13

数理解析研究所講究録
第2004巻 2016年 13-22

A family of the Seiberg‐Witten equations and configurations ofembedded surfaces in 4‐manifolds
Hokuto Konno

Graduate School of Mathematical Sciences, the University of Tokyo

Abstract
In this paper we consider constraints on configurations consisting of finitely many surfaces em‐

bedded in an oriented closed 4‐manifold and its genera. A study of a family of the Seiberg‐Witten

equations, namely, the (high‐dimensional wall crossing phenomenon plays a prominent role in our
method. The results in § 2 are based on [3].

The1adjunction inequalities and configurations of embedded

surfaces with positive intersection numbers
It is a fundamental problem in 4‐dimensional topology to find a lower bound for the
genus of an embedded surface which represents

dimensional manifold.

This

problemagiven second homology class of a 4‐

is often called the minimal genus

problem.

For

example, the minimal genus problem for \mathb {C}\mathb {P}^{2} is called the Thom conjecture and this isoneof the most classical problem in 4‐dimensional topology.

Gauge theory provides strong tools to answer the minimal genus problem and a certain
type of inequality for genus obtained by gauge theory is often called the adjunction in‐equality. Here we explain this terminology. Let X be a complex surface and C a smooth
algebraic curve in X.

Then it is easy to see that the Euler characteristic$\chi$(C)=2-2g(C)satisfies the equality- $\chi$(C)=c_{1}(X)\cdot C+C^{2},
where

means

the intersection number and C^{2}=C\cdot C

adJunction formula.

.

This

equality is called theWhen X is a C^{\infty}4 ‐manifold and a surface $\Sigma$ is embedded to X in

C^{\infty} sense, then, in general, we cannot determine g( $\Sigma$) by the homology class [ $\Sigma$]asurprising observation in Kronheimer‐Mrowka[4].

However,is that, for a suitable characteristicc\in H^{2}(X;\mathbb{Z}) one can expect the inequality,- $\chi$( $\Sigma$)\geq|c\cdot[ $\Sigma$]|+[ $\Sigma$]^{2}.
This type of inequality is called the adjunction inequality.

14

After Seiberg‐Witten theory appeared, it is successfully used to study the minimal genus
Kronheimer‐Mrowka

problem.

Witten equations.

[5] proved the Thom conjecture by using the Seiberg‐

They gave the wall crossing formula for the Seiberg‐Witten invariants

for 4‐manifolds with b^{+}=1 and use this formula for the proof of the Thom conjecture.
Hereb^{+}(X) is the maximal dimension of positive definite subspaces of H^{2}(X;\mathbb{R}) withrespect to the intersection form of X.
The direct consequence of arguments in Kronheimer‐Mrowka [5] is that the strong re‐lation between the

Seiberg‐Witten invariants and the adjunction inequalities.

For

an

oriented, closed smooth 4‐manifold X with b^{+}(X)\geq 2 and a spin \mathrm{c} structure \mathfrak{s} on X let,\mathrm{S}\mathrm{W}_{X}(\mathfrak{s})\in \mathbb{Z} denote the Seiberg‐Witten invariant of X with respect to \mathfrak{s} (More precisely,
we have to fix a homology orientation of X to determine thesign of \mathrm{S}\mathrm{W}_{X}(\mathfrak{s}) Here a.

.

homology orientation of X means an orientation of H^{0}(X;\mathbb{R})\oplus H^{1}(X;\mathbb{R})\oplus H^{+}(X;\mathbb{R}) ,where

H^{+}(X;\mathbb{R}) is a b^{+} ‐dimensional positive definite subspace of H^{2}(X;\mathbb{R}paper,

weIn thisconsider only surfaces which are oriented, closed and connected. Put$\chi$^{-}( $\Sigma$):=\displaystyle \max\{- $\chi$( $\Sigma$), 0\}for a surface $\Sigma$.
Theorem 1.1.

(Kronheimer‐Mrowka [5]) Let X be an oriented, closed smooth 4‐manifoldwith b^{+}(X)\geq 2 and $\Sigma$ be a surface embedded in X with [ $\Sigma$]^{2}\geq 0
with \mathrm{S}\mathrm{W}_{X}(\mathfrak{s})\neq 0.

