# Almost complex manifold with Betti number $b_{i}=0$ except $i=0,n/2,n$

and  Zhixu Su  Department of Mathematics, University of Washington Seattle
<EMAIL_ADDRESS>
###### Abstract.

This paper studies existence of $n=4k\ (k>1)$ dimensional simply-connected
closed almost complex manifold with Betti number $b_{i}=0$ except $i=0,n/2,n$.
We characterize all the rational cohomology rings of such manifolds and show
they must have even Euler characteristic and even signature, which is to say
the middle Betti number $b_{n/2}$ must be even. Parallel to the author’s
earlier work on realizing rational cohomology ring by smooth closed manifolds,
we state and prove Sullivan’s rational surgery realization theorem for almost
complex manifold and demonstrate its application in our context. A prescribed
rational cohomology ring can be realized by a simply connected almost complex
manifold if and only if the ring structure supports the intersection form of a
closed manifold, and it holds Chern numbers that satisfy the signature
equation and the Riemann–Roch integrality relations, and the top Chern number
equals the Euler characteristic. According to Stong’s characterization of $U$
and $SU$ cobordism, we explicitly compute the Riemann–Roch integrality
relations among Chern numbers in the case when only the middle and top Chern
classes can be nonzero. The necessary and sufficient conditions for
realization are expressed as a set of congruence relations among the signature
and Euler characteristic, we show that the lower bounds of the 2–adic order of
the signature and the Euler characteristic increase with respect to the
dimension of the realizing manifold.

##  1\. introduction

Let $H^{*}=\bigoplus_{i=0}^{n}H^{i}$ be a $1$–connected ${\mathbb{Q}}$
Poincaré duality algebra of dimension $n=4k\,(k>1)$, i.e.,
$H^{*}=\bigoplus_{i=0}^{n}H^{i}$ is a graded commutative algebra over
${\mathbb{Q}}$ with $H^{i}$ finite dimensional for each $i$, $H^{1}=0$,
$H^{0}\cong H^{n}={\mathbb{Q}}$, and the multiplication defines a
non–degenerate bilinear mapping $H^{i}\times H^{n-i}\to
H^{n}(\cong{\mathbb{Q}})$ sending $(\alpha,\beta)\mapsto\alpha\beta$, which
induces the Poincaré duality isomorphism $H^{n-i}\cong(H^{i})^{*}$.

Prescribing a $1$–connected ${\mathbb{Q}}$ Poincaré duality algebra $H^{*}$,
does there exist a simply connected smooth closed manifold $M^{n}$ whose
rational cohomology ring $H^{*}(M;{\mathbb{Q}})\cong H^{*}$? Sullivan’s
rational surgery realization theorem ([Sul77, Theorem 13.2], [Bar76, Theorem
1]) provided necessary and sufficient conditions to answer this question.
There has been work studying realization of certain prescribed algebras. In
[PL07], Papadima and Pǎunescu analyzed the smoothing problem of Artinian
complete intersection below dimension $12$. In the author’s PhD thesis [Su09]
under the supervision of Jim Davis, and later paper [Su14], Sullivan’s theorem
was applied to study existence dimensions of rational projective plane, which
is named for smooth closed manifolds with rational cohomology ring
$H^{*}(M^{n};{\mathbb{Q}})\cong{\mathbb{Q}}[\alpha]/\langle\alpha^{3}\rangle,|\alpha|=n/2$,
such manifolds have Betti numbers $b_{i}=0$ except $b_{0}=b_{n/2}=b_{n}=1$.
The author’s thesis gave a concrete proof illustrating how a rational degree
one normal map was obtained in rational surgery if the rational cohomology
ring supports Pontryagin numbers of a smooth manifold that satisfy the
Riemann–Roch integrality relations, these relations were computed explicitly
according to the Hattori–Stong Theorem that characterizes the smooth ($SO$)
cobordism group. Combining the relations with the signature equation, the
topological existence problem was reduced to a number–theoretic question
asking whether there exist solutions to a system of Diophantine equations with
rational coefficients involving Bernoulli numbers.

In [Su09] and [Su14], the author showed that above dimension 4, 8, 16, in
which the well-known complex, quaternionic, octonionic projective planes
(${\mathbb{CP}}^{2},{\mathbb{HP}}^{2},{\mathbb{OP}}^{2}$) exist respectively,
the smallest possible dimension a rational projective plane can exist is 32.
In later papers [FS16], [KS17], the author and her collaborators Jim Fowler
and Lee Kennard showed that (1) a rational projective plane of dimension
greater than 4 can only exist in dimensions $n=8k$ where $k=2^{a}$ or
$k=2^{a}+2^{b}$; (2) such manifold exist in dimension $n\leq 512$ if and only
if $n\in\\{4,8,16,32,128,256\\}$, and does not exist in any dimension
$512<n<2^{13}$ except for four possible exceptions; (3) such manifold can not
admit Spin structure if the dimension $n>16$. In 2017, Matthias Kreck
suggested to the author that the rational projective plane problem is an
example of the more general question of realizing prescribed Betti numbers, in
light of which we phrase the question being studied in this paper as in the
title.

In [AM19], Albanese and Milivojević showed that a closed almost complex
manifold with sum of Betti numbers three can only exist in dimension
$n=2^{a}$. Note that a closed smooth manifold with sum of Betti number three
is exactly what we called a rational projective plane, existence of such
smooth manifold was only confirmed in dimension $n=2^{a}$ with
$a\in\\{2,3,4,5,7,8\\}$. In [FS16] we showed that the signature equation
written in terms of the Pontryagin numbers $p_{n/4}$ and $p_{n/8}^{2}$ can
obstruct existence of such smooth manifold in any dimension other than
$n=8(2^{a})$ or $n=8(2^{a}+2^{b})$ when $n>4$, while Albanese and Milivojević
expressed the Pontryagin numbers in terms of the Chern numbers $c_{n/2}$ and
$c_{n/4}^{2}$ in the signature equation to obtain their result on almost
complex manifold.

In order to conclude if there exists any almost complex rational projective
plane in dimension $n=2^{a}$, the approach that worked for smooth projective
plane suggests combining the signature equation with the Riemann–Roch
integrality relations among Chern numbers of almost complex manifolds.
Following up Albanese and Milivojević’s result published on arXiv in 2018, the
author adopted one particular integrality condition of the Chern number
$c_{n/4}^{2}$ according to Stong’s theorem characterizing the almost complex
($U$) cobordism, it turned out that this condition combined with the signature
equation can rule out the existence of almost complex rational projective
planes in any dimension $n>4$ (see Corollary 14), i.e.,there does not exist
any closed almost complex manifold with sum of Betti numbers three above
dimension 4. The author did not publish this result but had announced it in a
conference in summer 2018.

Motivated by the negative result in realizing middle Betti number $b_{n/2}=1$,
this paper studies existence of $n$–dimensional almost complex manifold with
Betti number $b_{i}=0$ except $b_{0}=b_{n}=1$ and $b_{n/2}\geq 1$. Parallel to
the smooth manifold realization problem, we apply the version of Sullivan’s
theorem for almost complex manifold. In 2019, the author was informed by
Sullivan that his student Jiahao Hu had also obtained the nonexistence result
of almost complex manifold with sum of Betti numbers three ([JH21]), and his
student Aleksandar Milivojević will be writing his thesis on the version of
rational surgery realization theorem on almost complex manifolds. As both
parties had already obtained partial results at the time, the author and
Milivojević communicated with each other and shared ideas on the realization
theorem and its application. The author thank Sullivan and Milivojević for
their communication, in particular, for pointing out that in the proof of the
theorem, the pullback space for surgery is non-simply-connected when the input
map $X\to BU_{0}$ has vanishing $c_{1}$ class. Milivojević’s PhD thesis
[Mil21] gave an exposition of Sullivan’s theorem on realizing rational
homotopy types by closed almost complex manifold, including a careful
explanation on how surgery was performed on a rational degree one normal map
to obtain a rational homotopy equivalence. In this paper, we phrase Sullivan’s
theorem by stating the necessary and sufficient conditions for realizing a
rational cohomology ring by simply-connected closed almost complex manifold,
followed by a proof parallel to the one given in the author’s thesis for
smooth manifold, we explicitly compute the conditions under our Betti number
assumption for any dimension $n=4k$. The author would like to thank Jim Davis
for some helpful discussions.

In section 2, we state Sullivan’s theorem on realizing a rational cohomology
ring by $n=4k$ dimensional simply-connected closed almost complex manifold.
Compared to the original theorem for smooth manifold, the necessary and
sufficient conditions are expressed in terms of Chern numbers instead of
Pontryagin numbers. Parallel to Hattori-Stong integrality relations among
Pontryagin numbers, the Riemann–Roch integrality relations among Chern numbers
of almost complex manifolds are expressed according to Stong’s Theorem
characterizing $U$ and $SU$ cobordism. Depending on the first Chern class
being non-zero or zero, the input Chern classes are either maps from the
rational space $X$ to the classifying space $BU$ or $BSU$ respectively.

In section 3, we show that under the assumption when all Chern classes vanish
except possibly for the middle and top dimensional classes $c_{n/4}$ and
$c_{n/2}$, the Riemann–Roch integrality relations among Chern numbers of
$n=4k$–dimensional almost complex manifold are equivalent to a set of three
sub-relations, the rational coefficients of these relations are computed
explicitly. We also compute the additional relations among Chern numbers of
$SU$ manifold when the dimension $n\equiv 4\pmod{8}$, .

In section 4, we apply the realization theorem to formulate the necessary and
sufficient conditions for a rational cohomology ring under our Betti number
assumption to be realized by a simply-connected almost complex manifold. In
dimension $n=8k$ and $n=8k+4$ respectively, we characterize all realizable
rational cohomology rings by a set of congruence relations among the signature
and Euler characteristic. Examples in dimensions 8, 12, and 16 are computed
explicitly. Then we derive the following divisibility statements on the 2-adic
orders of the signature and Euler characteristic.

######  Theorem A (Theorem 11 and 12).

If $M^{n}$ is an $n=8k$–dimensional almost complex manifold with Betti number
$b_{i}=0$ except $b_{0}=b_{n}=1$ and $b_{n/2}\geq 1$, the 2–adic order of its
signature $\sigma$ and Euler characteristic $\chi$ must satisfy

$\displaystyle\nu_{2}(\sigma)\geq 4k-2\nu_{2}(k)-3,$
$\displaystyle\nu_{2}(\chi)\geq 4k-2\nu_{2}(k)-2\textup{wt}(k)-2.$

For example, in dimension $n=8$, $\sigma\equiv 0\bmod{2}$,

in dimension $n=16$, $\sigma\equiv 0\bmod{2^{3}}$ and $\chi\equiv
0\bmod{2^{2}}$,

in dimension $n=24$, $\sigma\equiv 0\bmod{2^{9}}$ and $\chi\equiv
0\bmod{2^{6}}$.

If $M^{n}$ is an $n=8k+4(k>0)$–dimensional almost complex manifold with Betti
number $b_{i}=0$ except $b_{0}=b_{n}=1$ and $b_{n/2}\geq 1$,

$\displaystyle\nu_{2}(\sigma)\geq 4k+4$ $\displaystyle\nu_{2}(\chi)\geq
4k-2\textup{wt}(k)$

######  Theorem B (Theorem 13 and Corollary 14).

An $n=4k(k>1)$–dimensional closed almost complex manifold with Betti number
$b_{i}=0$ except $b_{0}=b_{n}=1$,$b_{n/2}\geq 1$ must have even signature
$\sigma$ and even Euler characteristic $\chi$, i.e., the middle Betti number
$b_{n/2}$ must be even.

In dimension greater than 4, the rarely existing rational projective plane
(smooth manifold whose Betti number $b_{0}=b_{n/2}=b_{n}=1$) does not admit
any almost complex structure. Equivalently, there does not exist any closed
almost complex manifold whose sum of Betti numbers equals three.

