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Almost complex manifolds with total Betti number three

Jiahao Hu Stony Brook University, Department of Mathematics, 100 Nicolls Road, 11794 Stony Brook [email protected]
Abstract.

We show the minimal total Betti number of a closed almost complex manifold of dimension 2​n≥82n\geq 8 is four, thus confirming a conjecture of Sullivan except for dimension 66. Along the way, we prove the only simply connected closed complex manifold having total Betti number three is the complex projective plane.

1. Introduction

The group S​p​(2​n,ℝ)Sp(2n,\mathbb{R}) of symplectic transformations and the group G​L​(n,ℂ)GL(n,\mathbb{C}) of complex linear transformations of a 2​n2n–dimensional real vector space share a common maximal compact subgroup—the unitary group U​(n)U(n). Consequently smooth manifolds admitting almost symplectic structures are the same as those admitting almost complex structures. An almost symplectic structure is a non-degenerate two form ω\omega, while an almost complex structure is an automorphism JJ of the tangent bundle so that J2=−i​dJ^{2}=-id. With any given Riemannian metric, ω\omega and JJ generates one another in view of S​p​(2​n,ℝ)∩O​(2​n)=G​L​(n,ℂ)∩O​(2​n)=U​(n)Sp(2n,\mathbb{R})\cap O(2n)=GL(n,\mathbb{C})\cap O(2n)=U(n). Two different integrability conditions are usually imposed to remove the word almost. They are: the symplectic condition d​ω=0d\omega=0 yielding a local Darboux chart, or the complex condition that the Nijenhaus tensor NJ=0N_{J}=0 yielding a local holomorphic chart.

The symplectic condition is directly reflected in the topology of closed manifolds: the de Rham cohomology class of the symplectic form ω\omega generates a subring that is isomorphic to the de Rham cohomology of ℂ​ℙn\mathbb{C}\mathbb{P}^{n}. This implies:

Fact.

The minimal total Betti number (i.e. the sum of all Betti numbers) of a 2​n2n–dimensional closed symplectic manifold is n+1n+1.

In contrast, however, the topological consequence of the complex condition is widely unknown, except in dimension two and four, featuring the work of Riemann and Kodaira. Motivated by that the complex realm should be mirror to the symplectic realm, Dennis Sullivan proposes to study the minimal total Betti numbers of complex manifolds with respect to their dimensions. In mirror correspondence to the above fact, he formulated:

Conjecture (Sullivan).
111According to Sullivan, he was inspired by how information is beautifully organized through graphs in fluid dynamics, e.g. K41, and observing the minimal total Betti numbers of symplectic manifolds grow linearly with respect to dimension, fitting into a nice linear graph. This conjecture thus gives the corresponding graph for complex manifolds a particularly nice form: first grows linearly and then flats out. See also [1, Fig. 1].

The minimal total Betti number of a 2​n2n–dimensional closed complex manifold is min⁡{n+1,4}\min\{n+1,4\}.

The object of this paper is to confirm this conjecture except when nn is 33. We note total Betti number 44 is achieved by Hopf and Calabi-Eckmann manifolds in all even dimensions. Also a confirmation for n=3n=3 will imply S6S^{6} admits no complex structure.

It turns out quite surprisingly, even though the conjecture is made for complex manifolds, Albanese and Milivojević [1] proved this statement in fact holds in the almost complex category when nn is neither 33 nor of the form 2l2^{l} for l≥10l\geq 10. Our result below settles the latter case they left out.

Theorem 1.1.

Let MM be a closed almost complex manifold of dimension 2​n≥82n\geq 8. Then the total Betti number of MM is ≥4\geq 4.

By Poincaré duality, the total Betti number of a (nonempty positive dimensional) closed orientable manifold is at least two. In [1] it is shown (positive dimensional) almost complex manifolds having total Betti number exactly two must live in dimension 22 and 66. Theorem 1.1 is therefore achieved by studying almost complex manifolds with total Betti number 33. Our main theorem in this direction is:

Theorem 1.2.