.

Lets be a spin c structureThen, the inequality$\chi$^{-}( $\Sigma$)\geq|c_{1}(\mathfrak{s})\cdot[ $\Sigma$]|+[ $\Sigma$]^{2}holds.

However, there are many 4‐manifolds whose Seiberg‐Witten invariants vanish.

For ex‐ample, the Seiberg‐Witten invariants for 4‐manifolds obtained by connected sum vanish
under mildb^{+}(X_{i})\geq 1
weassumptions on b^{+} : let X_{i}(i=1,2) be oriented, closed 4‐manifolds with,then\mathrm{S}\mathrm{W}_{X_{1}\# X_{2} (\mathfrak{s})=0 for any spin \mathrm{c} structure \mathfrak{s} on X_{1}\# X_{2}.

Thereforecannot use the Seiberg‐Witten invariant to show the adjunction inequalities for such

4‐manifolds.

Infact, Nouh [7] proved that the adjunction inequality for a surface in\mathb {C}\mathb {P}^{2}\#\mathb {C}\mathb {P}^{2} does not hold in general. Nouhs result shows that, for such 4‐manifolds, notonly does one cannot use the Seiberg‐Witten invariants, but also one may find examples
of surfaces which violate the adjunction inequalities.
Thus a natural question is when one can show the adjunction inequality for 4‐manifolds

whose Seiberg‐Witten invariants vanish.

In Strles paper[12], he showed the following

adjunction inequalities for disjoint embedded surfaces with positive self‐intersection num‐bers. Let \mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(X) denote the signature of X.

15

Theorem 1.2.

(Strle [12]) Let X be an oriented closed smooth 4‐manifold with b_{1}(X)=0and c\in H^{2}(X;\mathbb{Z}) be a characteristic with c^{2}>\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(X)(A) In the case of b^{+}(X)=1$\alpha$\in H_{2}(X;\mathbb{Z}) be a homology class with $\alpha$^{2}>0 and

let

,

.

$\Sigma$\subset X be an embedded surface with [ $\Sigma$]= $\alpha$.

Then the inequality- $\chi$( $\Sigma$)\geq-|c\cdot $\alpha$|+$\alpha$^{2}(1)

holds.

(B) In the case of b^{+}(X)>1,$\alpha$_{i}^{2}>0(1\leq i\leq b^{+}) and $\Sigma$_{1}
Assume that $\Sigma$_{1}

,

.

.

.

,

$\alpha$_{b+}\in H_{2}(X;\mathbb{Z}) be homology classes with$\Sigma$_{b+}\subset X be embedded surfaces with [$\Sigma$_{i}]=$\alpha$_{i}.

let $\alpha$_{1} ,
,

.

.

.

,

.

.

.

,

$\Sigma$_{b+} are disjoint. Then the inequality- $\chi$($\Sigma$_{i})\geq-|c\cdot$\alpha$_{i}|+$\alpha$_{i}^{2}(2)holds at least one i\in\{1, . . . , b^{+}\}.
Note that Strles theorem can be applied to 4‐manifolds whose Seiberg‐Witten invariants
vanish.

His result suggests that

one

can

expect some constraints on configurations ofembedded surfaces in a 4‐manifold even when its Seiberg‐Witten invariant vanishes.
In the rest of this paper, we will explain two constraints on configurations of embedded

surfaces with self‐intersection number zero.

Our constraints can be also

applied to 4‐

manifolds whose Seiberg‐Witten invariants vanish. While Strles proof stands on a studyof the moduli space of the

Seiberg‐Witten equations on a 4‐manifold with cylindricalends, our method is to study only compact 4‐manifolds and use the high‐dimensional
wallcrossing phenomena.

In Seiberg‐Witten theory, the wall crossing phenomena

usually studied in the case when b^{+}=1
b^{+}

.

are

Li‐Liu[6] gave its generalizations for any

While in the usual wall crossing phenomena a

1‐parameter family of the Seiberg‐.