##  2\. The rational surgery realization theorem for almost complex manifold

Firstly, we give a brief review of the original rational surgery realization
theorem on smooth manifold ([Sul77, Theorem 13.2], [Bar76]). If a 1-connected
rational Poincaré duality algebra $H^{*}$ can be realized as the rational
cohomology ring of an $n=4k$–dimensional closed smooth manifold, it must
satisfy the following necessary conditions. The bilinear form
$(H^{2k},\lambda)$, being induced from the intersection form of a closed
manifold, must be isomorphic to an orthogonal sum of copies of $\langle
1\rangle$ and $\langle-1\rangle$ over ${\mathbb{Q}}$. The ring $H^{*}$ must
contain cohomology classes $p_{1},\ldots,p_{k}$, so that the pairing of their
products of total degree $n$ with a fundamental homology class $\mu$ agree
with the Pontryagin numbers of a smooth manifold, so the numbers must satisfy
the Hirzebruch signature equation and the Riemann-Roch Integrality relations
which characterize the smooth ($SO$) cobordism. Sullivan’s theorem showed that
these necessary conditions are sufficient to construct a simply-connected
closed smooth manifold realizing $H^{*}$ using rational surgery. The input
data for rational surgery is a ${\mathbb{Q}}$–local ${\mathbb{Q}}$ Poincaré
duality space $X^{n}$ with cohomology classes $p_{i}\in
H^{4i}(X;{\mathbb{Q}})$ for $i=1,\ldots,k$ and a fundamental class $\mu\in
H_{n}(X;{\mathbb{Z}})\cong{\mathbb{Q}}$. The pullback digram of the
classifying map
$p:X\xrightarrow{(p_{1},\ldots,p_{k})}BSO_{0}\simeq\prod\limits
K(\mathbb{Q},4i)$ and the localization $BSO\to BSO_{(0)}$ gives rise to a
normal map $f:M\to X$ such that $f^{*}(p_{i})=p_{i}(\tau_{M})$. If the
pairings $\langle p_{I},\mu\rangle$ satisfy the Riemann-Roch Integrality
relations, a normal map of degree one in the sense that $f_{*}[M]=\mu$ is
guaranteed to exist. Then surgery can be performed on the candidate normal map
to obtain a rational homotopy equivalence if the intersection form condition
and the signature equation are satisfied to give a vanishing surgery
obstruction.

Analogous to the smooth case, if a 1-connected rational Poincaré duality
algebra $H^{*}$ can be realized as the rational cohomology ring of a
$n=4k$–dimensional closed almost complex manifold, besides the condition on
the intersection form, the ring $H^{*}$ must contain cohomology classes
$c_{1},\ldots,c_{2k}$, so that the pairing of their products of total degree
$n$ with a fundamental homology class $\mu$ agree with the Chern numbers of an
almost complex manifold, therefore the numbers must satisfy the signature
equation and the Riemann-Roch Integrality relations which characterizes the
almost complex ($U$) cobordism, as well as the relations for $SU$ cobordism if
$c_{1}=0$. Moreover, the number $\langle c_{n/2},\mu\rangle$ should match with
the Euler characteristic of $H^{*}$. The following version of Sullivan’s
theorem states that these necessary conditions are also sufficient for the
realization.

######  Theorem 1.

Let $H^{*}=\bigoplus_{i=0}^{n}H^{i}$ be a $1$–connected ${\mathbb{Q}}$
Poincaré duality algebra of dimension $n=4k$ with $k>1$. There exists
$n=4k(k>1)$–dimensional, simply-connected, closed, almost complex manifold $M$
such that $H^{*}(M;{\mathbb{Q}})\cong H^{*}$ if and only if there exist choice
of total class $c=1+c_{1}+c_{2}+\cdots+c_{2k}\in\bigoplus_{i=0}^{2k}H^{2i}$,
and a fundamental class $\mu\in H_{0}=(H^{n})^{*}\cong{\mathbb{Q}}$ that
satisfy the following conditions:

  1. (i)

The bilinear form $\lambda_{\mu}:H^{2k}\times H^{2k}\rightarrow{\mathbb{Q}}$
defined as $(\alpha,\beta)\mapsto\langle\alpha\beta,\mu\rangle$ is isomorphic
to an orthogonal sum of copies of $\langle 1\rangle$ and $\langle-1\rangle$.

  2. (ii)

Let $p_{i}(c)=(-1)^{i}\sum_{j=0}^{2i}(-1)^{j}c_{j}c_{2i-j}$, the $k$–th
${\mathcal{L}}$–polynomial of $p_{i}(c)$ paired with $\mu$ equals the
signature of the bilinear form $(H^{2k},\lambda_{\mu})$,

$\langle{\mathcal{L}}_{k}(p_{1}(c),\ldots
p_{k}(c)),\mu\rangle=\sigma(H^{2k},\lambda_{\mu})$ (1)

  3. (iii)

The pairings $\langle c_{I},\mu\rangle=\langle c_{i_{1}}\cdots
c_{i_{r}},\mu\rangle$ over all the partitions $I=(i_{1},\ldots,i_{r})$ of $2k$
agree with the Chern numbers of an almost complex manifold, i.e., there exists
a $4k$-dimensional almost complex manifold $N$ such that $\langle
c_{I}(\tau_{N}),[N]\rangle=\langle c_{I},\mu\rangle$ for all partitions $I$ of
$2k$. By [Sto65I, Theorem 1], this is equivalent to say the numbers $\langle
c_{I},\mu\rangle$ must satisfy the integrality relations among Chern numbers
of almost complex manifold given by the Hirzebruch-Riemann-Roch Theorem, i.e.,

$\langle{\mathbb{Z}}[e^{c}_{1},e^{c}_{2},\ldots]\mathord{\cdot}\textup{Td},\mu\rangle\in{\mathbb{Z}}$
(2)

where $e^{c}_{l}$ is the $l$-th elementary symmetric polynomial of the
variables $e^{x_{i}}-1=\sum_{m=1}^{\infty}\frac{x_{i}^{m}}{m!}$. By the formal
expression $c=\prod_{i}(1+x_{i})$, each $e^{c}_{l}$ can be expressed as a
polynomial in $c_{1},c_{2},\ldots,c_{2k}$ with rational coefficients,

$e^{c}_{l}:=\sigma_{l}(e^{x_{1}}-1,e^{x_{2}}-1,\cdots)=c_{l}+\textup{higher
degree terms}.$

The Todd class Td is expressed as a polynomial in the $c_{i}$ classes:

$\textup{Td}=\prod_{i}\displaystyle\frac{x_{i}}{1-e^{-x_{i}}}=1+\frac{c_{1}}{2}+\frac{c_{1}^{2}+c_{2}}{12}+\textup{higher
degree terms}$

  4. (iv)

If $n\equiv 4\pmod{8}$,

Case 1. If $c_{1}\neq 0$, no additional condition besides (i)-(iv).

Case 2. If $c_{1}=0$, the parings $\langle c_{I},\mu\rangle$ over all the
partitions $I$ of $2k$ agree with the Chern numbers of a $SU$ manifold, i.e.,
there exists a $4k$-dimensional $SU$ manifold $N$ such that $\langle
c_{I}(\tau_{N}),[N]\rangle=\langle c_{I},\mu\rangle$ for all partitions $I$ of
$2k$. By [Sto65II, Theorem 1] and [Sto68, Page 259], this is equivalent to say
the numbers $\langle c_{I},\mu\rangle$ must satisfy the integrality relations
among Chern numbers of SU manifold given by the Riemann-Roch Theorem, i.e., in
additional to the relations (2) in (iii),

$\langle{\mathbb{Z}}[e^{p(c)}_{1},e^{p(c)}_{2},\ldots]\mathord{\cdot}{\hat{A}},\mu\rangle\in
2{\mathbb{Z}}$ (3)

where $e^{p}_{l}(c)$ is the $l$-th elementary symmetric polynomial of the
variables
$e^{x_{i}}+e^{-x_{i}}-2=2\sum_{m=1}^{\infty}\frac{x_{i}^{2m}}{(2m)!}$. By the
formal expression $c=\prod_{i}(1+x_{i})$ and $p(c)=\prod_{i}(1+x_{i}^{2})$,
each $e^{p(c)}_{l}$ can be expressed as a polynomial in
$c_{1},c_{2},\ldots,c_{2k}$ with rational coefficients,

$e^{p(c)}_{l}:=\sigma_{l}(e^{x_{1}}+e^{-x_{1}}-2,e^{x_{2}}+e^{-x_{2}}-2,\cdots)=p_{l}(c)+\textup{higher
degree terms}.$

The A–hat class ${\hat{A}}$ is expressed as a polynomial in the $c_{i}$
classes:

${\hat{A}}=\prod_{i}\displaystyle\frac{x_{i}}{2\sinh(x_{i}/2)}=1-\frac{p_{1}(c)}{24}+\frac{7p_{1}(c)^{2}+4p_{2}(c)}{5760}+\textup{higher
degree terms}.$

  5. (v)

The number $\langle
c_{2k},\mu\rangle=\chi(H^{*})=\sum_{i=0}^{n}(-1)^{i}\textup{rk}(H^{i})$, the
Euler characteristic of $H^{*}$.

Any choice of $c$ and $\mu$ satisfying the conditions corresponds to a
realizing manifold $M$ whose Chern numbers $\langle
c_{I}(\tau_{M}),[M]\rangle=\langle c_{I},\mu\rangle$. In particular, if the
choice has $c_{1}=0$, $H^{*}$ can be realized by an almost complex manifold
$M$ with integral first Chern class $c_{1}(\tau_{M})=0$, i.e., $H^{*}$ can be
realized by a SU manifold; if the choice has $c_{1}\neq 0$, $H^{*}$ can be
realized by an almost complex manifold $M$ with integral first Chern class
$c_{1}(\tau_{M})\neq 0$.

###### Proof.

($\Longrightarrow$) It is mostly straightforward that the conditions are
necessary. If $M$ is a simply-connected almost complex manifold such that
$H^{*}(M;{\mathbb{Q}})\cong H^{*}$, let $c$ be the rational total Chern class
$c(\tau_{M})\in\bigoplus_{i=0}^{2k}H^{2i}(M;{\mathbb{Q}})$, let $\mu$ be the
image of the fundamental class $[M]$ in $H_{n}(M;{\mathbb{Q}})$,

(i) The bilinear form $(H^{2k},\lambda_{\mu})$ is isomorphic to the rational
intersection form of $M$, since the intersection form of a $4k$–dimensional
manifold is a symmetric inner product space over ${\mathbb{Z}}$,
$(H^{2k},\lambda_{\mu})$ is in the image of $W({\mathbb{Z}})\to
W({\mathbb{Q}})$, which consists exactly of orthogonal sum of copies of
$\langle 1\rangle$ and $\langle-1\rangle$ ([MH73, IV. 2.6]).

(ii) The Pontryagin classes of the tangent bundle of $M$ can be written in
terms of the Chern classes by the formula
$p_{i}(\tau_{M})=(-1)^{i}\sum_{j=0}^{2i}(-1)^{j}c_{j}(\tau_{M})c_{2i-j}(\tau_{M})$.
By the Hirzebruch signature theorem, $\langle{\mathcal{L}}_{k}(p_{1}(c),\ldots
p_{k}(c)),\mu\rangle=\langle{\mathcal{L}}_{k}(p_{1}(\tau_{M}),\ldots
p_{k}(\tau_{M})),[M]\rangle=\sigma(M)=\sigma(H^{2k},\lambda_{\mu})$.

(iii) For all partitions $I=(i_{1},i_{2},\ldots,i_{r})$ of $2k$, the numbers
$\langle c_{I},\mu\rangle=\langle c_{I}(\tau_{M}),[M]\rangle$ must satisfy the
Riemann–Roch integrality relations among Chern numbers of almost complex
manifold. By Stong’s theorem ([Sto65I, Theorem 1]), these relations can be
formulated as (2).

(iv) To see Case 2, note that the realizing almost complex manifold $M$ is
assumed to be simply-connected, since $H^{2}(M;{\mathbb{Z}})$ is free, its
integral first Chern class is zero if and only if the rational first Chern
class is zero. If $M$ has the first Chern class $c_{1}(\tau_{M})=0$, $M$ is a
$SU$ manifold, the numbers $\langle c_{I},\mu\rangle=\langle
c_{I}(\tau_{M}),[M]\rangle$ satisfy the Riemann–Roch integrality relations
among Chern numbers of $SU$ manifold. By Stong’s theorem ([Sto65II, Theorem
1(a)], [Sto68, Page 259]), if $n\not\equiv 4\pmod{8}$, the $SU$ relations
agree with the $U$ relations (2) . If $n\equiv 4\pmod{8}$, the $SU$ relations
include both (2) and (3). The proof for the sufficiency of the conditions
provides some further explanations on these integrality relations.

