1 Introduction
This is a companion paper to our earlier work [Reference Akhmedov, Hughes and Park1] with M. C. Hughes and addresses the geography problem for closed simply connected nonspin symplectic
$4$
-manifolds with positive signature. For some background and history, we refer the reader to the introduction in [Reference Akhmedov, Hughes and Park1]. For the corresponding spin geography problem, we refer the reader to our papers [Reference Akhmedov and Park3, Reference Akhmedov and Park4].
We start by setting up some basic notation. Given a closed smooth
$4$
-manifold M, let
$e(M)$
and
$\sigma (M)$
denote the Euler characteristic and the signature of M, respectively. We define
$\chi _h(M) = \frac {1}{4}(e(M)+\sigma (M))$
and
$c_1^2(M) = 2e(M)+3\sigma (M)$
. When M is a complex surface,
$\chi _h(M)$
is the holomorphic Euler characteristic of
$M,$
while
$c_1^2(M)$
is the square of the first Chern class of M. Given an ordered pair of integers
$(a,b)$
, the geography problem asks whether there exists a closed smooth
$4$
-manifold M with the desired properties satisfying
$\chi _h(M) =a$
and
$c_1^2(M)= b$
. We note that such M must satisfy
$b=8a+\sigma (M)$
.
Given
$x\in \mathbb {R}$
, we define the ceiling function as
$$ \begin{align} \lceil x\rceil=\min\{k\in\mathbb{Z} \mid k\geq x\}. \end{align} $$
Next we recall the following definition from [Reference Akhmedov, Hughes and Park1, Definition 13].
Definition 1.1 Given an integer
$\sigma \geq 0$
, let
$\lambda (\sigma )$
be the smallest positive integer with the following properties.
-
(i)
$\lambda (\sigma )\geq \lceil (\sigma +1)/2\rceil $
. -
(ii) Every integral point
$(a,b)$
on the line
$b = 8a + \sigma $
satisfying
$a\geq \lambda (\sigma )$
is realized as
$(\chi _h(M_i),c_1^2(M_i))$
, where
$\{M_i \mid i\in \mathbb {Z} \}$
is an infinite family of homeomorphic but pairwise nondiffeomorphic closed simply connected nonspin irreducible
$4$
-manifolds such that
$M_i$
is symplectic for each
$i\geq 0$
and
$M_i$
is nonsymplectic for each
$i<0$
.
We also recall the following definition from [Reference Akhmedov and Park3, Definition 1].
Definition 1.2 We say that a
$4$
-manifold M has
$\infty ^2$
-property if there exist infinitely many pairwise nondiffeomorphic irreducible symplectic
$4$
-manifolds and infinitely many pairwise nondiffeomorphic irreducible nonsymplectic
$4$
-manifolds, all of which are homeomorphic to M.
Let
$\mathbb {CP}^2$
be the complex projective plane, and let
$\overline {\mathbb {CP}}{}^2$
be the underlying smooth
$4$
-manifold
$\mathbb {CP}^2$
equipped with the opposite orientation. By Freedman’s classification theorem (cf. [Reference Freedman11]), if k is any odd integer satisfying
$k\geq 2\lambda (\sigma )-1$
, then the nonspin
$4$
-manifold
$k\mathbb {CP}^2\#(k-\sigma )\overline {\mathbb {CP}}{}^2$
, the connected sum of k copies of
$\mathbb {CP}^2$
, and
$k-\sigma $
copies of
$\overline {\mathbb {CP}}{}^2$
, have
$\infty ^2$
-property. The following conjecture from [Reference Akhmedov, Hughes and Park1] remains open.
Conjecture 1.3
$\lambda (\sigma ) = \lceil (\sigma +1)/2\rceil $
for every integer
$\sigma \geq 0$
. Equivalently, given any integer
$\sigma \geq 0$
,
$k\mathbb {CP}^2\#(k-\sigma )\overline {\mathbb {CP}}{}^2$
has
$\infty ^2$
-property for every odd integer k satisfying
$$ \begin{align*} k\geq \begin{cases} \sigma & \textrm{when}\ \sigma\ \textrm{is}\ \textrm{odd,}\ \\ \sigma+1 & \textrm{when}\ \sigma\ \textrm{is}\ \textrm{even.} \end{cases} \end{align*} $$
We note that Conjecture 1.3 postulates that there would be no constraint on
$k\mathbb {CP}^2\#(k-\sigma )\overline {\mathbb {CP}}{}^2$
having
$\infty ^2$
-property other than the positive integer k being odd, which is necessary for
$k\mathbb {CP}^2\#(k-\sigma )\overline {\mathbb {CP}}{}^2$
to support a symplectic (and hence an almost complex) structure.