Witten equations is the main object, in Li‐Liu[6] s situationab^{+} ‐parameter family is

it. We call Li‐Liu [6] s generalizations the high‐dimensional wall crossing phenomena. To
use

the high‐dimensional wall crossing phenomena for constraints on configurations, in[3] the author gave a sufficient condition on a certain b^{+} ‐parameter family to catch thehigh‐dimensional wall crossing phenomenon in terms of embedded surfaces.
foundation of the proof of the results in this paper.

This is the16

Figure 1: An example of a quadrilateral including the origin of \mathbb{R}^{2}Constraints on configurations obtained by the high‐dimensional

2

wall crossing phenomena
In this section, we explain a special case of the result obtained by the high‐dimensional

wall crossing phenomena, namely, the adjunction inequalities for configurations of surfaces

2\mathbb{C}\mathbb{P}^{2}\# n(-\mathbb{C}\mathbb{P}^{2})

in

our

results,.

we canFor a generalization of this result, see [3].

give

a

(From the general form ofsimple alternative proof of Strles results: Thoerem 1.2.) We

will often use the identification H^{2}\simeq H_{2}obtained by Poincaré duality.

Let consider the 4‐manifold

X=2\mathbb{C}\mathbb{P}^{2}\# n(-\mathbb{C}\mathbb{P}^{2})=(\#_{p=1}^{2}\mathbb{C}\mathbb{P}_{p}^{2})\#(\#_{q=1}^{n}(-\mathbb{C}\mathbb{P}_{q}^{2}) (n>0).

H_{p} denote a generator of H_{2}(\mathb {C}\mathb {P}_{p}^{2};\mathb {Z}) and E_{q} a generator of H_{2}(-\mathbb{C}\mathbb{P}_{q}^{2};\mathbb{Z}) For acohomology class c\in H^{2}(X;\mathbb{Z}) and homology classes $\alpha$_{1}$\alpha$_{4}\in H_{2}(X;\mathbb{Z}) we define a
Let

.

,

.

.

.

,

,

line L_{i}(i=1, \ldots, 4) in \mathbb{R}^{2} by

L_{i} :=\{(x_{1}, x_{2})\in \mathbb{R}^{2}|(x_{1}H_{1}+x_{2}H_{2})\cdot$\alpha$_{i}=c\cdot$\alpha$_{i}\}For these lines, we will consider the condition that (parts of) lines L_{1}of a quadrilateral
sense.
can

Let

Lí,

.

.

.

,

(3)

.

,

by this order. Here we use the word (quadrilateral

L_{4}'

be four line segments in \mathbb{R}^{2}

.

.

.

.

,

L_{4} form sides

in the following

If an orientation of Lí is given,wedefine the initial point I(L_{i}') and the terminal point T(Lí) of Lí. We call the ordered(Lí,
L_{4}' ) a quadrilateral when there exists an orientation for each Lí such that
T(Lí) =I(L_{i+1}') holds for each i\in \mathbb{Z}/4 (We admit a point as a line segment. Thus atriangle also a quadrilateral in our definition.)
set

.

.

.

,

.

Theorem 2.1. For the 4‐manifold

X=2\mathbb{C}\mathbb{P}^{2}\# n(-\mathbb{C}\mathbb{P}^{2}) (n>0),

17

let c\in H^{2}(X;\mathbb{Z}) beacharacteristic with c^{2}>\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(X) and $\alpha$_{1}homology classes with $\alpha$_{i}^{2}=0(i=1, \ldots, 4)with [$\Sigma$_{i}]=$\alpha$_{i}.

.

Let $\Sigma$_{1} ,

.

.

.

,

.

,

.

.

,

$\alpha$_{4}\in H_{2}(X;\mathbb{Z}) be$\Sigma$_{4}\subset X be embedded surfaces

Assume that $\alpha$_{i} and c satisfy the following (A) and $\Sigma$_{i} satisfy (B) :(A) The lines L_{1}

,

.

.

.

,

L_{4} form sides of a quadrilateral including the origin of \mathbb{R}^{2} by thisorder.

(B) $\Sigma$_{i}\cap$\Sigma$_{i+1}=\emptyset(i\in \mathbb{Z}/4).

Then, the inequality- $\chi$($\Sigma$_{i})\geq|c\cdot$\alpha$_{i}|holds for at least one i\in\{1

.

,

.

.

,

4 \}.