(v) The top Chern class $c_{2k}(\tau_{M})$, i.e., the Euler class, paired with
$[M]$ is equal to the Euler characteristic of $M$, so $\langle
c_{2k},\mu\rangle=\langle c_{2k}(\tau_{M}),[M]\rangle=\chi(M)=\chi(H^{*})$.

($\Longleftarrow$) By [Quil69, page 210, Theorem 1], if
$H^{*}=\bigoplus_{i=0}^{n}H^{i}$ is a graded commutative algebra over
${\mathbb{Q}}$ with $H^{i}$ finite dimensional for each $i$ and $H^{1}=0$,
$H^{0}={\mathbb{Q}}$, then $H^{*}$ is isomorphic to the rational cohomology
ring of a simply connected space. Let $X$ be a $n=4k$–dimensional simply-
connected ${\mathbb{Q}}$–local (rational) space such that
$H^{*}(X;{\mathbb{Q}})\cong H^{*}$. Assume there exist total cohomology class
$c=1+c_{1}+c_{2}+\cdots+c_{2k}\in\bigoplus_{i=0}^{2k}H^{2i}(X;{\mathbb{Q}})$
and a fundamental class $\mu\in H_{n}(X;{\mathbb{Z}})\cong
H_{n}(X;{\mathbb{Q}})\cong{\mathbb{Q}}$, we apply rational surgery to show the
existence of an $n=4k(k>1)$–dimensional almost complex manifold $M^{n}$ and a
${\mathbb{Q}}$–homotopy equivalence $f:M\to X$ such that
$f^{*}(c_{i})=c_{i}(\tau_{M})$ and $f_{*}[M]=\mu$.

We state a proof in a similar fashion to the one given in the author’s thesis
[Su09] (and [Su14]) for Sullivan’s theorem on smooth manifold. Any choice of
cohomology class
$c=1+c_{1}\cdots+c_{2k}\in\bigoplus_{i=0}^{2k}H^{2i}(X;{\mathbb{Q}})$
corresponds to a map

$c:X\xrightarrow{(c_{1},\ldots,c_{2k},0,\ldots)}\prod_{i=1}^{m}K(\mathbb{Q},2i)\simeq
BU(m)_{0}$

where $BU(m)_{0}$ is the rationalization (localization) of
$BU(m)=G_{m}({\mathbb{C}}^{\infty})$ and $m>>n$. Let
$\gamma^{m}_{{\mathbb{C}}}=\gamma^{m}({\mathbb{C}}^{\infty})$ denote the
universal complex $m$–plane bundle, which is a real $2m$–plane bundle, define
the map

$\overline{c}(\gamma):BU(m)\xrightarrow{(\overline{c}_{1}(\gamma^{m}_{{\mathbb{C}}}),\ldots,\overline{c}_{m}(\gamma^{m}_{{\mathbb{C}}}))}\prod\limits_{i=1}^{m}K(\mathbb{Q},2i)\simeq
BU(m)_{0},$

where
$\overline{c}(\gamma^{m}_{{\mathbb{C}}})=1+\overline{c}_{1}(\gamma^{m}_{{\mathbb{C}}})+\cdots+\overline{c}_{m}(\gamma^{m}_{{\mathbb{C}}})+\cdots\in\prod
H^{2i}(BU(m);{\mathbb{Q}})$ is the inverse class determined algebraically by
$c(\gamma^{m}_{{\mathbb{C}}})\overline{c}(\gamma^{m}_{{\mathbb{C}}})=1$. The
map $\overline{c}(\gamma)$ induces isomorphism on
$\pi_{*}(-)\otimes{\mathbb{Q}}$ and $H_{*}(-;{\mathbb{Q}})$. As shown in the
diagram below, let $PB$ be the homotopy pullback space of $c$ and
$\overline{c}(\gamma)$, let
$\xi^{m}_{{\mathbb{C}}}=pr_{2}^{*}(\gamma^{m}_{{\mathbb{C}}})$ be the induced
bundle over $PB$. The map $pr_{1}:PB\to X$ induces isomorphism on
$\pi_{*}(-)\otimes{\mathbb{Q}}$ for $*>1$.

$\textstyle{\nu_{M}^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{g}}$$\textstyle{\xi^{m}_{{\mathbb{C}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{pr_{2}}}$$\textstyle{\gamma^{m}_{{\mathbb{C}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{PB\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{pr_{1}}$$\scriptstyle{pr_{2}}$$\textstyle{BU(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{c}(\gamma)}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\textstyle{\prod\limits_{i=1}^{m}K(\mathbb{Q},2i)\simeq
BU(m)_{0}}$

We would like to construct a normal map
$(g,\widehat{g}):(M,\nu_{M})\to(PB,\xi)$ for simply-connected surgery. In the
original realization theorem on smooth manifold, the pullback space $PB$ is
automatically simply-connected for any input map $p:X\to\prod\limits
K(\mathbb{Q},4i)\simeq BSO_{0}$. This is not the case for almost complex
manifold. This observation and the following argument was due to Aleksandar
Milivojević and was communicated to the author in 2019. Let $F$ denote the
homotopy fiber of the map $\overline{c}:BU\to BU_{0}$, the exact sequence
below shows that $PB$ is simply connected if and only if the map
$\pi_{2}(X)\to\pi_{2}(BU_{0})$ is surjective, which happens if and only if the
input class $c_{1}\in H^{2}(X;{\mathbb{Q}})$ is nonzero.

$\textstyle{\pi_{2}(PB)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{2}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(PB)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(X)=0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{2}(BU)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{2}(BU_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(F)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(BU)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}(BU_{0})}$

$\hskip 45.52458pt{\mathbb{Z}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\mathbb{Q}}\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ {\mathbb{Q}}/{\mathbb{Z}}\ \ \ \ \ \ \ \ \ \ \ \ \
\ 0$

The proof is then divided into two cases. If $c_{1}$ is nonzero, we show that
the above pullback diagram with classifying space $BU$ yields a normal map
$f:M\to X$ where $M$ is a stably almost complex manifold, the existence of a
degree 1 normal map is guaranteed by condition (iii). If $c_{1}$ is zero, the
classifying space $BU$ is replaced by $BSU$ in the pullback diagram, which
yields a normal map $f:M\to X$ where $M$ is a stably almost complex manifold
with $c_{1}(\tau_{M})=0$, in this case, condition (iii) and (iv) Case 2 are
sufficient to produce a degree 1 normal map.

Case 1\. If $c_{1}\neq 0$, the pullback space $PB$ is simply-connected. The
generalized Thom-Pontryagin construction ([Las63, page 264–266] and [Sto68,
page 18–23]) produces a surgery normal map as follows. Let $\alpha$ be any
class in $\pi_{n+2m}(T\xi^{m}_{{\mathbb{C}}})$, for the composite
$S^{n+2m}\xrightarrow{\alpha}T\xi^{m}_{{\mathbb{C}}}\xrightarrow{Tpr_{2}}T\gamma_{{\mathbb{C}}}^{m}$,
since $S^{n+2m}$ is compact, the image $Tpr_{2}\circ\alpha(S^{n+2m})$ lives in
$T\gamma^{m}({\mathbb{C}}^{r})$ for some finite $r$. By the transversality
theorem, $Tpr_{2}\circ\alpha$ can be deformed to a map $h$ that is
differentiable in the pre-image of some open neighborhood of
$Gr_{m}({{\mathbb{C}}}^{r})\subset T\gamma^{m}({\mathbb{C}}^{r})$ and is
transverse regular on $Gr_{m}({{\mathbb{C}}}^{r})$. Then we obtain a manifold
$M^{n}=h^{-1}(\text{Gr}_{m}({{\mathbb{C}}}^{r}))$ and $h|_{M}$ is the
classifying map of the stable normal bundle $\nu_{M}$. A complex structure on
the stable normal bundle of $M$ determines a complex structure on the stable
tangent bundle of $M$ up to homotopy, so $M$ is a stably almost complex
manifold. By the homotopy lifting property, the deformation of
$Tpr_{2}\circ\alpha$ to $h$ can be covered by a homotopy of $\alpha$ to a new
map $g$ such that $h=Tpr_{2}\circ g$, then
$M=g^{-1}(PB)=h^{-1}(Gr_{m}({{\mathbb{C}}}^{r}))$. We have obtained a normal
map $(g,\widehat{g}):(M,\nu_{M})\to(PB,\xi)$. Chasing the diagram, for the map
$f:=pr_{1}\circ g:M\rightarrow X$, $f^{*}(c_{i})=c_{i}(\tau_{M})\in
H^{2i}(M;{\mathbb{Q}})$, which is the rational Chern class of the stable
tangent bundle of $M$.

The existence of a degree 1 normal map in the sense that $f_{*}[M]=\mu$ is
guaranteed by condition (iii). We outline a proof similar to the argument
given in [Su09, Lemma 3.2.2.] and [Su14] for the theorem on smooth manifold.
If the numbers $\langle c_{I},\mu\rangle$ agree with the set of Chern numbers
of an almost complex manifold, i.e., there exists a $n$–dimensional almost
complex manifold $N$ such that for all partitions $I$ of $2k$,

$\langle c_{I}(\tau_{N}),[N]\rangle=\langle c_{I},\mu\rangle=\langle
c_{I}(\gamma^{m}_{{\mathbb{C}}}),c_{*}\mu\rangle=\langle\overline{c}_{I}(\gamma^{m}_{{\mathbb{C}}}),{\overline{c}(\gamma)}_{*}^{-1}(c_{*}\mu)\rangle.$

This identity implies that $\langle c_{I}(\nu_{N}),[N]\rangle=\langle
c_{I}(\gamma^{m}_{{\mathbb{C}}}),{\overline{c}(\gamma)}_{*}^{-1}(c_{*}\mu)\rangle$,
therefore the rational homology class
${\overline{c}(\gamma)}_{*}^{-1}(c_{*}\mu)$ lies in the image of the
homomorphism $\nu:\Omega_{n}^{U}\rightarrow H_{n}(BU;\mathbb{Q})$ defined by
$\nu([M])=(\nu_{M})_{*}([M])$. Since the homomorphism $\nu$ is obtainable as
the composition
$\Omega_{n}^{U}\cong\pi_{n+2m}(T\gamma^{m})\xrightarrow{h}H_{n+2m}(T\gamma^{m})\xrightarrow{\cap
U}H_{n}(BU;{\mathbb{Z}})\to H_{n}(BU;{\mathbb{Q}})$, which is exactly the
diagonal map in the upper right corner of the digram below, there must exist a
homotopy class $\beta\in\pi_{n+2m}(T\gamma^{m})=\pi_{n+2m}(MU(m))$ mapping to
$c_{*}\mu\in H_{n}(BU(m)_{0};{\mathbb{Q}})$. Note that the diagonal maps in
the lower left and lower right corner of the diagram are isomorphisms, this is
because both the Thom space of the rational spherical fibration and the base
space are ${\mathbb{Q}}$–local (rational), and the Hurewicz map on the Thom
space is a rational isomorphism as we assumed $m>>n$. It can be shown that the
outer square of Thom spaces is a homotopy Cartesian square (see [TW78, Lemma
6.1], [Quil69, Theorem 2.3], or a detailed proof in [Su09, Lemma 3.2.3]). Then
the existence of homotopy classes $\beta\in\pi_{n+2m}(T\gamma^{m})$ and
$c_{X}\in\pi_{n+2m}(T\widetilde{\nu}_{X})$ both mapping to $c_{*}\mu$ implies
the existence of a desired Spivak class
$\alpha\in\pi_{n+2m}(T\xi^{m}_{{\mathbb{C}}})$ in the upper left corner
mapping to $\mu$. Therefore $f_{*}[M]=\mu$.