In [Reference Akhmedov, Hughes and Park1, Reference Akhmedov and Park2, Reference Akhmedov and Sakallı5], numerical upper bounds for
$\lambda (\sigma )$
were given when
$0\leq \sigma \leq 100$
. In Section 3, we will present a new algorithm for constructing simply connected
$4$
-manifolds starting from a surface fibration over a surface with a section, which need not be a fiber bundle nor a Lefschetz fibration. Using this algorithm, we will construct two new infinite families of closed, simply connected, nonspin, irreducible, symplectic
$4$
-manifolds of positive signature, many of which have a smaller value of
$\chi _h$
than the currently known upper bounds on
$\lambda (\sigma )$
. We cannot currently show that all of these
$4$
-manifolds have
$\infty ^2$
-property, but we suspect that they all do (see Remark 3.4 and Corollary 3.6). The new building blocks in our construction are certain complex surfaces of general type found in [Reference Bauer and Catanese6, Reference Catanese and Dettweiler8, Reference Lee, Lönne and Rollenske16], and these will be reviewed in Section 2. In Section 4, we will also provide two explicit formulae for upper bounds on
$\lambda (\sigma )$
that work for every nonnegative integer
$\sigma $
(see Corollaries 4.2 and 4.4). Asymptotically as
$\sigma \rightarrow \infty $
, we will prove that
$$ \begin{align} \lambda(\sigma)\leq\frac{8}{5}\sigma +O(\sigma^{1/2}). \end{align} $$
Such an asymptotic upper bound has been missing in the literature, and we hope that our bound provides a useful benchmark for future works. Our ultimate goal is to decrease the coefficient of
$\sigma $
in (1.2) from
$1.6$
to a smaller number that is much closer to the coefficient
$0.5$
in Conjecture 1.3.
2 Building Blocks
In this section, we will collect all the
$4$
-manifold building blocks that we will need for our constructions later. Our first family of building blocks are the so-called
$BCD$
surfaces constructed by Bauer, Catanese, and Dettweiler in [Reference Bauer and Catanese6, Reference Catanese and Dettweiler8].
Lemma 2.1 For each positive integer
$n\geq 5$
that is coprime with
$6$
, there exists a minimal complex surface
$S(n)$
of general type with
$c_1^2(S(n))=5(n-2)^2$
,
$e(S(n))=2n^2-10n+15$
, and
$\sigma (S(n))=(n^2-10)/3$
. Each
$S(n)$
admits a genus
$n-1$
fibration over a genus
$(n-1)/2$
curve. Moreover,
$S(n)$
also contains four disjoint genus
$(n-1)/2$
curves of self-intersection
$-1$
, one of which is a section of the fibration, and each of the other three is contained in a singular fiber and hence disjoint from regular fibers.
Proof Recall from [Reference Catanese and Dettweiler8] that
$S(n)$
arises as a
$(\mathbb {Z}/n\mathbb {Z})^2$
Abelian Galois ramified cover (in the sense of [Reference Pardini18]) over a del Pezzo surface
$\mathbb {CP}^2\#4\overline {\mathbb {CP}}{}^2$
of degree 5. The branch divisor of this covering is a sum of ten rational curves, four of which are the exceptional divisors of the blow-ups. We note that the preimages of the exceptional divisors under this
$(\mathbb {Z}/n\mathbb {Z})^2$
covering map are disjoint genus
$(n-1)/2$
curves of self-intersection
$-1$
. The genus
$n-1$
fibration structure on
$S(n)$
and its singular fibers are discussed in [Reference Catanese and Dettweiler8, Proposition 4.2]. We recall that this fibration is obtained by lifting a pencil of lines going through a point of blow-up, and thus a section of the fibration is given by the inclusion of the preimage of the corresponding exceptional divisor. The characteristic numbers
$c_1^2(S(n))$
and
$e(S(n))$
were computed in [Reference Catanese and Dettweiler8, Proposition 4.3]. We can readily compute the signature of
$S(n)$
using the well-known formula
$c_1^2=2e+3\sigma $
. ▪
Let
$\Sigma _b$
denote a closed connected
$2$
-manifold with genus
$b\geq 0$
. Our second building block is a
$\Sigma _7$
bundle over
$\Sigma _5$
that was constructed in [Reference Lee, Lönne and Rollenske16].
Lemma 2.2 There exists a minimal complex surface Y of general type with
$e(Y)=96$
and
$\sigma (Y)=16$
such that Y is the total space of a surface bundle over a surface with base genus
$5$
and fiber genus
$7$
. Moreover, this surface bundle admits a section whose image in Y has self-intersection
$-8$
.
Proof In [Reference Lee, Lönne and Rollenske16, Example 6.9], such Y was constructed as the double cover of
$\Sigma _3\times \Sigma _3$
branched over
$4$
disjoint graphs of involutions on
$\Sigma _3$
. Each graph in the branch locus gives rise to a section of the bundle whose image in Y has self-intersection equal to
$2$
times the self-intersection of the graph in
$\Sigma _3\times \Sigma _3,$
which is
$-4$
. ▪
Our next family of building blocks are the homotopy elliptic surfaces constructed by Fintushel and Stern in [Reference Fintushel and Stern9]. Let
$E(1)=\mathbb {CP}^2\#9\overline {\mathbb {CP}}{}^2$
denote a rational elliptic surface that is the complex projective plane blown up nine times. For a positive integer r, let
$E(r)$
denote the fiber sum of r copies of
$E(1)$
. Then
$E(r)$
is a simply connected elliptic surface without any multiple fiber. Let F be a smooth torus fiber of
$E(r)$
and let K be a knot of genus
$g(K)$
in
$S^3$
. Let
$E(r)_K$
denote the result of performing a knot surgery on
$E(r)$
along F:
$$ \begin{align} E(r)_K = \big[ E(r)\backslash \nu(F) \big] \cup \big[ S^1\times(S^3\backslash\nu(K)) \big], \end{align} $$
where the
$\nu $
’s denote tubular neighborhoods. In (2.1), we glue the 3-torus boundaries in such a way that the meridians of F get identified with the longitudes of K.