Let MM be a positive dimensional closed almost complex manifold whose total Betti number is 33. Then dimM=4\dim M=4 and MM is complex cobordant to ℂ​ℙ2\mathbb{C}\mathbb{P}^{2}.

As consequences of Theorem 1.2, we immediately obtain:

Theorem 1.3.

Let MM be a simply connected closed almost complex manifold whose total Betti number is 33. Then MM is homeomorphic to ℂ​ℙ2\mathbb{C}\mathbb{P}^{2}.

Theorem 1.4.

Let MM be a simply connected closed complex manifold whose total Betti number is 33. Then MM is biholomorphic to ℂ​ℙ2\mathbb{C}\mathbb{P}^{2}.

Proof of Theorem 1.3.

The only zero–dimensional simply connected manifold is ℝ0\mathbb{R}^{0}, whose total Betti number is one, hence dimM>0\dim M>0. Then by Theorem 1.2, MM is a 44–manifold. Since MM is now assumed to be simply connected, H1​(M;ℤ)=0H_{1}(M;\mathbb{Z})=0. Thus by the universal coefficient theorem, H2​(M;ℤ)H^{2}(M;\mathbb{Z}) is free. Furthermore by Poincaré duality and the total Betti number three assumption, H2​(M;ℤ)H^{2}(M;\mathbb{Z}) is of rank one. So the intersection form of MM is determined by its signature, which is equal to one by Theorem 1.2 (because signature is a cobordism invariant). Our theorem then follows from Freedman’s theorem [2, Theorem 1.5] that the homeomorphism type of a simply connected closed smooth 44–manifold is determined by its intersection form. ∎

Proof of Theorem 1.4.

This follows from Theorem 1.3 and Yau’s theorem [7, Theorem 5] that any compact complex surface homotopy equivalent to ℂ​ℙ2\mathbb{C}\mathbb{P}^{2} is biholomorphic to ℂ​ℙ2\mathbb{C}\mathbb{P}^{2}. ∎

It should be noted the simply-connectedness assumption is essential to both Theorem 1.3 and Theorem 1.4. There do exist non-simply-connected complex surfaces, so called fake projective planes (see e.g. [6]), having the same Betti numbers as ℂ​ℙ2\mathbb{C}\mathbb{P}^{2}.

The heart of our proof of Theorem 1.2 lies in the integrality of the signature and (especially) the Todd genus of an almost complex manifold. Another important ingredient is, on an almost complex manifold the Todd class coincides with the A^\hat{A} class up to the exponential of half the first Chern class (see [3, pp. 197]):

(1.1) A^=ec1/2​Td.\hat{A}=e^{c_{1}/2}\operatorname{Td}.

In particular if all the Chern numbers that involves the first Chern class are zero, then the Todd genus is equal to the A^\hat{A} genus. Our major technical tool is therefore the intimate relation between the signature and the A^\hat{A} genus, which we will establish in Section 2. In Section 3 we prove Theorem 1.2 and derive Theorem 1.1 as an easy corollary.

Acknowledgements

Theorem 1.1 was also independently obtained by Zhixu Su and was announced by her in a conference in 2018; her argument differs from ours (see Remark 3.3 for a sketch of her proof). The author would like to thank Su for informing him on this matter and for sharing her proof. The author also wishes to thank Dennis Sullivan and Aleksandar Milivojević for their encouragement, helpful discussions and valuable comments on early versions of this paper.

2. Relation between the signature and A^\hat{A} genus

Let us introduce our notation. Throughout MM is a nonempty closed oriented smooth manifold of dimension dimM=2​n>0\dim M=2n>0. The total Betti number of MM is the sum of all Betti numbers of MM, namely ∑i≥0dimHi​(M;ℚ)\sum_{i\geq 0}\dim H^{i}(M;\mathbb{Q}). By pip_{i} we mean the ithi^{\text{th}} Pontryagin class of MM. If MM is further almost complex, cic_{i} will be its ithi^{\text{th}} Chern class. The symbol ∫M\int_{M} means pairing a cohomology class with the fundamental class of MM. The signature, Euler characteristic, A^\hat{A} genus and Todd genus (if defined) of MM will be denoted by σ​(M)\sigma(M), χ​(M)\chi(M), A^​(M)\hat{A}(M) and Td⁡(M)\operatorname{Td}(M) respectively.