---
$\textstyle{\alpha\in\pi_{n+2m}(T\xi^{m}_{{\mathbb{C}}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{Tpr_{1}}_{*}}$$\scriptstyle{{T_{pr_{2}}}_{*}}$$\textstyle{\pi_{n+2m}(T\gamma^{m}_{{\mathbb{C}}})\ni\beta\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\scriptstyle{T\overline{c}(\gamma)_{*}}$$\textstyle{H_{n}(PB)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{pr_{1}}_{*}}$$\scriptstyle{{pr_{2}}_{*}}$$\textstyle{H_{n}(BU(m))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{c}(\gamma)_{*}}$$\textstyle{\mu\in
H_{n}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c_{*}}$$\textstyle{H_{n}(BU(m)_{0})}$$\textstyle{c_{X}\in\pi_{n+2m}(T\widetilde{\nu}_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{{Tc}_{*}}$$\textstyle{\pi_{n+2m}(T\gamma^{m}_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$

Case 2\. If $c_{1}=0$, replace the classifying space $BU$ by $BSU$, the input
map becomes
$c:X\xrightarrow{(c_{1}=0,c_{2},\ldots,c_{2k},0,\ldots)}\prod_{i=2}^{m}K(\mathbb{Q},2i)\simeq
BSU(m)_{0}$. Since $\pi_{2}(BSU)=0$, the pullback space $PB$ is simply-
connected. Let $\eta_{{\mathbb{C}}}^{m}$ denote the universal bundle over
$BSU(m)$.

$\textstyle{\nu_{M}^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{g}}$$\textstyle{\xi^{m}_{{\mathbb{C}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{pr_{2}}}$$\textstyle{\eta^{m}_{{\mathbb{C}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\gamma^{m}_{{\mathbb{C}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{g}$$\textstyle{PB\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{pr_{1}}$$\scriptstyle{pr_{2}}$$\textstyle{BSU(m)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{c}(\gamma)}$$\textstyle{BU(m)}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c}$$\textstyle{\prod\limits_{i=2}^{m}K(\mathbb{Q},2i)\simeq
BSU(m)_{0}}$

Similar to the argument in Case 1, the Thom-Pontryagin construction produces a
normal map $(g,\widehat{g}):(M,\nu_{M})\to(PB,\xi)$ such that $M$ is a stably
almost complex manifold and $f^{*}(c_{i})=c_{i}(\tau_{M})\in
H^{2i}(M;{\mathbb{Q}})$, in particular, $c_{1}(\tau_{M})=f^{*}(c_{1})=0$. If
the numbers $\langle c_{I},\mu\rangle$ agree with the set of Chern numbers of
a $SU$ manifold, there exists a homotopy class
$\beta\in\pi_{n+2m}(T\eta^{m})=\pi_{n+2m}(MSU(m))\cong\Omega_{n}^{SU}$ mapping
to $c_{*}\mu\in H_{n}(BSU(m)_{0};{\mathbb{Q}})$, this guarantees the existence
of a Spivak class $\alpha\in\pi_{n+2m}(T\eta^{m}_{{\mathbb{C}}})$ mapping to
$\mu$, which implies that $f_{*}[M]=\mu$.

We elaborate on why the numbers $\langle c_{I},\mu\rangle$ agree with the set
of Chern numbers of an almost complex ($U$) or $SU$ manifold if and only if
they satisfy the integrality relations phrased as (2) and (3) in condition
(iii) and (iv). For $G=U$ or $SU$, since $H^{*}(BG;{\mathbb{Q}})$ is the
rational polynomial algebra on the universal Chern classes and
$H_{n}(BG;{\mathbb{Q}})\cong Hom(H^{n}(BG;{\mathbb{Q}}),{\mathbb{Q}})$, the
integrality relations among Chern numbers of $G$ manifold characterize the
group $\tau\Omega^{G}_{n}$, which is the image of the $G$ cobordism group
under the homomorphism $\tau:\Omega_{n}^{G}\to H_{n}(BG;{\mathbb{Q}})$ induced
by the stable tangent bundle $\tau_{M}:M\to BG$.

As stated in [Sto65I, Theorem 1], $\tau\Omega^{U}_{n}=\\{x\in
H_{n}(BU;{\mathbb{Q}})\,|\,\langle{\mathbb{Z}}[e^{c}_{1},e^{c}_{2},\ldots]\mathord{\cdot}\textup{Td},x\rangle\in{\mathbb{Z}}\\}$,
therefore the $U$ relations are phrased as (2) in (iii).

In [Sto65II, Theorem 1], [Sto68, Page 259], the $SU$ relations are described
in terms of the $s_{\omega}$–symmetric polynomials, since the
$s_{\omega}$–symmetric polynomials form an additive basis for the ring of
formal power series of elementary symmetric polynomials. It is equivalent to
phrase the relations in terms of the elementary symmetric polynomials as
follows.

If $n\not\equiv 4\pmod{8}$, $\tau\Omega^{SU}_{n}=\\{x\in
H_{n}(BSU;{\mathbb{Q}})\,|\,\langle{\mathbb{Z}}[e^{c}_{1},e^{c}_{2},\ldots]\mathord{\cdot}\textup{Td},x\rangle\in{\mathbb{Z}}\\},$
i.e., the $SU$ relations agree with the $U$ relations. Therefore when
$c_{1}=0$ and $n\not\equiv 4\pmod{8}$, the relations (2) in condition (iii) is
sufficient to produce a desired normal map.

If $n\equiv 4\pmod{8}$,
$\tau\Omega^{SU}_{n}\\!\\!\\!=\\!\\!\\{x\\!\in\\!H_{n}(BSU;{\mathbb{Q}})\,|\,\langle{\mathbb{Z}}[e^{c}_{1},e^{c}_{2},\ldots]\mathord{\cdot}\textup{Td},x\rangle\\!\in\\!{\mathbb{Z}}\textup{
and
}\langle{\mathbb{Z}}[e^{p(c)}_{1}\\!\\!,e^{p(c)}_{2}\\!\\!\\!,\ldots]\mathord{\cdot}{\hat{A}},x\rangle\\!\\!\in\\!\\!2{\mathbb{Z}}\\}.$
Therefore in (iv) Case 2, the additional relation (3) is required.

The next step is to perform surgery on the candidate degree 1 normal map,
condition (i) and (ii) guarantee a vanishing surgery obstruction so that a
rational homotopy equivalence can be obtained. The proof outlined in the
original theorem [Sul77, Page 326] on smooth manifold still applies in the
almost complex case. The book [And77] discussed surgery theory with
${\mathbb{Z}}_{P}$ coefficients (${\mathbb{Z}}_{P}={\mathbb{Q}}$ if $P$ is the
set of all primes). One can also refer to Milivojević’s PhD thesis [Mil21] for
an exposition on rational surgery. The candidate normal map
$(g,\widehat{g}):(M,\nu_{M})\to(PB,\xi)$ is normally cobordant to a rational
homotopy equivalence if and only if the obstruction vanishes in the $L$ group
$L_{4k}({\mathbb{Q}})\cong{\mathbb{Z}}\oplus\bigoplus_{\infty}{\mathbb{Z}}_{2}\oplus\bigoplus_{\infty}{\mathbb{Z}}_{4}$.
Condition (i) guarantees that the intersection form
$(H^{2k}(X;{\mathbb{Q}}),\lambda_{\mu})$ is contained in the image of the map
$W({\mathbb{Z}})\to W({\mathbb{Q}})$, and therefore has a vanishing
${\mathbb{Z}}_{2}$ and ${\mathbb{Z}}_{4}$ summands in $L_{4k}({\mathbb{Q}})$.
Since
$\langle{\mathcal{L}}_{k}(p_{i}(c)),\mu\rangle=\langle{\mathcal{L}}_{k}(p_{i}(c)),f_{*}([M])\rangle=\langle{\mathcal{L}}_{k}(p_{i}(f^{*}c)),[M]\rangle=\langle{\mathcal{L}}_{k}(p_{i}(c(\tau_{M}))),[M]\rangle=\sigma(M)$.
The signature equation in (ii) implies that
$\sigma(M)=\sigma(H^{2k}(X;{\mathbb{Q}}),\lambda_{\mu})$, so the obstruction
has a vanishing ${\mathbb{Z}}$ summand in $L_{4k}({\mathbb{Q}})$. After a
sequence of surgery performed, the resulting normal map, again denoted
$(g,\widehat{g}):(M,\nu_{M})\to(PB,\xi)$, still gives a complex structure on
the normal bundle of $M$, the map $f:M\to X$ is a desired rational homotopy
equivalence from a simply-connected stably almost complex manifold.

Condition (v) ensures that $\langle c_{2k}(\tau_{M}),[M]\rangle=\langle
f^{*}(c_{2k}),[M]\rangle=\langle c_{2k},\mu\rangle=\chi(H^{*})=\chi(X)$, which
is now equal to $\chi(M)$ as $f:M\to X$ is a rational homotopy equivalence,
then by [Tho67, Theorem 1.7] and [Sut65, Theorem 1.1], the $2k$–Chern number
of $M$ agrees with its Euler class implies that $M$ admits an almost complex
structure.

This finished the proof that conditions (i)-(v) are sufficient for the
existence of a simply-connected almost complex manifold $M$ and a rational
homotopy equivalence $f:M\to X$ such that $f^{*}(c_{i})=c_{i}(\tau_{M})$ and
$f_{*}[M]=\mu$, and therefore $M$ realizes the prescribed rational cohomology
ring $H^{*}$ and the Chern numbers $\langle c_{I}(\tau_{M}),[M]\rangle=\langle
c_{I},\mu\rangle$ for all partitions $I$ of $2k$. Note that the first rational
Chern class $c_{1}(\tau_{M})=f^{*}(c_{1})=0$ if and only if the input class
$c_{1}=0$. Since the realizing manifold $M$ is simply-connected,
$H^{2}(M;{\mathbb{Z}})$ is free, so the integral Chern class of $M$ is zero if
and only if the input class $c_{1}=0$. ∎

##  3\. Integrality relations among Chern numbers.

In order to apply Theorem 1 in our prescribed Betti number setting, we
explicitly calculate the integrality relations (2) and (3) under the
assumption that all the Chern classes $c_{i}$ vanish except possibly for the
middle and top dimensional classes $c_{k}$ and $c_{2k}$.

######  Proposition 2.

If the product of Chern classes $c_{\omega}=0$ except possibly for $c_{k}$,
$c_{k}^{2}$, and $c_{2k}$, the integrality relations among Chern numbers of
$n=4k(k>1)$–dimensional almost complex manifolds, expressed as (2)

$\langle{\mathbb{Z}}[e^{c}_{1},e^{c}_{2},\ldots]\mathord{\cdot}\textup{Td},\mu\rangle\in{\mathbb{Z}},$

are satisfied if and only if the following sub-relations are satisfied.

$\displaystyle\langle\textup{Td},\mu\rangle=\frac{1}{2}(t_{k}^{2}-t_{2k})\langle
c_{k}^{2},\mu\rangle+t_{2k}\langle c_{2k},\mu\rangle\in{\mathbb{Z}}\ \
{\scriptstyle\textup{with}\ \ t_{k}=\frac{B_{k}}{k!}}$ (4a)
$\displaystyle\langle
e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle=\left((-1)^{k+1}\displaystyle\frac{t_{k}}{(k-1)!}+\displaystyle\frac{1}{2(2k-1)!}\right)\langle
c_{k}^{2},\mu\rangle-\displaystyle\frac{1}{(2k-1)!}\langle
c_{2k},\mu\rangle\in{\mathbb{Z}}$ (4b) $\displaystyle\langle
e_{1}^{c}e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle=\displaystyle\frac{1}{[(k-1)!]^{2}}\langle
c_{k}^{2},\mu\rangle\in\delta_{k}{\mathbb{Z}}\ \ \textup{with}\
\delta_{k}=\left\\{\begin{array}[]{ll}1&k=2\\\ 2&k>2\end{array}\right.$ (4e)

###### Proof.

The proof is similar to the one given in [KS17, Lemma 2, 3, 4]. Since
$c_{\omega}=0$ except possibly for $c_{k}$, $c_{k}^{2}$, and $c_{2k}$, the
total Todd class is
$\textup{Td}=1+\textup{Td}_{k}+\textup{Td}_{2k}=1+t_{k}c_{k}+(t_{k,k}c_{k}^{2}+t_{2k}c_{2k})$
where $t_{k}=B_{k}/k!$ and $t_{k,k}=\frac{1}{2}(t_{k}^{2}-t_{2k})$.