We recall that
$E(r)_K$
is homeomorphic to
$E(r),$
so we have
$\pi _1(E(r)_K)=1$
,
$$ \begin{align*} e(E(r)_K) = e(E(r)) = 12r \quad \text{and} \quad \sigma(E(r)_K) = \sigma(E(r)) = -8r . \end{align*} $$
We also recall that
$E(r)$
and
$E(r)_K$
are spin if and only if r is even. If K is a fibered knot, then
$E(r)_K$
admits a symplectic structure, and a sphere section of
$E(r)$
and a Seifert surface of K can be glued together to form a symplectic submanifold
$\Sigma _K$
of genus
$g(K)$
and self-intersection
$-r$
inside
$E(r)_K$
. Given a nonnegative integer m, let
$F_K^m$
be the genus
$g(K)+m$
symplectic submanifold of
$E(r)_K$
with self-intersection
$2m-r$
that is obtained from the union of
$\Sigma _K$
and m copies of torus fiber by symplectically resolving their m intersection points. We note that
$F_K^0=\Sigma _K$
.
Lemma 2.3 Let
$m\geq 0$
and
$r>0$
be integers, and let K be a fibered knot in
$S^3$
. Let
$\nu (F_K^m)$
denote a tubular neighborhood of
$F_K^m$
in
$E(r)_K$
. Then the complement
$E(r)_K \backslash \nu (F_K^m)$
is simply connected. If
$r\geq 2$
, then write
$r=2\rho +\epsilon $
for integers
$\epsilon =0,1$
and
$\rho \geq 1$
. Then
$E(r)_K\backslash \nu (F_K^m)$
contains
$2\rho $
disjoint symplectic tori
$T_j (j=1,\dots ,2\rho )$
of self-intersection
$0$
such that
$\pi _1(E(r)_K\backslash (\nu (F_K^m) \cup (\cup _{j=1}^{2\rho } T_j)))=1$
.
Proof Each surface
$F_K^m$
transversely intersects once a topological sphere in
$E(r)_K$
coming from a cusp fiber of
$E(r)$
. Thus, any meridian of
$F_K^m$
is nullhomotopic in
$E(r)_K \backslash \nu (F_K^m)$
. Hence, we conclude that
$\pi _{1}(E(r)_K \backslash \nu (F_K^m))=\pi _1(E(r)_K)=1$
. Next, we recall from [Reference Gompf and Mrowka13] that
$E(2)$
contains
$3$
disjoint copies of the Gompf nucleus. If
$r\geq 2$
, then
$E(r)$
can be viewed as the fiber sum of
$\rho $
copies of
$E(2)$
and possibly a copy of
$E(1)$
. In each copy of
$E(2)$
, we have
$2$
copies of Gompf nuclei that are disjoint from the tori and sections used in the fiber sum, and thus
$E(r)_K$
contains
$2\rho $
Gompf nuclei that are all disjoint from
$\nu (F_K^m)$
. Let
$N_j$
denote one of these nuclei, and let
$T_j$
be a smooth torus fiber in
$N_j$
(
$j=1,\dots ,2\rho $
). By changing the symplectic form on the
$E(2)$
parts if necessary, we can arrange each
$T_j$
to be a symplectic submanifold of
$E(r)_K$
. Since
$T_j$
transversely intersects once a sphere section of
$N_j$
with self-intersection
$-2$
, every meridian of
$T_j$
is nullhomotopic. It follows that
$\pi _1(E(r)_K\backslash (\nu (F_K^m) \cup (\cup _{j=1}^{2\rho } T_j)))=\pi _1(E(r)_K\backslash \nu (F_K^m))=1$
. ▪
From the Seifert–Van Kampen theorem, we can also deduce that
$$ \begin{align*} \pi_1(E(r)_K\backslash (\nu(F_K^m) \cup (\cup_{j=1}^{\tau} T_j)))=1 \end{align*} $$
for any integer
$\tau $
satisfying
$0\leq \tau \leq 2\rho $
, i.e., we can choose to take out less tori and still have the complement remain simply connected. Our final set of building blocks are certain families of symplectic
$4$
-manifolds that were studied in [Reference Akhmedov, Hughes and Park1].
Lemma 2.4 For each positive integer u, there is a pair of closed simply connected nonspin irreducible symplectic
$4$
-manifolds
$Q(u)$
and
$\tilde {Q}(u)$
satisfying
$$ \begin{align*} \sigma(Q(u))&=26u-2\lceil u/2\rceil -2,\\ \chi_h(Q(u))&=27u+32\lceil u/2\rceil -2,\\ \sigma(\tilde{Q}(u))&=25u^2+u-2\lceil u/2\rceil -2,\\ \chi_h(\tilde{Q}(u))&=25u^2+\lceil u/2\rceil(30u+2) +2u -2, \end{align*} $$
where
$\lceil \ \rceil $
is given by (1.1). Let Q denote either
$Q(u)$
or
$\tilde {Q}(u)$
. Then each Q contains a disjoint pair of symplectic tori
$T_1^{\prime }$
and
$T_2^{\prime }$
of self-intersection
$0$
satisfying
$\pi _1(Q\backslash (T_1^{\prime } \cup T_2^{\prime }))=1$
.
