.The necessary and suﬃcient conditions for the existence of almost

complex structure on αCPn]βCPnare investigate by Sato and Suzuki [9, p. 102,

Proposition]. Unfortunately, their results are not correct.

The proof of Theorems 1.8 and 1.12 will be given in section 2.

2. Proof of Theorems 1.8 and 1.12

In order to prove our main results Theorems 1.8 and 1.12, we need some

preliminaries.

Let Mbe a 2n-dimensional oriented manifold with tangent bundle T M. For

a complex vector bundle ηover M, denote by ¯ηand ηRthe conjugate and real

reduction bundle of ηrespectively. It is known that

ηR=¯ηR.

If the complex vector bundle ηsatisﬁes that ηRis isomorphic (resp. stably iso-

morphic) to T M, it follows from the deﬁnition of almost (resp. stable almost)

complex structure that ηdetermines an almost (resp. a stable almost) complex

structure on M, and we denote it as Jη(resp. ˜

Jη).

For the complex projective space CPn, denote by γthe canonical complex

line bundle over CPn,T P the real tangent bundle of CPnand T P the real tangent

bundle of CPn.

Moreover, we should use the results and conventions of Kahn [6]. Let Jbe

an almost complex structure on M−D2nfor some embedded disc D2n. Denote by

o(M,J)∈H2n(M;π2n−1(S O(2n)/U(n)))

CONNECTED SUMS OF ALMOST COMPLEX MANIFOLDS 5

the obstruction to extending Jas an almost complex structure over M, and set

o[M,J]=ho(M,J),[M]i

where [M] is the fundamental class of Mand h,iis the Kronecker product.

Then we have the following statements from Kahn [6] (cf. Geiges [3]):

Lemma 2.1 (Kahn).o[S2n,J]is independent of J and will be written as o[S2n].

Lemma 2.2 (Kahn).Almost complex structure J on M−D2nand J0on M0−D2n

give rise to a natural almost complex structure J +J0on M]M0−D2n(which

coincides with J resp. J0along the (2n−1)-skeleton of M]M0) such that

o[M]M0,J+J0]=o[M,J]+o[M0,J0]−o[S2n].

Lemma 2.3 (Kahn).Let J be an almost complex structure on M−D2nthat extends

over M as a stable almost complex structure ˜

J. Then

o[M,J]=1

2χ(M)−cn[˜

J]o[S2n],

where cn[˜

J]is the top Chern number of ˜

J.

Now we are in position to prove the Theorems 1.8 and 1.12.

Proof of Theorem 1.8. Firstly, let us consider the stable almost complex structure

on CP2n. Let η=(2n−1)γ+2 ¯γ. It is known that

ηR=(2n+1)γR=(2n+1)¯γR

is stably isomorphic to T P. Hence ηdetermines a stable almost complex structure

˜

Jηon CP2nand the total Chern class of ˜

Jηis

c(˜

Jη)=c(η)=c(γ)2n−1c(¯γ)2=(1 +x)2n−1(1 −x)2,

where x∈H2(CP2n;Z) is the ﬁrst Chern class of γ. Therefore,

c2n[˜

Jη]=c2n[η]=2n−3.

Since CP2n−D4nis homotopic equivalent to CP2n−1and the coeﬃcient groups

πr(S O(4n)/U(2n)) for the obstructions to an almost complex structure are stable

for r<4n−1 (cf. Massey [7]), the stable almost complex structure ˜

Jηinduces an

almost complex structure Jηon CP2n−D4n. Then it follows from Lemma 2.3 that

o[CP2n,Jη]=2o[S4n].

Now denote by Jithe given almost complex structures on Mi,1≤i≤α. It

is clearly that

o[Mi,Ji]=0.

Consequently, Lemma 2.2 implies that

o[]α

i=1Mi](α−1)CP2n,Σα

i=1Ji+(α−1)Jη]

= Σα

i=1o[Mi,Ji]+(α−1)o[CP2n,Jη]−(2α−2)o[S4n]

=0.

6 HUIJUN YANG

This completes the proof.

Proof of Theorem 1.12. Firstly, let us consider the stable almost complex struc-

tures on CPn. In the stable range, it is obviously that the tangent bundle T P of

CPnis stably isomorphic to the tangent bundle T P of CPn. Thus, let η=nγ+¯γ,

it is follows that

ηR=(n+1)γR=(n+1)¯γR

is stably isomorphic to T P. Hence ηdetermines a stable almost complex structure

˜

Jηon CPnand the total Chern class of ˜

Jηis

c(˜

Jη)=c(η)=c(γ)nc(¯γ)=(1 +x)n(1 −x).

Therefore,

cn[˜

Jη]=cn[η]=hcn(η),[CPn]i=n−1.

Then, as in the proof of the Theorem 1.8, the stable almost complex structure

˜

Jηinduces an almost complex structure Jηon CPn−D2n. Hence it follows from

Lemma 2.3 that

o[CPn,Jη]=o[S2n].

Now denote by Jthe given almost complex structures on M. It is clearly

that

o[M,J]=0.

Consequently, Lemma 2.2 implies that

o[M]CPn,J+Jη]=o[M,J]+o[CPn,Jη]−o[S2n]=0.

This completes the proof.

Acknowledgment. The author would like to thank the University of Melbourne

where parts of this work to be carried out and also Diarmuid Crowley for his

hospitality. The author is partially supported by the China Scholarship Council

(File No. 201708410052).

References

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CONNECTED SUMS OF ALMOST COMPLEX MANIFOLDS 7

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School of Mathematics and Statistics, Henan University, Kaifeng 475004, Henan, China

E-mail address:yhj@amss.ac.cn