In our notation, $e_{l}^{c}$ is the $l$–th elementary symmetric polynomial of
the variables $e^{x_{i}}-1=\sum_{m=1}^{\infty}\frac{x_{i}^{m}}{m!}$. Since
each $e_{l}^{c}$ class can be expressed as a linear combination of $c_{k}$,
$c_{k}^{2}$ and $c_{2k}$ with rational coefficients, any monomial
$e_{\omega}^{c}=0$ if the degree $|\omega|>2$, so (2) holds true if and only
if $\langle\textup{Td},\mu\rangle\in{\mathbb{Z}}$, $\langle
e_{l}^{c}\mathord{\cdot}\textup{Td},\mu\rangle\in{\mathbb{Z}}$, and $\langle
e_{l}^{c}e_{m}^{c}\mathord{\cdot}\textup{Td},\mu\rangle\in{\mathbb{Z}}$ for
all $l,m\geq 1$, we will show the later two conditions hold true if and only
if (4b) and (4e) hold true. An explicit formula for the $e_{l}^{c}$ class will
be derived via the power sum symmetric polynomial.

Let $S_{m}(x_{i})$ denote the $m$-th power sum of the variables $x_{i}$, i.e.,
$S_{m}(x_{i})=\sum_{i}x_{i}^{m}$. Apply the Newton-Girard formula, the power
sum symmetric polynomial $S_{m}(x_{i})$ can be written in terms of the
elementary symmetric polynomials $c_{j}=\sigma_{j}(x_{i})$,

$S_{m}(x_{i})=\begin{cases}(-1)^{k+1}kc_{k}&\text{if }m=k\\\
k(c_{k}^{2}-2c_{2k})&\text{if }m=2k\\\ 0&\text{otherwise}\end{cases}.$ (5)

Then we can write

$\displaystyle
e_{1}^{c}=\displaystyle\sum_{i}e^{x_{i}}-1=\sum_{i}\sum_{m=1}^{\infty}\frac{x_{i}^{m}}{m!}$
$\displaystyle=$ $\displaystyle\sum_{m=1}^{\infty}\frac{1}{m!}S_{m}(x_{i})$
(6) $\displaystyle=$
$\displaystyle\displaystyle\frac{1}{k!}S_{k}(x_{i})+\displaystyle\frac{1}{(2k)!}S_{2k}(x_{i})$
$\displaystyle\stackrel{{\scriptstyle}}{{=}}$
$\displaystyle\displaystyle\frac{(-1)^{k+1}}{(k-1)!}c_{k}+\displaystyle\frac{1}{2(2k-1)!}(c_{k}^{2}-2c_{2k}).$

Let $S_{l}^{c}$ denote the $l$-th power sum of the variables $e^{x_{i}}-1$.
Let $\\{-\\}_{m}$ denote the degree $m$ terms of the variable $x_{i}$’s in an
expression. Let $C_{l}(m):=\sum_{j=0}^{l-1}(-1)^{j}\binom{l}{j}(l-j)^{m}$.

$\displaystyle\\{S_{l}^{c}\\}_{m}=\left\\{\displaystyle\sum_{i}(e^{x_{i}}-1)^{l}\right\\}_{m}$
$\displaystyle=$
$\displaystyle\left\\{\displaystyle\sum_{i}\displaystyle\sum_{j=0}^{l}(-1)^{j}\binom{l}{j}e^{x_{i}(l-j)}\right\\}_{m}$
$\displaystyle=$
$\displaystyle\displaystyle\sum_{i}\displaystyle\sum_{j=0}^{l}(-1)^{j}\binom{l}{j}\displaystyle\frac{x_{i}^{m}(l-j)^{m}}{m!}$
$\displaystyle=$
$\displaystyle\left[\sum_{j=0}^{l-1}(-1)^{j}\binom{l}{j}(l-j)^{m}\right]\frac{S_{m}(x_{i})}{m!}=C_{l}(m)\\{e_{1}^{c}\\}_{m}.$

Note that $C_{l}(m)=l!S(m,l)$ where $S(m,l)$ is the Stirling number of the
second kind. If $m<l$, $C_{l}(m)=0$. By (3) and (6),

$\displaystyle S_{l}^{c}=\\{S_{l}^{c}\\}_{k}+\\{S_{l}^{c}\\}_{2k}$
$\displaystyle=$ $\displaystyle
C_{l}(k)\\{e_{1}^{c}\\}_{k}+C_{l}(2k)\\{e_{1}^{c}\\}_{2k}$ $\displaystyle=$
$\displaystyle
C_{l}(2k)\left(\\{e_{1}^{c}\\}_{k}+\\{e_{1}^{c}\\}_{2k}\right)+\left[C_{l}(k)-C_{l}(2k)\right]\\{e_{1}^{c}\\}_{k}$
$\displaystyle=$ $\displaystyle
C_{l}(2k)e_{1}^{c}+[C_{l}(k)-C_{l}(2k)]\displaystyle\frac{(-1)^{k+1}}{(k-1)!}\,c_{k}.$
(8)

Apply Newton-Girard again to relate the power sum symmetric polynomial
$S_{l}^{c}=S_{l}^{c}(e^{x_{i}}-1)$ with the elementary symmetric polynomials
$e_{j}^{c}=\sigma_{j}(e^{x_{i}}-1)$, since any monomial $e_{\omega}^{c}=0$ if
the degree $|\omega|>2$, we can solve that

$\displaystyle e_{l}^{c}$ $\displaystyle=$
$\displaystyle\displaystyle\frac{(-1)^{l+1}}{l}S_{l}^{c}+\displaystyle\frac{1}{2}\displaystyle\sum_{i=1}^{l-1}e_{i}^{c}e_{l-i}^{c}.$
(9)

Multiply $e_{l}^{c}$ with the total Todd class
$\textup{Td}=1+\textup{Td}_{k}+\textup{Td}_{2k}=1+\displaystyle\frac{B_{k}}{k!}c_{k}+\textup{Td}_{4k}$,
then evaluate on the fundamental class $\mu$, by (9) and (8),

$\langle
e_{l}^{c}\mathord{\cdot}\textup{Td},\mu\rangle=(-1)^{l+1}\underbrace{\displaystyle\frac{C_{l}(2k)}{l}\langle
e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle}_{(a)}+(-1)^{l+k}\underbrace{{\displaystyle\frac{[C_{l}(k)-C_{l}(2k)]}{l(k-1)!}\langle
c_{k}\mathord{\cdot}\textup{Td},\mu\rangle}}_{(b)}+\displaystyle\frac{1}{2}\underbrace{\displaystyle\sum_{i=1}^{l-1}\langle
e_{i}^{c}e_{l-i}^{c}\mathord{\cdot}\textup{Td},\mu\rangle}_{(c)}.$ (10)

By (9) and (3),

$\displaystyle\langle e_{l}^{c}e_{m}^{c}\mathord{\cdot}\textup{Td},\mu\rangle$
$\displaystyle=$
$\displaystyle(-1)^{l+m}\displaystyle\frac{C_{l}(k)}{l}\displaystyle\frac{C_{m}(k)}{m}\langle
e_{1}^{c}e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle.$ (11)

If we assume $\langle
e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle\in{\mathbb{Z}}$ and $\langle
e_{1}^{c}e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle\in 2{\mathbb{Z}}$,
then $\langle
e_{l}^{c}e_{m}^{c}\mathord{\cdot}\textup{Td},\mu\rangle\in{\mathbb{Z}}$
because $C_{i}(k)/i$ is an integer for any $i$. In the expression of $\langle
e_{l}^{c}\mathord{\cdot}\textup{Td},\mu\rangle$, $(a)\in{\mathbb{Z}}$ for the
same reason. For $(b)$, since $\langle
e_{1}^{c}e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle=\displaystyle\frac{1}{[(k-1)!]^{2}}\langle
c_{k}^{2},\mu\rangle$, $\langle
c_{k}\mathord{\cdot}\textup{Td},\mu\rangle=\displaystyle\frac{B_{k}}{k!}[(k-1)!]^{2}\langle
e_{1}^{c}e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle$,

$\displaystyle(b)$ $\displaystyle=$
$\displaystyle\displaystyle\frac{[C_{l}(k)-C_{l}(2k)]}{l}\displaystyle\frac{B_{k}}{k}\langle
e_{1}^{c}e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{l-1}(-1)^{j+1}\underbrace{\frac{1}{l}\binom{l}{j}(l-j)}_{(b)_{1}}\underbrace{(l-j)^{k-1}\left[(l-j)^{k}-1\right]\displaystyle\frac{B_{k}}{k}}_{(b)_{2}}\langle
e_{1}^{c}e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle,$

so $(b)_{1}\in{\mathbb{Z}}$ from the fact that $a/\gcd(a,b)$ divides
$\binom{a}{b}$ for any integer $a,b$. By a stronger version of the Lipschitz-
Sylvester theorem that $a^{\lfloor\log_{2}m\rfloor+1}(a^{m}-1)B_{m}/m$ is an
integer for any $a\in{\mathbb{Z}}$ ([Sla95]), the expression
$(b)_{2}\in{\mathbb{Z}}$ if $k>2$. $(c)\in 2{\mathbb{Z}}$ because $\langle
e_{i}^{c}e_{l-i}^{c}\mathord{\cdot}\textup{Td},\mu\rangle=(-1)^{l}\frac{C_{i}(k)}{i}\frac{C_{l-i}(k)}{l-i}\langle
e_{1}^{c}e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle\in 2{\mathbb{Z}}$ for
$i=1,2\ldots l-1$. Therefore the entire expression of $\langle
e_{l}^{c}\mathord{\cdot}\textup{Td},\mu\rangle\in{\mathbb{Z}}$.

If we assume $\langle
e_{l}^{c}\mathord{\cdot}\textup{Td},\mu\rangle\in{\mathbb{Z}}$ and $\langle
e_{l}^{c}e_{m}^{c}\mathord{\cdot}\textup{Td},\mu\rangle\in{\mathbb{Z}}$ for
all $l,m\geq 1$, in the expression for $\langle
e_{2}^{c}\mathord{\cdot}\textup{Td},\mu\rangle$, since $(a)\in{\mathbb{Z}}$,
and $(b)\in{\mathbb{Z}}$ when $k>2$, $(c)/2$ must be an integer, hence
$\langle e_{1}^{c}e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle\in
2{\mathbb{Z}}$ when $k>2$.

For $k=2$, i.e., the $8$–dimensional case, the only nontrivial $e_{l}^{c}$
classes are $e_{1},e_{2},e_{3},e_{4}$. It suffices to show that $\langle
e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle\in{\mathbb{Z}}$ and $\langle
e_{1}^{c}e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle\in{\mathbb{Z}}$
together implies $\langle
e_{l}^{c}\mathord{\cdot}\textup{Td},\mu\rangle\in{\mathbb{Z}}$ for $l=2,3,4$.
By (10), (3), (6), $\textup{Td}=1+\frac{c_{2}}{12}+\textup{Td}_{4}$,
$C_{l}(2)=2,0,0$ and $C_{l}(4)=14,36,24$ for $l=2,3,4$, one can compute that

$\displaystyle\langle
e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle=\langle-\frac{c_{4}}{6},\mu\rangle,\
\ \ \ \langle e_{1}^{c}e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle=\langle
c_{2}^{2},\mu\rangle,$ $\displaystyle\langle
e_{2}^{c}\mathord{\cdot}\textup{Td},\mu\rangle=\langle\frac{7c_{4}}{6},\mu\rangle,\
\ \langle
e_{3}^{c}\mathord{\cdot}\textup{Td},\mu\rangle=\langle-2c_{4},\mu\rangle,\ \
\langle e_{4}^{c}\mathord{\cdot}\textup{Td},\mu\rangle=\langle
c_{4},\mu\rangle.$

Therefore the statement is true for $k=2$. Alternatively, one can verify that
when $k=2$, in the expression (10) (and (3)) of $\langle
e_{l}^{c}\mathord{\cdot}\textup{Td},\mu\rangle$,
$(b)+\displaystyle\frac{1}{2}(c)=0$ for $l=2,3,4$, therefore $\langle
e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle\in{\mathbb{Z}}$ itself already
guarantees $\langle
e_{l}^{c}\mathord{\cdot}\textup{Td},\mu\rangle\in{\mathbb{Z}}$ for all $l>1$
when $k=2$.