Example 2.2. Let X=2\mathbb{C}\mathbb{P}^{2}\# 19(-\mathbb{C}\mathbb{P}^{2}),

c=H_{1}-3H_{2}-\displaystyle \sum_{q=1}^{19}E_{q} The homology classes.

$\alpha$_{1} :=3H_{1}+3H_{2}-\displaystyle \sum_{q=1}^{3}E_{q}+\sum_{q=4}^{10}E_{q}+2(E_{11}+E_{12})$\alpha$_{2} :=-3H_{1}+2H_{2}+\displaystyle \sum_{q=1}^{3}E_{q}+\sum_{q=13}^{18}E_{q}+2E_{19},$\alpha$_{3}:=H_{1}+H_{2}+E_{12}-E_{13},$\alpha$_{4} :=2H_{1}-H_{2}-\displaystyle \sum_{q=1}^{3}E_{q}-E_{13}+E_{14}satisfy that $\alpha$_{i}^{2}=0 and $\alpha$_{i}\cdot$\alpha$_{i+1}=0(i\in \mathbb{Z}/4).

It is easy to check that these $\alpha$_{i} and csatisfy (A) in Theorem 2.1.

(See Figure 2.)
Thus, by Theorem 2.1, for embedded surfaces $\Sigma$_{i} satisfying [$\Sigma$_{i}]=$\alpha$_{i} if they also satisfy,

that $\Sigma$_{i}\cap$\Sigma$_{i+1}=\emptyset(i\in \mathbb{Z}/4),- $\chi$($\Sigma$_{i})\geq|c\cdot$\alpha$_{i}|holds for at least one i\in\{1 ,

.

.

.

,

4 \}

.

This means that the genus boundg($\Sigma$_{i})\geq 2
holds for at least one i\in\{1 ,

.

.

.

,

4 \}.

Under certain assumptions on geometric intersections with embedded surfaces violating
the adjunction inequalities, we can derive the adjunction inequality for a single surface.
Before giving an example, we mention an easy method to make surfaces with small genera.
For a homology classcurvesC\subset \mathbb{C}\mathbb{P}_{2}^{2}$\beta$=aH_{2}+\displaystyle \sum_{q=1}^{n}b_{q}E_{q}\in H_{2}(\mathbb{C}\mathbb{P}_{2}^{2}\# n(-\mathbb{C}\mathbb{P}^{2});\mathbb{Z}) considering algebraic
C_{q}\subset \mathb {C}\mathb {P}_{q}^{2} and reversing orientations of them if need,
,

and

we

we

can

easily construct the surface S\subset \mathbb{C}\mathbb{P}_{2}^{2}\# n(-\mathbb{C}\mathbb{P}^{2}) by

S:=C\#(\#_{q=1}^{n}C_{q})

(4)

18

Figure 2: L_{1}

,

.

.

.

,

L_{4} in Example 2.2

satisfing[S]= $\beta$, g(S)=\displaystyle \frac{(|a|-1)(|a|-2)}{2}+\sum_{q=1}^{n}\frac{(|b_{q}|-1)(|b_{q}|-2)}{2}.

For example, on the characteristic

H_{2}-\displaystyle \sum_{q=1}^{n}E_{q} such naive construction is sufficient to,give many examples of surfaces which violate the adjunction inequality on this character‐istic.

Example 2.3. Let give natural numbers d_{1}\geq 4, d_{2}\geq 1, d_{3}\geq 2 and n\displaystyle \geq d_{1}^{2}+\max\{d_{2}^{2}, d_{3}^{2}\}.
For

X=2\mathbb{C}\mathbb{P}^{2}\# n(-\mathbb{C}\mathbb{P}^{2}) let consider the homology classes,$\alpha$:=d_{1}H_{1}-\displaystyle\sum_{q=1}^{d_{1}^{2}E_{q},$\beta$_{1}:=d_{2}H_{2}+\displayst le\sum_{q=d_{1}^{2}+1}^{d_1}^{2}+d_{2}^{2}E_{q},$\beta$_{2}:=d_{3}H_{2}-\displayst le\sum_{q=d_{1}^{2}+1}^{d_1}^{2}+d_{3}^{2}E_{q}.
Let S_{i}\subset \mathbb{C}\mathbb{P}_{2}^{2}\# n(-\mathbb{C}\mathbb{P}^{2})\backslash ( disk) \subset X be surfaces with [S_{i}]=$\beta$_{i} obtained as (4). For anembedded surface $\Sigma$\subset X satisfying [ $\Sigma$]= $\alpha$ and$\Sigma$\cap S_{i}=\emptyset(i=1,2)g( $\Sigma$)\displaystyle \geq\frac{(d_{1}-1)(d_{1}-2)}{2},

we canshow that(5)from Theorem 2.1.