Since the appearance of Hirzebruch’s celebrated work on multiplicative sequences and his signature theorem, σ​(M)\sigma(M) and A^​(M)\hat{A}(M) are known to be rational linear combinations of Pontryagin numbers. That is,

(2.1) σ​(M)=∑hi1,…,ir​∫Mpi1​⋯​pir,A^​(M)=∑ai1,…,ir​∫Mpi1​⋯​pir.\sigma(M)=\sum h_{i_{1},\dots,i_{r}}\int_{M}p_{i_{1}}\cdots p_{i_{r}},\quad\hat{A}(M)=\sum a_{i_{1},\dots,i_{r}}\int_{M}p_{i_{1}}\cdots p_{i_{r}}.

The coefficients of ∫Mpm\int_{M}p_{m} can be obtained by applying the Cauchy formula to the characteristic power series associated to the signature and A^\hat{A} genus, which are Qσ​(z)=ztanh⁡zQ_{\sigma}(z)=\frac{\sqrt{z}}{\tanh\sqrt{z}} and QA^​(z)=z/2sinh⁡(z/2)Q_{\hat{A}}(z)=\frac{\sqrt{z}/2}{\sinh(\sqrt{z}/2)} respectively (see [4, pp. 16,19], [3, pp.9-11]). The results are well-known (cf. [3, pp. 12-13]):

hm=22​m​(22​m−1−1)(2​m)!​Bm,am=−Bm2​(2​m)!.h_{m}=\frac{2^{2m}(2^{2m-1}-1)}{(2m)!}B_{m},\quad a_{m}=\frac{-B_{m}}{2(2m)!}.

Here

(2.2) Bm=(2​m)!⋅ζ​(2​m)22​m−1​π2​mB_{m}=\frac{(2m)!\cdot\zeta(2m)}{2^{2m-1}\pi^{2m}}

is the mthm^{\text{th}} nontrivial Bernoulli number without sign, ζ\zeta is the Riemann zeta function (see [4, pp. 129-131]). For instance

B1=16,h1=13,a1=−124;B2=130,h2=745,a2=−11440.B_{1}=\frac{1}{6},h_{1}=\frac{1}{3},a_{1}=-\frac{1}{24};B_{2}=\frac{1}{30},h_{2}=\frac{7}{45},a_{2}=-\frac{1}{1440}.

Clearly hmh_{m} and ama_{m} are related by

(2.3) hm=−22​m+1​(22​m−1−1)​am.h_{m}=-2^{2m+1}(2^{2m-1}-1)a_{m}.

This immediately proves the following well-known lemma (cf. [4, pp. 90]).

Lemma 2.1.

Let MM be a manifold of dimension 4​k4k with ∫Mpk\int_{M}p_{k} being the only possibly nonzero Pontryagin number. Then

σ​(M)+22​k+1​(22​k−1−1)​A^​(M)=0.\sigma(M)+2^{2k+1}(2^{2k-1}-1)\hat{A}(M)=0.

We will need a generalization of Lemma 2.1 in the following situation.

Lemma 2.2.

Let MM be a manifold of dimension 8​k8k with ∫Mpk2\int_{M}p_{k}^{2} and ∫Mp2​k\int_{M}p_{2k} being the only possibly nonzero Pontryagin numbers. Then

σ​(M)+24​k+1​(24​k−1−1)​A^​(M)=24​k​(22​k−1)2​(Bk2​(2​k)!)2​∫Mpk2.\sigma(M)+2^{4k+1}(2^{4k-1}-1)\hat{A}(M)=2^{4k}(2^{2k}-1)^{2}\Big{(}\frac{B_{k}}{2(2k)!}\Big{)}^{2}\int_{M}p_{k}^{2}.
Proof.