∎

######  Proposition 3.

If the product of Chern classes $c_{\omega}=0$ except possibly for $c_{k}$,
$c_{k}^{2}$, and $c_{2k}$, the additional integrality relations (besides (2))
among Chern numbers of $SU$ manifolds of dimension $n=4k(k>1)$ with $k$ odd,
which is expressed as (3)

$\langle{\mathbb{Z}}[e^{p(c)}_{1},e^{p(c)}_{2},\ldots]\mathord{\cdot}{\hat{A}},\mu\rangle\in
2{\mathbb{Z}},$

are satisfied if the following sub-relations are satisfied.

$\displaystyle\langle\textup{Td},\mu\rangle=\frac{1}{2}t_{2k}[-\langle
c_{k}^{2},\mu\rangle+2\langle c_{2k},\mu\rangle]\in 2{\mathbb{Z}}\ \
{\scriptstyle\textup{with}\ \ t_{2k}=\frac{B_{2k}}{(2k)!}}$ (13a)
$\displaystyle\langle
e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle=\left(\displaystyle\frac{1}{2(2k-1)!}\right)\langle
c_{k}^{2},\mu\rangle-\displaystyle\frac{1}{(2k-1)!}\langle
c_{2k},\mu\rangle\in{\mathbb{Z}}$ (13b)

###### Proof.

$\textup{Td}=\prod_{i}\frac{x_{i}}{1-e^{-x_{i}}}=\left(\prod_{i}e^{x_{i}/2}\right)\left(\prod_{i}\frac{x_{i}}{e^{x_{i}/2}-e^{-x_{i}/2}}\right)=e^{c_{1}/2}{\hat{A}}$.
Since $c_{1}=0$, the Td class coincides with the ${\hat{A}}$ class and the odd
degree (of $c_{j}$) terms of Td vanish for $SU$ manifold. Since $c_{\omega}=0$
except possibly for $c_{k}$, $c_{k}^{2}$, and $c_{2k}$, $k$ is odd and
$t_{k}=\frac{B_{k}}{k!}=0$,

${\hat{A}}=\textup{Td}=1+\textup{Td}_{2k}=1+\frac{-1}{2}t_{2k}c_{k}^{2}+t_{2k}c_{2k}.$

In our notation, $e_{l}^{p(c)}$ is the $l$–th elementary symmetric polynomial
of the variables
$e^{x_{i}}+e^{-x_{i}}-2=\sum_{m=1}^{\infty}\frac{2x_{i}^{2m}}{(2m)!}$, which
contains only even degree terms of $x_{i}$, since $k$ is odd, each
$e_{l}^{p(c)}$ class contains only degree $2k$ terms and is therefore
expressed as a linear combination of $c_{k}^{2}$ and $c_{2k}$ with rational
coefficients, then any monomial $e_{\omega}^{p(c)}=0$ if the degree
$|\omega|>1$, so (3) holds true if and only if
$\langle\textup{Td},\mu\rangle\in 2{\mathbb{Z}}$ and $\langle
e_{l}^{p(c)}\mathord{\cdot}\textup{Td},\mu\rangle\in 2{\mathbb{Z}}$ for all
$l\geq 1$. We will show the almost complex relation $\langle
e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle\in{\mathbb{Z}}$ guarantees the
$SU$ relation $\langle e_{l}^{p(c)}\mathord{\cdot}\textup{Td},\mu\rangle\in
2{\mathbb{Z}}$ for all $l\geq 1$ in this case.

In particular, by (5),

$\displaystyle
e_{1}^{p(c)}=\left\\{\sum_{i}\sum_{m=1}^{\infty}\frac{2x_{i}^{2m}}{(2m)!}\right\\}_{2k}=\frac{2}{(2k)!}S_{2k}(x_{i})=\displaystyle\frac{1}{(2k-1)!}(c_{k}^{2}-2c_{2k}).$

Comparing with the expression of $e_{1}^{c}$ in (6), note that
$e_{1}^{p(c)}=2\\{e_{1}^{c}\\}_{2k}$.

Let $S_{l}^{p(c)}$ denote the $l$-th power sum of the variables
$e^{x_{i}}+e^{-x_{i}}-2=\sum_{m=1}^{\infty}\frac{2x_{i}^{2m}}{(2m)!}$, which
also only contains degree $2k$ terms. Let
$M_{l}(k):=\sum_{j=0}^{l-1}(-1)^{j}\binom{2l}{j}(l-j)^{2k}$.

$\displaystyle
S_{l}^{p(c)}=\displaystyle\sum_{i}(e^{x_{i}}+e^{-x_{i}}-2)^{l}=\displaystyle\sum_{i}\left(e^{x_{i}/2}-e^{-x_{i}/2}\right)^{2l}\\!\\!\\!\\!$
$\displaystyle=$
$\displaystyle\\!\\!\\!\\!\left\\{\displaystyle\sum_{i}\displaystyle\sum_{j=0}^{2l}(-1)^{j}\binom{2l}{j}e^{x_{i}(l-j)}\right\\}_{\\!\\!2k}$
(14) $\displaystyle=$
$\displaystyle\displaystyle\sum_{i}\displaystyle\sum_{j=0}^{2l}(-1)^{j}\binom{2l}{j}\displaystyle\frac{x_{i}^{2k}(l-j)^{2k}}{(2k)!}$
$\displaystyle=$ $\displaystyle
2\left[\sum_{j=0}^{l-1}(-1)^{j}\binom{2l}{j}(l-j)^{2k}\right]\frac{S_{2k}(x_{i})}{(2k)!}$
$\displaystyle=$ $\displaystyle M_{l}(k)e_{1}^{p(c)}.$

Apply the Newton-Girard formula, since any monomial $e_{\omega}^{p(c)}=0$ if
the degree $|\omega|>1$,

$\displaystyle e_{l}^{p(c)}$ $\displaystyle=$
$\displaystyle\displaystyle\frac{(-1)^{l+1}}{l}S_{l}^{p(c)}=(-1)^{l+1}\displaystyle\frac{M_{l}(k)}{l}e_{1}^{p(c)}.$
(15)

By the observation that $e_{1}^{p(c)}=2\\{e_{1}^{c}\\}_{2k}$ and
$\textup{Td}=1+\textup{Td}_{2k}$,

$\langle
e_{l}^{p(c)}\mathord{\cdot}\textup{Td},\mu\rangle=(-1)^{l+1}\displaystyle\frac{M_{l}(k)}{l}\
2\langle e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle.$ (16)

As addressed in [KS17, Lemma 3], $M_{l}(k)$ is divisible by $l$, therefore
$\langle e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle\in{\mathbb{Z}}$
implies $\langle e_{l}^{p(c)}\mathord{\cdot}\textup{Td},\mu\rangle\in
2{\mathbb{Z}}$.

∎

##  4\. almost complex manifold with Betti number $b_{i}=0$ except
$i=0,n/2,n$.

We apply Theorem 1 under our Betti number assumption. If a rational cohomology
ring $H^{*}$ is realizable, it is necessary that condition (i) is satisfied,
so we prescribe $H^{*}$ to have a bilinear form $(H^{2k},\lambda)$ isomorphic
(over ${\mathbb{Q}}$) to an orthogonal sum of $\langle 1\rangle$’s and
$\langle-1\rangle$’s. Let $(H^{*},\sigma,\chi)$ denote any $n=4k$–dimensional
rational cohomology ring

$H^{*}\cong\left\\{\begin{array}[]{ll}{\mathbb{Q}}&\ast=0,4k;\\\
{\mathbb{Q}}^{r}&\ast=2k;\\\ 0&\hbox{ otherwise}\end{array}\right.$

with signature $\sigma$ and Euler characteristic $\chi$, such that for some
fundamental class $\mu\in H_{0}\cong{\mathbb{Q}}$, the bilinear form
$\lambda:H^{2k}\times H^{2k}\rightarrow{\mathbb{Q}}$ defined as
$\langle\cdot\cup\cdot,\mu\rangle$ is isomorphic to $a\langle 1\rangle\oplus
b\langle-1\rangle$ for some non-negative integers $a,b$. Then there exist
basis classes
$\alpha_{1},\ldots\alpha_{a},\overline{\alpha}_{1},\ldots\overline{\alpha}_{b}\in
H^{n/2}$ such that $\langle\alpha_{i}\alpha_{j},\mu\rangle=\delta_{ij}$,
$\langle\overline{\alpha}_{i}\overline{\alpha}_{j},\mu\rangle=-\delta_{ij}$,
and $\langle\alpha_{i}\overline{\alpha}_{j},\mu\rangle=0$. Since
$(\sigma,\chi)=(a-b,a+b+2)$, we also want to prescribe
$a=\frac{\chi+\sigma-2}{2}\geq 0,b=\frac{\chi-\sigma-2}{2}\geq 0$, and at
least one of $a,b$ is nonzero.

By Theorem 1, there exists a simply-connected almost complex manifold $M^{4k}$
such that $H^{*}(M;{\mathbb{Q}})\cong(H^{*},\sigma,\chi)$ if and only if there
exist choice of class $c=1+c_{k}+c_{2k}\in H^{*}$ with $c_{k}\in
H^{2k}\cong{\mathbb{Q}}^{r}$ and $c_{2k}\in H^{4k}\cong{\mathbb{Q}}$, and a
fundamental class $\mu\in H_{0}\cong{\mathbb{Q}}$ that satisfy conditions
(i)–(v),

  1. (i)

(intersection form) By our definition, $(H^{*},\sigma,\chi)$ already satisfied
this condition and $a=\frac{\chi+\sigma-2}{2}\geq
0,b=\frac{\chi-\sigma-2}{2}\geq 0$. Then any class $c_{k}$ and $c_{2k}$ can be
expressed as

$\begin{cases}c_{k}=\sum_{i=1}^{a}w_{i}\alpha_{i}+\sum_{j=1}^{b}\overline{w}_{j}\overline{\alpha_{j}},\
\ \ \mbox{ for some }w_{i},\overline{w}_{j}\in{\mathbb{Q}}\\\
c_{2k}=y\alpha_{1}^{2}\ \ \mbox{ or }-y\overline{\alpha_{1}}^{2}\ \ \ \mbox{
for some }y\in{\mathbb{Q}}\end{cases}.$

Then for all partitions $I$ of $2k$, the parings $\langle c_{I},\mu\rangle$
are zero except possibly for

$x=\langle
c_{k}^{2},\mu\rangle=\sum_{i=1}^{a}w_{i}^{2}-\sum_{j=1}^{b}\overline{w}_{j}^{2}\
\ \mbox{ and }\ \ y=\langle c_{2k},\mu\rangle\ \ \ \mbox{ for some
}w_{i},\overline{w}_{j},y\in{\mathbb{Q}}.$

  2. (ii)

(signature equation)

    1. (a)

When $n=4k\equiv 0\pmod{8}$, $p_{i}(c)=0$ for all $i$ except
$p_{k/2}=(-1)^{\frac{k}{2}}2c_{k}$ and $p_{k}=c_{k}^{2}+2c_{2k}$. The $k$–th
${\mathcal{L}}$ class is
${\mathcal{L}}_{k}(p)=s_{\frac{k}{2},\frac{k}{2}}p_{\frac{k}{2}}^{2}+s_{k}p_{k}$,
where the coefficients $s_{\frac{k}{2}}=\frac{2^{k}(2^{k-1}-1)|B_{k}|}{k!}$
and $s_{\frac{k}{2},\frac{k}{2}}=\frac{1}{2}(s_{\frac{k}{2}}^{2}-s_{k})$. Then
condition (ii) says

$\displaystyle\ \ \ \langle L_{k}(p),\mu\rangle=\langle
4s_{\frac{k}{2},\frac{k}{2}}c_{k}^{2}+s_{k}(c_{k}^{2}+2c_{2k}),\mu\rangle=(2s_{\frac{k}{2}}^{2}-s_{k})x+2s_{k}y=\sigma$
(17)

    2. (b)

When $n=4k\equiv 4\pmod{8}$, $p_{i}(c)=0$ for all $i$ except
$p_{k}=c_{k}^{2}-2c_{2k}$. The $k$–th ${\mathcal{L}}$ class is
${\mathcal{L}}_{k}(p)=s_{k}p_{k}$, where the coefficient
$s_{k}=\frac{2^{2k}(2^{2k-1}-1)|B_{2k}|}{2k!}$. Then (ii) says

$\displaystyle\langle{\mathcal{L}}_{k}(p),\mu\rangle=s_{k}(x-2y)=\sigma$ (18)

  3. (iii)

and (iv) (integrality relations from $U$ and $SU$ cobordism)

    1. (a)

When $n=4k\equiv 0\pmod{8}$, condition (iii) says $x=\langle
c_{k}^{2},\mu\rangle$ and $y=\langle c_{2k},\mu\rangle$ are integers that
satisfy the integrality relations (2), which have been simplified in
Proposition 2 to $\langle\textup{Td},\mu\rangle\in{\mathbb{Z}}$, $\langle
e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle\in{\mathbb{Z}}$ and $\langle
e_{1}^{c}e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle\in\delta_{k}{\mathbb{Z}}$.