Proof We let
$Q(u)=Q_n^m(W^{p,v}_{u_1,u_2})$
in [Reference Akhmedov, Hughes and Park1, Example 12] with
$m=1$
,
$p=5$
,
$u_1=u$
,
$u_2=1$
,
$v=1$
,
$t=\lceil u/2 \rceil $
, and
$n=16\lceil u/2\rceil +u+1\geq 18$
. We let
$\tilde {Q}(u)=Q_n^m(W^{p,v}_{u_1,u_2})$
in [Reference Akhmedov, Hughes and Park1, Example 12] with
$m=1$
,
$p=5$
,
$u_1=u$
,
$u_2=u$
,
$v=1$
,
$t=\lceil u/2 \rceil $
, and
$n=\lceil u/2\rceil (15u+1)+u+1\geq 18$
. The existence of
$T_1^{\prime }$
and
$T_2^{\prime }$
follows from [Reference Akhmedov, Hughes and Park1, Theorem 9]. ▪
3 New Symplectic
$4$
-manifolds
We start the section with a general algorithm for producing simply connected
$4$
-manifolds from a symplectic fibration. Let X be a closed symplectic
$4$
-manifold that is the total space of a fibration
$f: X\rightarrow \Sigma _b$
whose regular fiber is a
$2$
-manifold
$\Sigma _a$
with genus
$a\geq 0$
. Assume that this fibration has a section
$s:\Sigma _b \rightarrow X$
whose image
$s(\Sigma _b)$
has self-intersection equal to d in X. Next let t and
$\delta $
be nonnegative integers. By symplectically resolving the double points of the union of
$s(\Sigma _b)$
and t copies of the fiber
$\Sigma _a$
, we obtain a symplectic submanifold
$\Sigma _{ta+b}$
in X with genus
$ta+b$
and self-intersection
$2t+d$
. By symplectically blowing up
$\delta $
points of
$\Sigma _{ta+b}$
in X, we obtain a genus
$ta+b$
symplectic submanifold
$\Sigma _{ta+b}^{\prime }$
in the blow-up
$X\#\delta \overline {\mathbb {CP}}{}^2$
with self-intersection
$2t+d-\delta $
.
Let
$E(r)_K$
and
$F_K^m$
be as in Section 2. Let
$E(X)_{K,m,r}^{t,\delta }$
denote the symplectic normal sum (cf. [Reference Gompf12, Reference McCarthy and Wolfson17]) of
$X\#\delta \overline {\mathbb {CP}}{}^2$
and
$E(r)_K$
along symplectic submanifolds
$\Sigma _{ta+b}^{\prime }$
and
$F_K^m$
:
$$ \begin{align*} E(X)_{K,m,r}^{t,\delta}=\big[(X\#\delta\overline{\mathbb{CP}}{}^2)\backslash\nu(\Sigma_{ta+b}^{\prime})\big]\cup\big[E(r)_K\backslash\nu(F_K^m)\big]. \end{align*} $$
For this symplectic normal sum to be well-defined, we require the genera of submanifolds to be equal and their self-intersections to have opposite signs, i.e.,
$$ \begin{align} ta+b=g(K)+m \quad \mathrm{and} \quad 2t+d-\delta=-(2m-r). \end{align} $$
Theorem 3.1 Assume that both conditions in (3.1) hold. Then
$E(X)_{K,m,r}^{t,\delta }$
is a closed symplectic
$4$
-manifold with
$$ \begin{align*} e(E(X)_{K,m,r}^{t,\delta})&=e(X)+\delta+12r+4ta+4b-4, \\ \sigma(E(X)_{K,m,r}^{t,\delta})&=\sigma(X)-\delta-8r, \\ \chi_h(E(X)_{K,m,r}^{t,\delta})&= \chi_h(X)+r+ta+b-1. \end{align*} $$
If
$\delta>0$
, then
$E(X)_{K,m,r}^{t,\delta }$
is nonspin. If
$t>0$
, then
$E(X)_{K,m,r}^{t,\delta }$
is simply connected. If
$t>0$
and
$r\geq 2$
, then
$E(X)_{K,m,r}^{t,\delta }$
contains two disjoint symplectic tori,
$T_1$
and
$T_2,$
of self-intersection
$0$
such that
$\pi _1(E(X)_{K,m,r}^{t,\delta } \backslash (T_1\cup T_2))=1$
.