From Equation 2.1 we have

σ​(M)=h2​k​∫Mp2​k+hk,k​∫Mpk2,A^​(M)=a2​k​∫Mp2​k+ak,k​∫Mpk2.\sigma(M)=h_{2k}\int_{M}p_{2k}+h_{k,k}\int_{M}p_{k}^{2},\quad\hat{A}(M)=a_{2k}\int_{M}p_{2k}+a_{k,k}\int_{M}p_{k}^{2}.

By [3, Lemma 1.4.1] hk,kh_{k,k} and ak,ka_{k,k} can be expressed as

(2.4) hk,k=12​hk2−12​h2​k,ak,k=12​ak2−12​a2​k.h_{k,k}=\frac{1}{2}h_{k}^{2}-\frac{1}{2}h_{2k},\quad a_{k,k}=\frac{1}{2}a_{k}^{2}-\frac{1}{2}a_{2k}.

Therefore we get

σ​(M)\displaystyle\sigma(M) +24​k+1​(24​k−1−1)​A^​(M)\displaystyle+2^{4k+1}(2^{4k-1}-1)\hat{A}(M)
=(h2​k+24​k+1​(24​k−1−1)​a2​k)​∫Mp2​k+(hk,k+24​k+1​(24​k−1−1)​ak,k)​∫Mpk2\displaystyle=(h_{2k}+2^{4k+1}(2^{4k-1}-1)a_{2k})\int_{M}p_{2k}+(h_{k,k}+2^{4k+1}(2^{4k-1}-1)a_{k,k})\int_{M}p_{k}^{2}
=24​k​(22​k−1)2​ak2​∫Mpk2\displaystyle=2^{4k}(2^{2k}-1)^{2}a_{k}^{2}\int_{M}p_{k}^{2}
=24​k​(22​k−1)2​(Bk2​(2​k)!)2​∫Mpk2.\displaystyle=2^{4k}(2^{2k}-1)^{2}\Big{(}\frac{B_{k}}{2(2k)!}\Big{)}^{2}\int_{M}p_{k}^{2}.

Here the second to last equality follows from Equation 2.3 and Equation 2.4. ∎

Next we turn to almost complex manifolds.

Lemma 2.3.

Let MM be an almost complex manifold of dimension 8​k8k with ∫Mc2​k2\int_{M}c_{2k}^{2} and ∫Mc4​k\int_{M}c_{4k} being the only possibly nonzero Chern numbers. Then

24​k+1​[(24​k−1−1)​(1−rk)+(3−22​k+1)]⋅Td⁡(M)=(1+rk)⋅σ​(M)−24​k+1​(22​k−1)2​B2​k(4​k)!⋅χ​(M),2^{4k+1}\Big{[}(2^{4k-1}-1)(1-r_{k})+(3-2^{2k+1})\Big{]}\cdot\operatorname{Td}(M)=(1+r_{k})\cdot\sigma(M)-2^{4k+1}(2^{2k}-1)^{2}\frac{B_{2k}}{(4k)!}\cdot\chi(M),

where rkr_{k} is defined by (4​k2​k)​Bk2⋅rk=B2​k\binom{4k}{2k}B_{k}^{2}\cdot r_{k}=B_{2k}.

Proof.

Since MM is assumed to be almost complex, its top Chern class coincides with its Euler class, therefore ∫Mc4​k=χ​(M)\int_{M}c_{4k}=\chi(M). Also its Pontryagin numbers can be recovered from its Chern numbers. The only possibly nonzero Pontryagin numbers are:

∫Mpk2=∫M[(−1)k​2​c2​k]2=4​∫Mc2​k2,∫Mp2​k=∫Mc2​k2+2​∫Mc4​k.\int_{M}p_{k}^{2}=\int_{M}[(-1)^{k}2c_{2k}]^{2}=4\int_{M}c_{2k}^{2},\quad\int_{M}p_{2k}=\int_{M}c_{2k}^{2}+2\int_{M}c_{4k}.

Notice all the Chern numbers involving c1c_{1} vanish, so by Equation 1.1 we have A^​(M)=Td⁡(M)\hat{A}(M)=\operatorname{Td}(M).