    2. (b)

When $n=4k\equiv 4\pmod{8}$, since $H^{2}=0$, we are in the case $c_{1}=0$, so
condition (iii) and (iv) says $x=\langle c_{k}^{2},\mu\rangle$ and $y=\langle
c_{2k},\mu\rangle$ are integers that satisfy the integrality relations (2) and
(3). Combining Proposition 2 and 3, the relations are satisfied if and only if
$\langle\textup{Td},\mu\rangle\in 2{\mathbb{Z}}$, $\langle
e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle\in{\mathbb{Z}}$ and $\langle
e_{1}^{c}e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle\in\delta_{k}{\mathbb{Z}}$.

  4. (v)

(Euler characteristic) condition (v) requires $y=\langle
c_{2k},\mu\rangle=\chi$.

######  Remark 4.

Note that because of condition (v), we can not presumably choose an
orientation $\mu$ to prescribe $a\geq b$ or $b\geq a$. This is essentially
because any complex vector bundle gives the underlying real vector bundle a
canonical preferred orientation. One can recall the fact that
${\mathbb{CP}}^{2}$ admits an almost complex structure but
$\overline{{\mathbb{CP}}^{2}}$ does not.

###  4.1. Realization theorem

We phrase the realization theorem in dimension $n\equiv 0\pmod{8}$ and
$n\equiv 4\pmod{8}$ respectively. A rational cohomology ring $H^{*}$ under our
Betti number assumption is realizable by an almost complex manifold if and
only if the signature $\sigma$ and the Euler characteristic $\chi$ satisfy a
set of integrality conditions.

######  Theorem 5.

There exists an $n=8k$–dimensional simply-connected closed almost complex
manifold $M$ with Betti number $b_{i}=0$ except $b_{0}=b_{n}=1$ and
$b_{n/2}\geq 1$, signature $\sigma$, and Euler characteristic $\chi$ if and
only if $a=\frac{\chi+\sigma-2}{2}\geq 0,b=\frac{\chi-\sigma-2}{2}\geq 0$, and
the following integrality relations are satisfied for some integer $x$.

$\displaystyle\langle
L_{2k},\mu\rangle=(2s_{k}^{2}-s_{2k})x+2s_{2k}\chi=\sigma\ \
{\scriptstyle\textup{with}\ \ s_{k}=\frac{2^{2k}(2^{2k-1}-1)|B_{2k}|}{(2k)!}}$
(19a)
$\displaystyle\langle\textup{Td},\mu\rangle=\frac{1}{2}(t_{2k}^{2}-t_{4k})x+t_{4k}\chi\in{\mathbb{Z}}\
\ {\scriptstyle\textup{with}\ \ t_{k}=\frac{B_{k}}{k!}}$ (19b)
$\displaystyle\langle
e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle=\left[\displaystyle\frac{-t_{2k}}{(2k-1)!}+\displaystyle\frac{1}{2(4k-1)!}\right]x-\displaystyle\frac{\chi}{(4k-1)!}\in{\mathbb{Z}}$
(19c) $\displaystyle\langle
e_{1}^{c}e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle=\displaystyle\frac{x}{[(2k-1)!]^{2}}\in\delta_{k}{\mathbb{Z}}\
\ {\small\textup{with}\ \delta_{k}=\left\\{\begin{array}[]{ll}1&k=1\\\
2&k>1\end{array}\right.}$ (19f)

######  Theorem 6.

There exists an $n=8k+4(k>0)$–dimensional simply-connected closed almost
complex manifold $M$ with Betti number $b_{i}=0$ except $b_{0}=b_{n}=1$ and
$b_{n/2}\geq 1$, signature $\sigma$, and Euler characteristic $\chi$ if and
only if $a=\frac{\chi+\sigma-2}{2}\geq 0,b=\frac{\chi-\sigma-2}{2}\geq 0$, and
the following integrality relations are satisfied for some integer $x$.

$\displaystyle\langle L_{2k+1},\mu\rangle=s_{2k+1}(x-2\chi)=\sigma$ (20a)
$\displaystyle\langle\textup{Td},\mu\rangle=-\frac{1}{2}t_{4k+2}(x-2\chi)\in
2{\mathbb{Z}}$ (20b) $\displaystyle\langle
e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle=\displaystyle\frac{1}{2(4k+1)!}(x-2\chi)\in{\mathbb{Z}}$
(20c) $\displaystyle\langle
e_{1}^{c}e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle=\displaystyle\frac{x}{[(2k)!]^{2}}\in
2{\mathbb{Z}}$ (20d)

###### Proof of Theorem 5 and 6.

We have explained in the beginning of this section that conditions (ii)-(v) in
Theorem 1 applied in our prescribed rational cohomology ring result in
(19a)–(19f) in dimension $n=8k$ and (20a)–(20d) in dimension $n=8k+4$, where
the integer $x$ equals the Chern number $\langle
c_{2k}^{2}(\tau_{M}),[M]\rangle$ of the realizing manifold $M$. We are only
left to show that any integer $x$ satisfying these conditions can be written
as $\sum_{i=1}^{a}w_{i}^{2}-\sum_{j=1}^{b}\overline{w}_{j}^{2}$ for some
rational numbers $w_{i},\overline{w}_{j}$, this is true as long as $a\neq 0$
and $b\neq 0$ because any integer $x$ can be written as a difference of two
rational squares. In fact, if $\sigma$ and $\chi$ are integers that support
any solution $x$ satisfying the integrality conditions, the integers
$a=\frac{\chi+\sigma-2}{2}$ and $b=\frac{\chi-\sigma-2}{2}$ are automatically
nonzero. For dimension $n=8$, this is shown in the proof of Proposition 7. For
any dimension $n=8k$ with $k>1$ or $n=8k+4$ with $k>0$, this can be seen from
Theorem 11 and Theorem 12 respectively, which says the integrality conditions
requires $\sigma$ and $\chi$ to be divisible by 4, therefore $a$ and $b$ must
be nonzero.

∎

###  4.2. Examples in dimension 8, 12, and 16

For dimension $n=4k$ with $k=1,2,3$, we calculate the congruence relations
among $\sigma$ and $\chi$, and work out some examples.

######  Proposition 7.

There exists an $8$–dimensional simply-connected almost complex manifold with
$b_{i}=0$ except $b_{0}=b_{8}=1$ and $b_{4}\geq 1$, signature $\sigma$, and
Euler characteristic $\chi$ if and only if $\sigma\equiv 0\bmod{2}$,
$\chi\equiv 0\bmod{6}$, $3\sigma-\chi\equiv 0\bmod{48}$, and
$a=\frac{\chi+\sigma-2}{2}>0,b=\frac{\chi-\sigma-2}{2}>0$.

###### Proof.

For dimension $n=8$, conditions (19a)–(19f) in Theorem 5 says:

$\begin{cases}\langle L_{2},\mu\rangle=\frac{1}{45}(3x+14\chi)=\sigma\\\
\langle\textup{Td},\mu\rangle=\frac{1}{720}(3x-\chi)\in{\mathbb{Z}}\\\ \langle
e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle=-\frac{1}{6}\chi\in{\mathbb{Z}}\\\
\langle
e_{1}^{c}e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle=x\in{\mathbb{Z}}\end{cases}\Longleftrightarrow\begin{cases}3x+14\chi=45\sigma\\\
3x-\chi=720m\text{ for }m\in{\mathbb{Z}}\\\ \chi=6s\text{ for
}s\in{\mathbb{Z}}^{+}\end{cases}.$

The system has integer solution $x$ if and only if $\chi=6s$, $\sigma=2s+16m$
for $s\in{\mathbb{Z}}^{+},m\in{\mathbb{Z}}$, and $x=2s+240m$. The rational
intersection form $a\langle 1\rangle\oplus b\langle-1\rangle$ requires
$a=\frac{\chi+\sigma-2}{2}=4s+8m-1\geq 0$ and
$b=\frac{\chi-\sigma-2}{2}=2s-8m-1\geq 0$, which is equivalent to
$\frac{-4s+1}{8}\leq m\leq\frac{2s-1}{8}$ and $s$ can take any positive
integer value. Note that $a,b\neq 0$ for any $s\in{\mathbb{Z}}^{+}$ and
$m\in{\mathbb{Z}}$.∎

Below is a list of realizable rational cohomology ring $(H^{*},\sigma,\chi)$
with the smallest few possible values of Euler characteristic. Note that
$\chi=48$ would be the smallest value to realize signature $\sigma=0$, with
$(a,b)=(23,23)$.

$\displaystyle\chi=6,\sigma=2;\ \ \ (a,b)=(3,1)^{*};$
$\displaystyle\chi=12,\sigma=4;\ \ \ (a,b)=(7,3)^{*};$
$\displaystyle\chi=18,\sigma=-10,6;\ \ \ (a,b)=(3,13),(11,5)^{*};$
$\displaystyle\chi=24,\sigma=-8,8;\ \ \ (a,b)=(7,15),(15,7)^{*};$
$\displaystyle\chi=30,\sigma=-22,-6,10,26;\ \ \
(a,b)=(3,25),(11,17),(19,9)^{*},(27,1);$

######  Remark 8.

Note that in dimension 8, the prescribed rational cohomology
$(H^{*},\sigma,\chi)$ agree with that of
$\\#_{a}{\mathbb{HP}}^{2}\\#_{b}\overline{{\mathbb{HP}}^{2}}$. One can compare
the realizable examples listed above with the discussion in [Mil21, 6.1],
which calculated that the connected sums of quaternionic projective planes
$\\#_{a}{\mathbb{HP}}^{2}\\#_{b}\overline{{\mathbb{HP}}^{2}}$ admits almost
complex structure if and only if $(a,b)=(4n+3,2n+1)$ for some
$n\in{\mathbb{Z}}$, that is to say the integral (no–torsion) version of the
rational cohomology ring $(H^{*},\sigma,\chi)$ is realizable if and only if
$\sigma\equiv 0\bmod{2}$, $\chi\equiv 0\bmod{6}$, and $3\sigma-\chi=0$. These
examples are marked with $*$ in the above list of $(a,b)$.

######  Proposition 9.

There exists a $16$–dimensional simply-connected almost complex manifold with
$b_{i}=0$ except $b_{0}=b_{16}=1$ and $b_{8}\geq 1$, signature $\sigma$, and
Euler characteristic $\chi$ if and only if $\sigma\equiv 0\bmod{8}$,
$\chi\equiv 0\bmod{120}$, $15\sigma-\chi\equiv 0\bmod{1952}$, and
$a=\frac{\chi+\sigma-2}{2}>0,b=\frac{\chi-\sigma-2}{2}>0$.

###### Proof.

For dimension $n=16$, (19a)–(19f) in Theorem 5 says:

$\begin{cases}\langle L_{2},\mu\rangle=\frac{1}{14175}(305x+762\chi)=\sigma\\\
\langle\textup{Td},\mu\rangle=\frac{1}{3628800}(5x-3\chi)\in{\mathbb{Z}}\\\
\langle
e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle=\frac{1}{15120}(5x-3\chi)\in{\mathbb{Z}}\\\
\langle
e_{1}^{c}e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle=\frac{x}{36}\in
2{\mathbb{Z}}\end{cases}\Longleftrightarrow\begin{cases}305x+762\chi=3^{4}5^{2}7\sigma\\\
5x-3\chi=2^{8}3^{4}5^{2}7m\ \text{ for }m\in{\mathbb{Z}}\\\ x=2^{3}3^{2}\ell\
\text{ for }\ell\in{\mathbb{Z}}\end{cases}$

The system has integer solution for $x$ if and only if $\chi=120s$,
$\sigma=8(s+1952m)$ for $s\in{\mathbb{Z}}^{+},m\in{\mathbb{Z}}$, and
$x=72s+725760m$. The rational intersection form requires
$a=\frac{\chi+\sigma-2}{2}=64s+7808m-1\geq 0$ and
$b=\frac{\chi-\sigma-2}{2}=56s-7808m-1\geq 0$, which is equivalent to
$\frac{-64s+1}{7808}\leq m\leq\frac{56s-1}{7808}$ and $s$ can take any
positive integer value. Note that $\chi$ and $\sigma$ are both divisible by 4,
so $a,b\neq 0$ for any value of $s$ and $m$.