Proof We compute that
$e(E(X)_{K,m,r}^{t,\delta })=e(X\#\delta \overline {\mathbb {CP}}{}^2)+e(E(r)_K)-2e(\Sigma _{ta+b}^{\prime })$
and
$\sigma (E(X)_{K,m,r}^{t,\delta })=\sigma (X\#\delta \overline {\mathbb {CP}}{}^2)+\sigma (E(r)_K)$
. When
$\delta>0$
, we have a punctured
$2$
-sphere in the
$[(X\#\delta \overline {\mathbb {CP}}{}^2)\backslash \nu (\Sigma _{ta+b}^{\prime })]$
half coming from an exceptional divisor of a blow-up. We can glue this disk to a punctured torus fiber in the
$[E(r)_K\backslash \nu (F_K^m)]$
half and obtain a torus with self-intersection
$-1$
, which implies that the intersection form of
$E(X)_{K,m,r}^{t,\delta }$
is not even.
Since we know from Lemma 2.3 that
$$ \begin{align} \pi_1(E(r)_{K} \backslash \nu(F_K^m))= 1, \end{align} $$
the Seifert–Van Kampen theorem implies that
$$ \begin{align} \pi_1(E(X)_{K,m,r}^{t,\delta}) \cong \frac{\pi_1((X\#\delta\overline{\mathbb{CP}}{}^2)\backslash \nu (\Sigma_{ta+b}^{\prime}))}{\langle\pi_1(\partial\nu(\Sigma_{ta+b}^{\prime}))\rangle}, \end{align} $$
where
$\partial \nu (\Sigma _{ta+b}^{\prime })$
is the boundary of
$\nu (\Sigma _{ta+b}^{\prime })$
and
$\langle \pi _1(\partial \nu (\Sigma _{ta+b}^{\prime }))\rangle $
is the normal subgroup of
$\pi _1((X\#\delta \overline {\mathbb {CP}}{}^2)\backslash \nu (\Sigma _{ta+b}^{\prime }))$
generated by the image of
$\pi _1(\partial \nu (\Sigma _{ta+b}^{\prime }))$
under the inclusion induced homomorphism.
Note that
$\partial \nu (\Sigma _{ta+b}^{\prime })$
is a circle bundle over
$\Sigma _{ta+b}^{\prime }$
with Euler number
$2t+d-\delta $
. It is well known (cf. [Reference Fomenko and Matveev10, Proposition 10.4]) that
$$ \begin{align*} \pi_1(\partial\nu(\Sigma_{ta+b}^{\prime})) = \Big\langle \alpha_i, \beta_i, \mu \bigm| {\textstyle \prod\limits_{i=1}^{ta+b}}[\alpha_i,\beta_i]=\mu^{2t+d-\delta},\, \alpha_i \mu \alpha_i^{-1} = \mu,\, \beta_i\mu\beta_i^{-1} = \mu \Big\rangle, \end{align*} $$
where the index i ranges over
$1,\dots ,ta+b$
. Here,
$\mu $
is represented by a fiber circle that is a meridian of
$\Sigma _{ta+b}^{\prime }$
, and
$\alpha _i,\beta _i$
are the parallel push-offs of the standard generators of
$\pi _1(\Sigma _{ta+b}^{\prime })$
.
In (3.3), we have
$\mu =1$
in the quotient group, since
$\mu \in \langle \pi _1(\partial \nu (\Sigma _{ta+b}^{\prime }))\rangle $
. Thus, we can write
$$ \begin{align} \pi_1(E(X)_{K,m,r}^{t,\delta}) \cong \frac{\pi_1(X\#\delta\overline{\mathbb{CP}}{}^2)}{\langle\pi_1(\Sigma_{ta+b}^{\prime})\rangle} \cong \frac{\pi_1(X)}{\langle\pi_1(\Sigma_{ta+b})\rangle}. \end{align} $$
From the long exact sequence of the fibration, we have an exact sequence
$$ \begin{align*} \pi_1(\Sigma_a) \longrightarrow \pi_1(X) \longrightarrow \pi_1(\Sigma_b) \longrightarrow 1, \end{align*} $$
where the first and second arrows are induced by the inclusion of a regular fiber and the fibration map, respectively. When
$t>0$
, the image of
$\pi _1(\Sigma _{ta+b})$
in
$\pi _1(X)$
contains all the generators of the images of
$\pi _1(\Sigma _a)$
and
$\pi _1(s(\Sigma _b))$
under the inclusion induced homomorphisms. Thus, we can conclude that the quotient group (3.4) is trivial.
When
$r\geq 2$
, Lemma 2.3 tells us that the
$[E(r)_K\backslash \nu (F_K^m)]$
half contains (at least) two disjoint symplectic tori
$T_1$
and
$T_2$
of self-intersection
$0$
such that
$$ \begin{align} \pi_1([E(r)_K\backslash\nu(F_K^m)]\backslash (T_1\cup T_2))=1. \end{align} $$
To show that
$\pi _1(E(X)_{K,m,r}^{t,\delta } \backslash (T_1\cup T_2))=1$
, we can apply the above argument to show that
$\pi _1(E(X)_{K,m,r}^{t,\delta })=1$
with the only change being the replacement of (3.2) with (3.5). ▪
Next we apply Theorem 3.1 to the
$BCD$
surface
$S(n)$
from Lemma 2.1, now viewed as a symplectic
$4$
-manifold.