Therefore from the proof of Lemma 2.2 we have

(2.5) σ​(M)+24​k+1​(24​k−1−1)​Td⁡(M)=24​k+2​(22​k−1)2​ak2​∫Mc2​k2.\sigma(M)+2^{4k+1}(2^{4k-1}-1)\operatorname{Td}(M)=2^{4k+2}(2^{2k}-1)^{2}a_{k}^{2}\int_{M}c_{2k}^{2}.

On the other hand, we also have

(2.6) Td⁡(M)=A^​(M)=a2​k​∫Mp2​k+ak,k​∫Mpk2=(a2​k+4​ak,k)​∫Mc2​k2+2​a2​k​χ​(M).\operatorname{Td}(M)=\hat{A}(M)=a_{2k}\int_{M}p_{2k}+a_{k,k}\int_{M}p_{k}^{2}=(a_{2k}+4a_{k,k})\int_{M}c_{2k}^{2}+2a_{2k}\chi(M).

Combining Equation 2.5 and Equation 2.6 to eliminate ∫Mc2​k2\int_{M}c_{2k}^{2}, we get

(2.7) 24​k+2​(22​k−1)2​ak2​[Td⁡(M)−2​a2​k​χ​(M)]=(a2​k+4​ak,k)​[σ​(M)+24​k+1​(24​k−1−1)​Td⁡(M)].2^{4k+2}(2^{2k}-1)^{2}a_{k}^{2}[\operatorname{Td}(M)-2a_{2k}\chi(M)]=(a_{2k}+4a_{k,k})[\sigma(M)+2^{4k+1}(2^{4k-1}-1)\operatorname{Td}(M)].

We note

a2​k+4​ak,k2​ak2=1+rk.\frac{a_{2k}+4a_{k,k}}{2a_{k}^{2}}=1+r_{k}.

Then the desired identity is a straightforward consequence of Equation 2.7 by dividing by 2​ak22a_{k}^{2} and reorganizing the terms. ∎

We end this section by estimating the sizes and the relative sizes of the coefficients appearing in Lemma 2.3. For simplicity, let us denote

CTd=24​k+1​[(24​k−1−1)​(1−rk)+(3−22​k+1)],Cσ=1+rk,Cχ=24​k+1​(22​k−1)2​B2​k(4​k)!.C_{\operatorname{Td}}=2^{4k+1}[(2^{4k-1}-1)(1-r_{k})+(3-2^{2k+1})],\quad C_{\sigma}=1+r_{k},\quad C_{\chi}=2^{4k+1}(2^{2k}-1)^{2}\frac{B_{2k}}{(4k)!}.

Then the conclusion of Lemma 2.3 reads as

CTd⋅Td⁡(M)=Cσ⋅σ​(M)−Cχ⋅χ​(M).C_{\operatorname{Td}}\cdot\operatorname{Td}(M)=C_{\sigma}\cdot\sigma(M)-C_{\chi}\cdot\chi(M).
Lemma 2.4.

We have the estimates

1<Cσ<32,0<Cχ<8​(2π)4​k,CTd≥35⋅28​k−3.1<C_{\sigma}<\frac{3}{2},\quad 0<C_{\chi}<8(\frac{2}{\pi})^{4k},\quad C_{\operatorname{Td}}\geq\frac{3}{5}\cdot 2^{8k-3}.
Proof.

By Equation 2.2 and the definition of rkr_{k}, we see rk=12​ζ​(4​k)ζ​(2​k)2r_{k}=\frac{1}{2}\frac{\zeta(4k)}{\zeta(2k)^{2}}. Since

ζ​(2​k)2=(∑m≥11m2​k)​(∑m≥11m2​k)>∑m≥11m4​k=ζ​(4​k),\zeta(2k)^{2}=\Big{(}\sum_{m\geq 1}\frac{1}{m^{2k}}\Big{)}\Big{(}\sum_{m\geq 1}\frac{1}{m^{2k}}\Big{)}>\sum_{m\geq 1}\frac{1}{m^{4k}}=\zeta(4k),

we have

1<Cσ=1+rk<32.1<C_{\sigma}=1+r_{k}<\frac{3}{2}.

