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Connected sums of almost complex manifolds

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Abstract

In this paper, firstly, for some $4n$-dimensional almost complex manifolds $M_{i}, ~1\le i \le \alpha$, we prove that $\left(\sharp_{i=1}^{\alpha} M_{i}\right) \sharp (\alpha{-}1) \mathbb{C} P^{2n}$ must admits an almost complex structure, where $\alpha$ is a positive integer. Secondly, for a $2n$-dimensional almost complex manifold $M$, we get that $M\sharp \overline{\mathbb{C} P^{n}}$ also admits an almost complex structure. At last, as an application, we obtain that $\alpha\mathbb{C} P^{2n}\sharp \beta\overline{\mathbb{C} P^{2n}}$ admits an almost complex structure if and only if $\alpha$ is odd.
CONNECTED SUMS OF ALMOST COMPLEX MANIFOLDS
HUIJUN YANG
Abstract. In this paper, firstly, for any 4n-dimensional almost complex man-
ifolds Mi,1iα, we prove that ]α
i=1Mi](α1)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 αCP2nCP2n
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 kT 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 n1 (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,1iα, be 4n-dimensional almost
complex manifolds (may be dierent). Denote by αM1the connected sum of α
copies of M1, and ]α
i=1Mithe connected sum of Mi,1iα. Then we have
χ(]α
i=1Mi)= Σα
i=1χ(Mi)2(α1),(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,1iα, 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,1iαare 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 dieomorphism to CP2n+1](α1)CP2n+1which
is a blow up of CP2n+1in (α1) 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,1iα, 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 α1 mod 1152.
In fact, if we set M4mbe the set of (4m1)-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 MiM4m,1iα,]α
i=1Miadmits an almost complex
structure if and only if α=1.
CONNECTED SUMS OF ALMOST COMPLEX MANIFOLDS 3
Proof. For a (4m1)-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×S4M4,
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,1iα, the connected sum
]α
i=1Mi](α1)CP2n
must admits an almost complex structure.
Remark 1.9.For n2 and 2n-dimensional almost complex manifolds Mi,1
iα, Geiges has proved in [3, Lemma 2] that ]α
i=1Mi](α1)S2×S2n2must
admits an almost complex structure. However, S2×S2n2does not admits an
almost complex structure whence n4 (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 k4.
However, for odd n,S2n]CPn=CPnis dieomorphic 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. αCP2nCP2nadmits 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 sucient conditions for the existence of almost
complex structure on αCPnCPnare 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 MD2nfor some embedded disc D2n. Denote by
o(M,J)H2n(M;π2n1(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 MD2nand J0on M0D2n
give rise to a natural almost complex structure J +J0on M]M0D2n(which
coincides with J resp. J0along the (2n1)-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 MD2nthat 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=(2n1)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)2n1c(¯γ)2=(1 +x)2n1(1 x)2,
where xH2(CP2n;Z) is the first Chern class of γ. Therefore,
c2n[˜
Jη]=c2n[η]=2n3.
Since CP2nD4nis homotopic equivalent to CP2n1and the coecient groups
πr(S O(4n)/U(2n)) for the obstructions to an almost complex structure are stable
for r<4n1 (cf. Massey [7]), the stable almost complex structure ˜
Jηinduces an
almost complex structure Jηon CP2nD4n. Then it follows from Lemma 2.3 that
o[CP2n,Jη]=2o[S4n].
Now denote by Jithe given almost complex structures on Mi,1iα. It
is clearly that
o[Mi,Ji]=0.
Consequently, Lemma 2.2 implies that
o[]α
i=1Mi](α1)CP2n,Σα
i=1Ji+(α1)Jη]
= Σα
i=1o[Mi,Ji]+(α1)o[CP2n,Jη](2α2)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=n1.
Then, as in the proof of the Theorem 1.8, the stable almost complex structure
˜
Jηinduces an almost complex structure Jηon CPnD2n. 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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ele Audin, Exemples de vari´et ´es presque complexes, Enseign. Math. (2) 37 (1991),
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CONNECTED SUMS OF ALMOST COMPLEX MANIFOLDS 7
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[10] W. A. Sutherland, A note on almost complex and weakly complex structures, J. London
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School of Mathematics and Statistics, Henan University, Kaifeng 475004, Henan, China
E-mail address:yhj@amss.ac.cn
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