Corollary 3.2 For any positive integer
$n\geq 5$
such that
$n\equiv \pm 1\pmod {6}$
and any fibered knot
$K\subset S^3$
of genus
$3(n-1)/2$
, there is a simply connected irreducible symplectic
$4$
-manifold
$M(n)_K$
that is homeomorphic to
$$ \begin{align} \bigg(\frac{7}{6}n^2-2n+\frac{11}{6}\bigg)\mathbb{CP}^2 \#\bigg(\frac{5}{6}n^2-2n+\frac{79}{6}\bigg)\overline{\mathbb{CP}}{}^2. \end{align} $$
Proof An integer n is coprime with
$6$
if and only if
$n\equiv \pm 1\pmod {6}$
. We let
$M(n)_K=E(X)_{K,m,r}^{t,\delta }$
with
$X=S(n)$
,
$a=n-1$
,
$b=(n-1)/2$
,
$d=-1$
,
$t=1$
,
$\delta =0$
,
$g(K)=3(n-1)/2$
,
$m=0$
, and
$r=1$
. We can easily check that both conditions in (3.1) are satisfied. We note that
$M(n)_K$
is nonspin, since it contains three curves of square
$-1$
in the
$[(X\#\delta \overline {\mathbb {CP}}{}^2)\backslash \nu (\Sigma _{ta+b}^{\prime })]$
half by Lemma 2.1.
Since
$e(M(n)_K)=2n^2-4n+17$
and
$\sigma (M(n)_K)=(n^2-34)/3$
, Freedman’s classification theorem (cf. [Reference Freedman11]) implies that
$M(n)_K$
must be homeomorphic to (3.6). Since
$S(n)$
is minimal, the symplectic normal sum
$M(n)_K$
is also minimal by Usher’s theorem in [Reference Usher19]. We recall from [Reference Hamilton and Kotschick14, Reference Kotschick15] that any simply connected minimal symplectic
$4$
-manifold is irreducible. ▪
We note that
$M(n)_K$
has positive signature except when
$n=5$
. For many values of n, Corollary 3.2 gives a new symplectic (and thus exotic) smooth structure on (3.6). For example, when
$n=7,11,13,17$
, we get an exotic smooth structure on each of
$45\mathbb {CP}^2\#40\overline {\mathbb {CP}}{}^2$
,
$121\mathbb {CP}^2\#92\overline {\mathbb {CP}}{}^2$
,
$173\mathbb {CP}^2\#128\overline {\mathbb {CP}}{}^2,$
and
$305\mathbb {CP}^2\#220\overline {\mathbb {CP}}{}^2$
. These
$4$
-manifolds have signature equal to
$5$
,
$29$
,
$45,$
and
$85$
, and
$\chi _h$
equal to
$23$
,
$61$
,
$87,$
and
$153$
, respectively. For comparison, we showed in [Reference Akhmedov, Hughes and Park1, Table 2] that
$\lambda (5)\leq 47$
,
$\lambda (29)\leq 87$
,
$\lambda (45)\leq 85$
and
$\lambda (85)\leq 166$
. Thus these exotic smooth structures are new solutions to the symplectic geography problem when
$n=7,11,17$
as far as we know.
Similarly, we can apply Theorem 3.1 to the surface bundle Y from Lemma 2.2 and obtain the following corollary.
Corollary 3.3 For any fibered knot
$K\subset S^3$
of genus
$8$
, there is a simply connected irreducible symplectic
$4$
-manifold
$Z_K$
that is homeomorphic to
$79\mathbb {CP}^2\#72\overline {\mathbb {CP}}{}^2$
.
Proof We let
$Z_K=E(X)_{K,m,r}^{t,\delta }$
with
$X=Y$
,
$a=7$
,
$b=5$
,
$d=-8$
,
$t=1$
,
$\delta =1$
,
$g(K)\,{=}\,8$
,
$m=4$
, and
$r=1$
. We have
$e(Z_K)=153$
and
$\sigma (Z_K)=7$
. The rest of the proof is similar to that of Corollary 3.2 and is left to the reader. ▪
In [Reference Akhmedov, Hughes and Park1], we showed that
$\lambda (7)\leq 49$
. Since
$\chi _h(Z_K)=40$
, the symplectic smooth structure in Corollary 3.3 is new.
Remark 3.4 It is well known (cf. [Reference Burde7]) that for a fixed genus
$g>1$
, there are infinitely many genus g fibered (and nonfibered) knots that are distinguished by their Alexander polynomials. By varying the knot K while fixing the genus
$g(K)$
, we expect the resulting collection of
$M(n)_K$
’s and
$Z_K$
’s to provide infinitely many distinct smooth structures on (3.6) and
$79\mathbb {CP}^2\#72\overline {\mathbb {CP}}{}^2$
. At present, it is not clear to us how to compute the Seiberg–Witten invariants of these
$4$
-manifolds completely so as to distinguish their smooth structures.
We end this section by constructing another family of simply connected irreducible symplectic
$4$
-manifolds.