By Equation 2.2 and ζ​(4​k)<ζ​(2)=π2/6<2\zeta(4k)<\zeta(2)=\pi^{2}/6<2 we get

0<Cχ=4​(22​k−1π2​k)2​ζ​(4​k)<8​(2π)4​k.0<C_{\chi}=4\Big{(}\frac{2^{2k}-1}{\pi^{2k}}\Big{)}^{2}\zeta(4k)<8(\frac{2}{\pi})^{4k}.

Finally, for k≥2k\geq 2

CTd>24​k+1​[12​(24​k−1−1)+(3−22​k+1)]>24​k+1​[24​k−2−22​k+1]≥28​k−2>35⋅28​k−3.C_{\operatorname{Td}}>2^{4k+1}[\frac{1}{2}(2^{4k-1}-1)+(3-2^{2k+1})]>2^{4k+1}[2^{4k-2}-2^{2k+1}]\geq 2^{8k-2}>\frac{3}{5}\cdot 2^{8k-3}.

As for k=1k=1, one can directly verify r1=15r_{1}=\frac{1}{5} and CTd=35⋅25C_{\operatorname{Td}}=\frac{3}{5}\cdot 2^{5}.

∎

Corollary 2.5.

We have the further estimates

CTdCσ+Cχ>4k,CσCχ≥3k.\frac{C_{\operatorname{Td}}}{C_{\sigma}+C_{\chi}}>4^{k},\quad\frac{C_{\sigma}}{C_{\chi}}\geq 3^{k}.
Proof.

By Lemma 2.4

CTdCσ+Cχ>3⋅28​k−315/2+40​(2/π)4​k≥3⋅28​k−315/2+40​(2/π)4>28​k−35>4k,\frac{C_{\operatorname{Td}}}{C_{\sigma}+C_{\chi}}>\frac{3\cdot 2^{8k-3}}{15/2+40(2/\pi)^{4k}}\geq\frac{3\cdot 2^{8k-3}}{15/2+40(2/\pi)^{4}}>\frac{2^{8k-3}}{5}>4^{k},

and

CσCχ>18​(π2)4​k≥3kfor ​k≥3.\frac{C_{\sigma}}{C_{\chi}}>\frac{1}{8}(\frac{\pi}{2})^{4k}\geq 3^{k}\quad\text{for }k\geq 3.

For k=1,2k=1,2 one can directly verify CσCχ=3,15\frac{C_{\sigma}}{C_{\chi}}=3,15 respectively, so the desired estimate holds. ∎

The estimates in Lemma 2.4 and Corollary 2.5 are not sharp. Nevertheless they will be sufficient for our applications.

3. Proof of main theorems

In this section we prove Theorem 1.1 and Theorem 1.2. Recall the following theorem of Milnor (cf. [5]).

Theorem 3.1 (Milnor).

Two stably almost complex manifolds are complex cobordant if and only if they have the same Chern numbers.

Proposition 3.2.

If MM is an almost complex manifold of dimension 8​k8k with ∫Mc2​k2\int_{M}c_{2k}^{2} and ∫Mc4​k\int_{M}c_{4k} being the only possibly nonzero Chern numbers. Then either MM is complex cobordant to the empty manifold or the total Betti number of MM is at least 3k3^{k}.

Proof.

If Td⁡(M)≠0\operatorname{Td}(M)\neq 0, then since Td⁡(M)\operatorname{Td}(M) is an integer, by Lemma 2.3 and Corollary 2.5 we get

Total Betti number≥max⁡{|σ​(M)|,|χ​(M)|}>CTdCσ+Cχ​|Td⁡(M)|>4k.\text{Total Betti number}\geq\max\{|\sigma(M)|,|\chi(M)|\}>\frac{C_{\operatorname{Td}}}{C_{\sigma}+C_{\chi}}|\operatorname{Td}(M)|>4^{k}.

If Td⁡(M)=0\operatorname{Td}(M)=0 but σ​(M)≠0\sigma(M)\neq 0, then again by Lemma 2.3 and Corollary 2.5 we have

Total Betti number≥|χ​(M)|=CσCχ​|σ​(M)|≥3k.\text{Total Betti number}\geq|\chi(M)|=\frac{C_{\sigma}}{C_{\chi}}|\sigma(M)|\geq 3^{k}.