By the adjunction formula for \mathbb{C}\mathbb{P}_{1}^{2}\# n(-\mathbb{C}\mathbb{P}^{2})
by

a

,

the homology class $\alpha$ can be represented

surface $\Sigma$ of genus (d_{1}-1)(d_{1}-2)/2 satisfying $\Sigma$\cap S_{i}=\emptyset

is the optimal bound under the condition $\Sigma$\cap S_{i}=\emptyset.

.

Thus the inequality (5)193Constraints on configurations obtained by the high‐dimensional
wall crossing phenomena and the gluing technique
To obtain the results in §nomena2, the author used the high‐dimensional wall crossing phe‐in [3]. On the other hand, Ruberman ([8],[9] and [10]) studied the combinationof the usual (\mathrm{i}.\mathrm{e}. b^{+}=1) wall crossing phenomena and the gluing technique. The gluing

technique is a deep analytical tool in gauge theory. A typical application of the gluing
technique is the proof the blowup formula, which describes the behavior of the Seiberg‐
Witten invariants (or Donaldson invariants) under blowups of 4‐manifolds. Rubermansarguments can be regarded as a 1‐parameter version of the proof of the blowup formula.

Namely, Ruberman considered the gluing argument as the proof of the blowup formula forthe 1‐parameter family to use the wall crossing argument. This argument can be gener‐

alized to higher‐dimensional families to use the high‐dimensional wall crossing argument.
This generalization gives new constraints on configurations. In this section, we give theformulation of these results.
To describe the results, foraspin c4‐manifold, we introduce an abstract simplicial

complex which consists of surfaces violating the adjunction inequalities. Before the defi‐
nition of this simplicial complex, we need an (ambient

simplicial complex

simplicial complex. This ambient

introduced to the author by Mikio Furuta.

was(Furuta) For an oriented, closed 4‐manifold X
simplicial complex \mathcal{K}=\mathcal{K}(X) as follows:
Definition 3.1.

,

wedefine the abstract

The set of vertices V(\mathcal{K}) is given as the set of smooth embeddings of surfaces with

self‐intersection number zero:V(\mathcal{K}):=\{ $\Sigma$\mapsto X|[ $\Sigma$]^{2}=0\}.
Here we consider only oriented, closed, connected surfaces. We denote each vertex( $\Sigma$\leftarrow+X)\in V(\mathcal{K}) briefly by $\Sigma$.
For k\geq 1 ,acollection of (k+1) vertices $\Sigma$_{0} ,and only if $\Sigma$_{0}

,

.

.

.

,

.

.

.

,

$\Sigma$_{k}\in V(\mathcal{K}) spans a k ‐simplex if$\Sigma$_{k} are disjoint.

We call \mathcal{K} the complex of surfaces of X.

Of course, any abstract simplicial complex is a CW complex, thus \mathcal{K} is a CW complex

although \mathcal{K} is a huge space. Wetopologize \mathcal{K} as a CW complex, i.e., by the weak topology.
Remark 3.2. The complex of surfaces is a 4‐dimensional analog of the complex of curves
due to Harvey

surfaces,we[2] in 2‐dimensional topology.

In the above definition of the complex of

do not consider the isotopy classes of embeddings of surfaces. On the other20

hand, in the same way of the definition of the complex of curves, one can define an ab‐
stract simplicial complex whose vertices are the isotopy classes of embeddings of surfaces

and whose simplices are spanned by collections of such isotopy classes which can be real‐
ized disjointly.

However, to give certain applications to the adjunction inequalities using

Seiberg‐Witten theory, the first definition of the complex of surfaces might be appropriate.
Here we consider the special phenomena in 4‐dimensional topology, namely, the adjunc‐tion inequalities.