Corollary 3.5 For any positive integer
$n\geq 5$
such that
$n\equiv \pm 1\pmod {6}$
and any fibered knot
$K\subset S^3$
of genus
$\frac {3}{2}(n-1)-1$
, there is a simply connected irreducible symplectic
$4$
-manifold
$X(n)_K$
that is homeomorphic to
$$ \begin{align} \bigg(\frac{7}{6}n^2-2n+\frac{23}{6}\bigg)\mathbb{CP}^2 \#\bigg(\frac{5}{6}n^2-2n+\frac{145}{6}\bigg)\overline{\mathbb{CP}}{}^2. \end{align} $$
Moreover, each
$X(n)_K$
contains two disjoint symplectic tori,
$T_1$
and
$T_2,$
of self-intersection
$0$
such that
$\pi _1(X(n)_K\backslash (T_1\cup T_2))=1$
.
Proof We let
$X(n)_K=E(X)_{K,m,r}^{t,\delta }$
with
$X=S(n)$
,
$a=n-1$
,
$b=(n-1)/2$
,
$d=-1$
,
$t=1$
,
$\delta =1$
,
$g(K)=\frac {3}{2}(n-1)-1$
,
$m=1$
, and
$r=2$
. We have
$e(X(n)_K)=2n^2-4n+30$
and
$\sigma (X(n)_K)=\frac {1}{3}(n^2-61)$
. The rest of the proof is similar to the proof of Corollary 3.2 and is left to the reader. ▪
We note that the signature of
$X(n)_K$
is positive when
$n\geq 11$
. By performing knot surgeries along
$T_1$
(and/or
$T_2$
) on
$X(n)_K$
, we can obtain infinitely many distinct smooth structures on (3.7).
Corollary 3.6 For any positive integer
$n\geq 5$
such that
$n\equiv \pm 1\pmod {6}$
, the
$4$
-manifold (3.7) in Corollary 3.5 has
$\infty ^2$
-property (cf. Definition 1.2).
Proof This follows immediately from [Reference Akhmedov, Hughes and Park1, Theorem 16]. ▪
For example, when
$n=17$
, we obtain infinitely many exotic smooth structures on
$307\mathbb {CP}^2\#231\overline {\mathbb {CP}}{}^2$
, which have signature equal to
$76$
and
$\chi _h$
equal to
$154$
. For comparison, we only showed in [Reference Akhmedov, Hughes and Park1] that
$\lambda (76)\leq 167$
, so these exotic smooth structures are new (cf. Remark 4.5).
4 Upper Bounds on
$\lambda (\sigma )$
The goal of this section is to exhibit concrete formulae for upper bounds on
$\lambda (\sigma )$
that are valid for any nonnegative integer
$\sigma $
. First, we recall the following theorem, which was proved in [Reference Akhmedov, Hughes and Park1, Corollary 17].
Theorem 4.1 Let X be a closed, simply connected, nonspin, minimal, symplectic
$4$
-manifold with
$b_2^+(X)>1$
and
$\sigma (X)\geq 0$
. Assume that X contains disjoint symplectic tori
$T_1$
and
$T_2$
of self-intersection
$0$
such that
$\pi _1(X\backslash (T_1 \cup T_2))=1$
. Suppose
$\sigma $
is a fixed integer satisfying
$0\leq \sigma \leq \sigma (X)$
. If
$\lceil x\rceil =\min \{k\in \mathbb {Z} \mid k\geq x\}$
and if we define
$$ \begin{align*} \ell(\sigma)=\Big\lceil \frac{\sigma(X)-\sigma}{8} -1 \Big\rceil , \end{align*} $$
then
$$ \begin{align*} \lambda(\sigma)\leq \chi_h(X) + \ell(\sigma) + 1. \end{align*} $$
Now we apply Theorem 4.1 to the
$4$
-manifolds
$Q(u)$
in Lemma 2.4 and obtain the following corollary.
Corollary 4.2 If
$\lambda (\sigma )$
is as in Definition 1.1, then we have
$$ \begin{align} \lambda(\sigma)< \frac{43}{25}\sigma+\frac{6813}{100} =1.72\sigma+68.13. \end{align} $$
Proof Given a nonnegative integer
$\sigma $
, let u be the smallest positive integer such that
$\sigma \leq \sigma (Q(u))=26u-2\lceil u/2\rceil -2$
. It follows that when
$u>1$
,
$$ \begin{align} \sigma(Q(u))-\sigma < \sigma(Q(u))-\sigma(Q(u-1)) = 26 -2\big(\lceil u/2\rceil -\lceil (u-1)/2\rceil\big) \leq 26, \end{align} $$
since
$\lceil u/2\rceil -\lceil (u-1)/2\rceil $
is either
$0$
or
$1$
depending on whether u is even or odd. Note that
$\sigma (Q(1))=22$
so that we still have
$\sigma (Q(u))-\sigma < 26$
even when
$u=1$
. Thus, we have
$\ell (\sigma )<(\sigma (Q(u))-\sigma )/8 < 13/4$
. Since
$$ \begin{align*} \sigma>\sigma(Q(u))-26=26u-2\lceil u/2\rceil -28 \end{align*} $$
by (4.2) and
$2\lceil u/2\rceil $
is u or
$u+1$
depending on whether u is even or odd, we conclude that
$\sigma>26u-(u+1)-28=25u-29$
. Thus, we have
$u<(\sigma +29)/25$
and
$$ \begin{align*} \lambda(\sigma)&\leq \chi_h(Q(u)) + \ell(\sigma) + 1 <27u+32\Big\lceil \frac{u}{2}\Big\rceil -2+\frac{13}{4}+1\\ &\leq 43u+\frac{73}{4} <\frac{43}{25}\sigma+\frac{6813}{100}, \end{align*} $$
since
$32\lceil u/2\rceil $
is
$16u$
or
$16u+16$
depending on whether u is even or odd. ▪
Remark 4.3 For a specific value of
$\sigma $
, (4.1) may not provide the optimal bound procured from
$Q(u)$
. For example, when
$\sigma =76$
, we can apply Theorem 4.1 to
$Q(4)$
directly and obtain
$\lambda (76)\leq 173$
, which is better than the bound
$\lambda (76)\leq 198$
coming from (4.1).