If Td⁡(M)=σ​(M)=0\operatorname{Td}(M)=\sigma(M)=0, then from Equation 2.5 and Equation 2.6 we have ∫Mc2​k2=0\int_{M}c_{2k}^{2}=0 and ∫Mc4​k=χ​(M)=0\int_{M}c_{4k}=\chi(M)=0. So all the Chern numbers of MM are zero, hence MM is complex cobordant to the empty manifold by Theorem 3.1. ∎

We are ready to prove our main theorems. See 1.2

Proof.

By [1, Theorem 2.2, Theorem 3.3, Remark 4.2] dimM\dim M is either 44 or divisible by 2112^{11}. We must show the latter cannot happen. Assume otherwise dimM=8​k\dim M=8k for some k≥2k\geq 2. Since the total Betti number of MM is assumed to be three, by Poincaré duality the only non-trivial rational cohomology groups are H0H^{0}, H4​kH^{4k} and H8​kH^{8k}, each of which is one-dimensional. Therefore the only possibly nonzero Chern numbers are ∫Mc2​k2\int_{M}c_{2k}^{2} and ∫Mc4​k\int_{M}c_{4k}. Then by Proposition 3.2, either MM is complex cobordant to the empty manifold which contradicts with ∫Mc4​k=χ​(M)=3≠0\int_{M}c_{4k}=\chi(M)=3\neq 0 (if it is complex cobordant to the empty manifold then all its Chern numbers are zero by Theorem 3.1), or the total Betti number is at least 3k≥93^{k}\geq 9 which contradicts the total Betti number three assumption. This completes the proof of our first assertion.

To prove the second assertion, notice now ∫Mc2=χ​(M)=3\int_{M}c_{2}=\chi(M)=3 and

σ​(M)=∫Mp13=∫M13​(c12−2​c2)=13​∫Mc12−2.\sigma(M)=\int_{M}\frac{p_{1}}{3}=\int_{M}\frac{1}{3}(c_{1}^{2}-2c_{2})=\frac{1}{3}\int_{M}c_{1}^{2}-2.

Meanwhile (cf. [3, pp. 14]),

Td⁡(M)=∫M112​(c12+c2)=112​∫Mc12+14.\operatorname{Td}(M)=\int_{M}\frac{1}{12}(c_{1}^{2}+c_{2})=\frac{1}{12}\int_{M}c_{1}^{2}+\frac{1}{4}.

It follows that σ​(M)+3=4⋅Td⁡(M)\sigma(M)+3=4\cdot\operatorname{Td}(M), so σ​(M)≡1\sigma(M)\equiv 1 modulo 44. Similar as before, by Poincaré duality and the total Betti number three assumption, the middle cohomology is one-dimensional. So σ​(M)\sigma(M) must be one. This in turn implies ∫Mc12=9\int_{M}c_{1}^{2}=9. We observe the Chern numbers of MM are the same as those of ℂ​ℙ2\mathbb{C}\mathbb{P}^{2}. So by Theorem 3.1 we conclude MM is complex cobordant to ℂ​ℙ2\mathbb{C}\mathbb{P}^{2}. ∎

Remark 3.3.

We here communicate another proof of that 8​k8k-manifolds with total Betti number three cannot admit almost complex structures, independently obtained by Zhixu Su222Our proof above was attained in summer 2019; later the author learned from Su that she had also proved this and announced it in a conference in 2018.. Briefly, in such case the Chern classes c1,…,c2​k−1c_{1},\dots,c_{2k-1} are torsion, then from Hattori-Stong relations one can show ∫Mc2​k2\int_{M}c_{2k}^{2} is divisible by [(2​k−1)!]2[(2k-1)!]^{2}. This extra divisibility combined with a careful 22-adic examination of the signature equation gives the result.

See 1.1

Proof.

This follows from Theorem 1.2 and [1, Theorem 2.2] that almost complex manifolds with total Betti number two exist only in dimension 22 and 66. ∎

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