Definition 3.3. Let \mathfrak{s} be a spin \mathrm{c} structure on X.

Then, the complex of surfaces violating

the adjunction inequality \mathcal{K}_{V}=\mathcal{K}_{V}(X, \mathfrak{s}) is the subcomplex of \mathcal{K}(X) defined as the set ofvertices is given by

V(\mathcal{K}_{V}):=\{ $\Sigma$\in V(\mathcal{K})|$\chi$^{-}( $\Sigma$)<|c_{1}(\mathfrak{s})\cdot[ $\Sigma$]|\}and having the induced structure of an abstract$\Sigma$_{0}

.

,

.

.

,

$\Sigma$_{k}\in V(\mathcal{K}_{V})simplicial complex from \mathcal{K}spans a k ‐simplex if and only if $\Sigma$_{0} ,

.

.

.

,

.

Namely,$\Sigma$_{k} span a k ‐simplex in \mathcal{K}.

Let \mathrm{s}^{+} be a spin \mathrm{c} structure on \mathb {C}\mathb {P}^{2} such that c_{1}(\mathfrak{s}^{+}) is a generator in H_{2}(\mathbb{C}\mathbb{P}^{2};\mathbb{Z}) and\mathfrak{s}^{-}

be a spin \mathrm{c} structure on -\mathbb{C}\mathbb{P}^{2} such that c_{1}(\mathfrak{s}^{-}) is a generator in H_{2}(-\mathbb{C}\mathbb{P}^{2};\mathbb{Z}).

(We

have two choices of each \mathfrak{s}^{+} and \mathfrak{s}The main result in this section is the following statement.
Theorem 3.4. Let(X, \mathrm{s}_{X}) be an oriented, closed spin c4 ‐manifold with b^{+}(X)\geq 2.

Suppose that \mathrm{S}\mathrm{W}_{X}(\mathfrak{s})\neq 0 and d(\mathfrak{s}_{X})=0 , where d(\mathrm{s}) :=(c_{1}(\mathfrak{s})^{2}-2 $\chi$(X)-3\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(X))/4.
Put

Z:=X\# m\mathbb{C}\mathbb{P}^{2}\# n(-\mathbb{C}\mathbb{P}^{2}) (m\geq 1, n\geq 4m)\mathfrak{s}_{Z}:=\mathfrak{s}_{X}\#(\#_{p=1}^{m}\mathfrak{s}_{p}^{+})\#(\#_{q=1}^{n}\mathfrak{s}_{q}^{-})

,

.

Then

\tilde{H}_{m-1}(\mathcal{K}_{V}(Z, \mathfrak{s}_{Z});\mathbb{Z})\neq 0
holds.

Remark 3.5. More precisely, we can give a concrete non‐trivial element of\tilde{H}_{m-1}(\mathcal{K}_{V}(Z, \mathfrak{s}_{Z});\mathbb{Z}).

To prove Theorem 3.4, we define a group homomorphism

SW =\mathrm{S}\mathrm{W}_{Z,\mathfrak{s}_{Z} :\mathcal{H}_{*}(Z,\mathfrak{s}_{Z})\rightar ow \mathbb{Z}and show that this map is non‐tivial. Here \mathcal{H}_{*}(Z,\mathfrak{s}_{Z}) is a certain subgroup of \tilde{H}_{*}(\mathcal{K}_{V}(Z, \mathfrak{s}_{Z});\mathbb{Z})The map SW is defined, roughly speaking, by counting the parametrized moduli spaceof the Seiberg‐Witten equations for parameter space introduced in[3]. (To justify the.

21

Figure 3: Some boundings for $\gamma$

counting argument, we use Ruans virtual neighborhood technique and its family version.)
This parameter space is obtained by stretching neighborhoods of embedded surfaces whichforms the element of \mathcal{H}_{*}.

This construction of the parameter space is a slight generaliza‐tion of one due to \mathrm{F}\mathrm{r}\emptyset yshov [ 1].