Similarly, we apply Theorem 4.1 to the
$4$
-manifolds
$\tilde {Q}(u)$
in Lemma 2.4 and obtain the following corollary.
Corollary 4.4 If
$\lambda (\sigma )$
is as in Definition 1.1, then we have
$$ \begin{align} \lambda(\sigma)< \frac{8}{5}\sigma +\frac{417}{20}\sqrt{\sigma +4}+\frac{1353}{20} =1.6\sigma +20.85\sqrt{\sigma +4}+67.65. \end{align} $$
Proof Given a nonnegative integer
$\sigma $
, let u be the smallest positive integer such that
$\sigma \leq \sigma (\tilde {Q}(u))= 25u^2+u-2\lceil u/2\rceil -2$
. Note that
$$ \begin{align*} \sigma(\tilde{Q}(u))-\sigma(\tilde{Q}(u-1)) =50u-24-2\big(\lceil u/2\rceil -\lceil (u-1)/2\rceil\big) \leq 50u-24. \end{align*} $$
It follows that
$$ \begin{align} \sigma(\tilde{Q}(u))-\sigma < 50u-24. \end{align} $$
Note that
$\sigma (\tilde {Q}(1))=22$
so that (4.4) still holds when
$u=1$
. Thus, we have
$\ell (\sigma )<(\sigma (\tilde {Q}(u))-\sigma )/8 < \frac {25}{4}u-3$
. Since
$\lceil u/2\rceil \leq (u+1)/2$
, we get
$$ \begin{align} \lambda(\sigma)&\leq \chi_h(\tilde{Q}(u))+\ell(\sigma)+1\nonumber \\ &< 25u^2+\lceil u/2\rceil(30u+2) +2u -2+\frac{25}{4}u-2\nonumber \\ &\leq 25u^2+(u+1)(15u+1)+\frac{33}{4}u-4 =40u^2+\frac{97}{4}u-3. \end{align} $$
From (4.4), we also obtain
$$ \begin{align*} \sigma &>\sigma(\tilde{Q}(u))-50u+24= 25u^2 -2\lceil u/2\rceil -49u+22\\ &\geq 25u^2 -(u+1) -49u+22 =25u^2-50u+21. \end{align*} $$
Thus, we must have
$u<1+\frac {1}{5}\sqrt {\sigma +4}$
, and plugging this into (4.5), we obtain (4.3). ▪
We observe that (4.1) is a better (i.e., lower) upper bound than (4.3) when
$\sigma \leq 30185$
and (4.3) is better than (4.1) when
$\sigma \geq 30186$
.
Remark 4.5 If we apply Theorem 4.1 to our
$4$
-manifolds
$X(n)_K$
in Corollary 3.5 with
$n\geq 11$
and argue as in the proof of Corollary 4.4, then we can deduce an upper bound
$$ \begin{align*} \lambda(\sigma)< \frac{7}{4}\sigma +4\sqrt{3\sigma+61}+45, \end{align*} $$
which is always worse than (4.1). However, we note that it is still possible to get a new and better upper bound for
$\lambda (\sigma )$
from
$X(n)_K$
for individual
$\sigma $
. For example, by applying Theorem 4.1 to
$X(17)_K$
, we obtain
$$ \begin{align} \lambda(75)\leq 155 \text{ and } \lambda(76)\leq 154, \end{align} $$
which are better than the bounds
$\lambda (75)\leq 197$
and
$\lambda (76)\leq 198$
coming from (4.1). The upper bounds in (4.6) are also better than the bound
$\lambda (\sigma )\leq 173$
for
$\sigma =75,76$
that is obtained by applying Theorem 4.1 to
$Q(4)$
(cf. Remark 4.3), and the bound
$\lambda (\sigma )\leq 167$
for
$\sigma =75,76$
in [Reference Akhmedov, Hughes and Park1, Table 2], which was obtained by applying Theorem 4.1 to
$\tilde {Q}(2)$
.
We finish our paper by observing that (4.1) does not give the least known upper bound on
$\lambda (\sigma )$
for very low values of
$\sigma $
. For example, (4.1) gives
$\lambda (0)\leq 68$
, whereas we already know from [Reference Akhmedov and Sakallı5] that
$\lambda (0)\leq 12$
. We still hope that (4.1) and (4.3) provide baselines of comparison for future research.
Acknowledgment
The authors thank F. Catanese for valuable e-mail exchanges regarding the
$BCD$
surfaces and S. Sakallı for her interest in this work.