If $H^{*}$ is realizable, the smallest two possible values of $\chi$ are
$\chi=120$, with $\sigma=8,(a,b)=(64,56)$, and $\chi=240$, with
$\sigma=16,(a,b)=(127,111)$. ∎

######  Proposition 10.

There exists a $12$–dimensional simply-connected almost complex manifold with
$b_{i}=0$ except $b_{0}=b_{12}=1$ and $b_{6}\geq 1$, signature $\sigma$, and
Euler characteristic $\chi$ if and only if the signature $\sigma\equiv
0\bmod{(2^{8})(31)}$, $\chi\equiv 0\bmod{4}$, and
$a=\frac{\chi+\sigma-2}{2}>0,b=\frac{\chi-\sigma-2}{2}>0$.

###### Proof.

For dimension $n=12$, conditions (20a)–(20d) in Theorem 6 says:

$\begin{cases}\langle L_{3},\mu\rangle=\frac{62}{945}(x-2\chi)=\sigma\\\
\langle\textup{Td},\mu\rangle=\frac{-1}{60480}(x-2\chi)\in 2{\mathbb{Z}}\\\
\langle
e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle=\frac{1}{240}(x-2\chi)\in{\mathbb{Z}}\\\
\langle e_{1}^{c}e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle=\frac{x}{4}\in
2{\mathbb{Z}}\end{cases}\Longleftrightarrow\begin{cases}62(x-2\chi)=3^{3}5^{1}7\sigma\\\
x-2\chi=2^{7}3^{3}5^{1}7m\text{ for }m\in{\mathbb{Z}}\\\ x=8\ell\text{ for
}\ell\in{\mathbb{Z}}\end{cases}.$

The system has solution if and only if $\chi=4s$, $\sigma=(2^{8})(31)m$ for
$s\in{\mathbb{Z}}^{+},m\in{\mathbb{Z}}$, and $x=8s+(2^{7})(3^{3})(5)(7)m$. The
rational intersection form $a\langle 1\rangle\oplus b\langle-1\rangle$
requires $a=\frac{\chi+\sigma-2}{2}=2s+2968m-1\geq 0$ and
$b=\frac{\chi-\sigma-2}{2}=2s-2968m-1\geq 0$, which is equivalent to $s\geq
1984|m|+1$ and $m$ can take any integer value. Again note that $\chi$ and
$\sigma$ are both divisible by 4, so $a,b\neq 0$ for any value of $s$ and $m$.

If $\sigma=0$, $H^{*}$ is realizable if and only if $\chi=4s$ for $s\geq 1$.
The smallest nonzero absolute value of the signature is $|\sigma|=7936$, in
this case, $H^{*}$ is realizable if and only if $\chi=4s$ for $s\geq 1985$. ∎

###  4.3. Divisibility of the signature and Euler characteristic

We obtain the following divisibility results on $\sigma$ and $\chi$ in
dimension $n=8k$ and $n=8k+4$ respectively. The approach is similar to that
applied in [FS16] and [KS17], by examining the coefficients in the signature
equation, Todd genus, and the other integrality conditions of Chern numbers.
Note that these divisibility conditions are necessary for any, not only
simply-connected, almost complex manifold realizing the prescribed rational
cohomology ring.

######  Theorem 11.

If $M^{n}$ is an $n=8k$–dimensional almost complex manifold with Betti number
$b_{i}=0$ except $b_{0}=b_{n}=1$ and $b_{n/2}\geq 1$, its signature $\sigma$
and Euler characteristic $\chi$ must satisfy the following divisibility
conditions, where $\nu_{2}(k)$ denotes the 2-adic order of $k$,
$\textup{wt}(k)$ denotes the number of ones in the binary representation of
$k$.

$\displaystyle\nu_{2}(\sigma)\geq 4k-2\nu_{2}(k)-3,$ (21)
$\displaystyle\nu_{2}(\chi)\geq 4k-2\nu_{2}(k)-2\textup{wt}(k)-2.$ (22)

For example, in dimension $n=8$, $\sigma\equiv 0\bmod{2}$,

in dimension $n=16$, $\sigma\equiv 0\bmod{2^{3}}$ and $\chi\equiv
0\bmod{2^{2}}$,

in dimension $n=24$, $\sigma\equiv 0\bmod{2^{9}}$ and $\chi\equiv
0\bmod{2^{6}}$.

###### Proof.

It has been shown in Proposition 7 that any $n=8$-dimensional almost complex
manifold realizing $H^{*}$ must have the signature $\sigma\equiv 0\bmod{2}$,
so 21 holds true for $k=1$.

If $(H^{*},\sigma,\chi)$ is realizable by an $n=8k$–dimensional almost complex
manifold, conditions (19a), (19b) and (19f) in Theorem 5 says there must exist
an integer $x$ such that

$\displaystyle 2s_{k}^{2}x-s_{2k}(x-2\chi)=\sigma$ (23a) $\displaystyle
t_{2k}^{2}x-t_{4k}(x-2\chi)=2m\ \ \text{ for }m\in{\mathbb{Z}}$ (23b)
$\displaystyle x=[(2k-1)!]^{2}\,2\ell\text{ for }\ell\in{\mathbb{Z}}$ (23c)

Since $s_{2k}=-2^{4k}(2^{4k-1}-1)t_{4k}$, we multiply 23b by
$2^{4k}(2^{4k-1}-1)$ and add it to 23a to eliminate the $(x-2\chi)$ term,
after simplifying and plugging in 23c, we have

$\underbrace{2^{4k}(2^{4k}-2^{2k+1}+1)\left(\displaystyle\frac{|B_{2k}|}{2k}\right)^{2}2}_{L1}\ell=\underbrace{2^{4k+1}(2^{4k-1}-1)}_{R1}m+\sigma.$

By the 2-adic order formula
$\nu_{2}\left(\frac{|B_{2k}|}{2k}\right)=-(\nu_{2}(k)+2)$ ([KS17, section 3]),
we have

$\nu_{2}(L1)=4k-2\nu_{2}(k)-3,\ \ \nu_{2}(R1)=4k+1>\nu_{2}(L1).$

If $\sigma\neq 0$, then
$\nu_{2}(\sigma)\geq\textup{min}\\{\nu_{2}(L1),\nu_{2}(R1)\\}=\nu_{2}(L1)$,
which proved (21). To show the Euler characteristic divisibility in (22), we
use conditions (19c) and (19f),

$\displaystyle\left[\displaystyle\frac{-t_{2k}}{(2k-1)!}+\displaystyle\frac{1}{2(4k-1)!}\right]x-\displaystyle\frac{\chi}{(4k-1)!}=m\
\ \text{ for }m\in{\mathbb{Z}}$ (24a) $\displaystyle x=[(2k-1)!]^{2}\,2\ell\ \
\text{ for }\ell\in{\mathbb{Z}}$ (24b)

Multiply (24a) by $(4k-1)!$ and plugin (24b), we get

$\underbrace{-\displaystyle\frac{|B_{2k}|}{2k}(4k-1)!\,2}_{L1}\ell+\underbrace{[(2k-1)!]^{2}}_{L2}\ell-\chi=\underbrace{(4k-1)!}_{R1}m.$

By the 2-adic order formula
$\nu_{2}\\!\left(\\!\frac{|B_{2k}|}{2k}\\!\right)=-(\nu_{2}(k)+2)$,
$\nu_{2}[(2k-1)!]=2k-\nu_{2}(k)-\textup{wt}(k)-1$, and
$\nu_{2}[(4k-1)!]=4k-\nu_{2}(k)-\textup{wt}(k)-2$ (computed in [KS17, section
3]), we have

$\displaystyle\nu_{2}(L1)=4k-2\nu_{2}(k)-\textup{wt}(k)-3,\ \
\nu_{2}(L2)=4k-2\nu_{2}(k)-2\textup{wt}(k)-2,$ $\displaystyle\
\nu_{2}(R1)=4k-\nu_{2}(k)-\textup{wt}(k)-2.$

Then
$\nu_{2}(\chi)\geq\textup{min}\\{\nu_{2}(L1),\nu_{2}(L2),\nu_{2}(R1)\\}=\nu_{2}(L2)$.
∎

######  Theorem 12.

If $M^{n}$ is an $n=8k+4\ (k>0)$–dimensional almost complex manifold with
Betti number $b_{i}=0$ except $b_{0}=b_{n}=1$ and $b_{n/2}\geq 1$, its
signature $\sigma$ and Euler characteristic $\chi$ must satisfy the following
divisibility conditions:

$\displaystyle\sigma\equiv 0\bmod{2^{4k+4}(2^{4k+1}-1)}$ (25)
$\displaystyle\chi\equiv 0\bmod{[(2k)!]^{2}}$ (26)

In particular, the 2–adic order of $\sigma$ and $\chi$ satisfy:

$\displaystyle\nu_{2}(\sigma)\geq 4k+4$ (27) $\displaystyle\nu_{2}(\chi)\geq
4k-2\textup{wt}(k)$ (28)

###### Proof.

The Todd genus condition (20b) says $-t_{4k+2}(x-2\chi)=4m$ for some
$m\in{\mathbb{Z}}$. Since $s_{2k+1}=2^{4k+2}(2^{4k+1}-1)t_{4k+2}$, multiplying
this equation by $2^{4k+2}(2^{4k+1}-1)$ and adding it to the signature
equation (20a) yields $0=\sigma+2^{4k+2}(2^{4k+1}-1)4m$, which proved the
claim on $\sigma$.

By (20d), $x=2[(2k)!]^{2}\ell$ for some $\ell\in{\mathbb{Z}}$, plug this into
the Todd genus condition (20b) which says $x-2\chi=2(4k+1)!m$ for some
$m\in{\mathbb{Z}}$, we have $[(2k)!]^{2}\ell-\chi=(4k+1)!m$, since
$\frac{(4k+1)!}{[(2k)!]^{2}}=(4k+1)\binom{4k}{2k}\in{\mathbb{Z}}$, (26) is
proved. $\nu_{2}((2k)!)=2k-\textup{wt}(2k)=2k-\textup{wt}(k)$ gives (28).

∎

Theorem 11 and Theorem 12 imply the following.

######  Theorem 13.

An $n=4k(k>1)$–dimensional closed almost complex manifold with Betti number
$b_{i}=0$ except $b_{0}=b_{n}=1$,$b_{n/2}\geq 1$ must have even signature
$\sigma$ and even Euler characteristic $\chi$, i.e., the middle Betti number
$b_{n/2}$ must be even.

######  Corollary 14.

In dimension greater than 4, the rarely existing rational projective plane
(smooth manifold whose Betti number $b_{0}=b_{n/2}=b_{n}=1$) does not admit
any almost complex structure. Equivalently, there does not exist any closed
almost complex manifold whose sum of Betti numbers equals three.

######  Remark 15.

One can refer to [Su09], [Su14], [FS16], [KS17] for results on existence of
rational projective planes. In summer 2018, upon seeing the arXiv paper [AM19]
showing that an almost complex manifold with sum of Betti numbers three can
only exists in dimension $n=2^{a}$, the author had worked out the nonexistence
result in Corollary 14 using the signature equation and the integrality
condition that $\langle
e_{1}^{c}e_{1}^{c}\mathord{\cdot}\textup{Td},\mu\rangle=\frac{c_{2k}^{2}[M^{8k}]}{[(2k-1)!]^{2}}\in{\mathbb{Z}}$.
In 2019, the author was informed that Jiahao Hu had also obtained this result
independently using the signature equation and the integrality of Todd genus
[JH21]. Now we know more generally, there does not exist almost complex
manifold (above dimension 4) with Betti numbers concentrated in the middle
dimension and the sum of Betti numbers being odd. The lower bound of the
2-adic order of the Euler characteristic increases with respect to the
dimension of the manifold.

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