The proof of the non‐triviality of this map is given by the

combination of the high‐dimensional wall crossing phenomena and the gluing technique.
Here we explain why a non‐trivial element of\tilde{H}_{*}(\mathcal{K}_{V}(Z, \mathfrak{s}_{Z});\mathbb{Z}) is useful to givestraints on configurations of embedded surfaces and its genera. For example, let $\Sigma$_{1} ,

.

.

con‐

.

,

$\Sigma$_{4}\inV(\mathcal{K}_{V}) be embedded surfaces with $\Sigma$_{i}\cap$\Sigma$_{i+1}=\emptyset(i\in \mathbb{Z}/4) Then the collection\{($\Sigma$_{i}, $\Sigma$_{i+1}\}\}_{i\in \mathrm{Z}/4} forms a 1‐cycle $\gamma$=\{$\Sigma$_{1}, $\Sigma$_{2}\}+\cdots+\langle$\Sigma$_{4}, $\Sigma$_{1}\rangle in \mathcal{K}_{V} Assume that [ $\gamma$]\neq 0
in H_{1}(\mathcal{K}_{V};\mathbb{Z}) For $\Sigma$\in V(\mathcal{K}) with $\Sigma$\cap$\Sigma$_{i}=\emptyset(i\in \mathbb{Z}/4) the collection \{\langle $\Sigma,\ \Sigma$_{i}, $\Sigma$_{i+1})\}_{i\in \mathbb{Z}/4}
can be regarded as the cone of this 1‐cycle. (See Figure 3.) If $\Sigma$\in V(\mathcal{K}_{V}) holds, thiscone is contained in \mathcal{K}_{V} thus [ $\gamma$]=0 in H_{1}(\mathcal{K}_{V};\mathbb{Z}) This contradicts our assumption,
therefore we have $\Sigma$\not\in V(\mathcal{K}_{V}) In conclusion, we have the adjunction inequality for anembedded surface $\Sigma$ with $\Sigma$\cap$\Sigma$_{i}=\emptyset(i\in \mathbb{Z}/4)
In the same way, for embedded surface $\Sigma$, $\Sigma$'\in V(\mathcal{K}) if $\Sigma$ and $\Sigma$' satisfies
.

.

.

,

.

,

.

.

,

$\Sigma$\cap$\Sigma$_{i}=\emptyset(i=1,2,3) , $\Sigma$'\cap$\Sigma$_{i}=\emptyset(i=1,3,4) , $\Sigma$\cap$\Sigma$'=\emptyset,then $\Sigma$\not\in V(\mathcal{K}_{V}) or $\Sigma$'\not\in V(\mathcal{K}_{V}) holds.

$\Sigma$'.

Namely, the adjuction inequality holds for $\Sigma$ or(See Figure 3.)

As in these example, if we find a non‐trivial element in H_{*}(\mathcal{K}_{V};\mathbb{Z})
on

,

we

obtain constraints

genera for infinitly many configurations of surfaces.

Acknowledgement. The author would like to express his deep gratitude to Mikio Furuta
for the numerous comments on this work. The author was supported by JSPS KAKENHI
Grant Number 16\mathrm{J}05569 and the Program for Leading Graduate Schools, MEXT, Japan.

22

References[1] Kim A. \mathrm{F}\mathrm{r}\emptyset yshov, An inequality for the h ‐invariant in instanton Floer theory, Topology 43 (2004), no. 2, 407‐432,
DOI 10. 1016/\mathrm{S}0040-9383(03)00049-1 MR2052970 (2005c:57043)[2] W. J. Harvey, Boundary structure of the modular group, Riemann surfaces and related topics: Proceedings of the 1978
Stony Brook Conference (State Univ. New York, Stony Brook, N Y., 1978), Ann. of Math Stud., vol. 97, Princeton
Univ.

Press, Princeton, N.J., 1981, pp. 245‐251. MR624817[3] H. Konno, Bounds on genus and configurations of embedded surfaces in 4‐manifolds, available at arXiv: 1507. 00139.
[4] P. B. Kronheimer and T. S. Mrowka, Embedded surfaces and the structure of Donaldsons polynomial invariants, \mathrm{J}
Differential Geom. 41 (1995), no. 3, 573‐734. MR1338483 (96e:570l9)[5]The genus of embedded surfaces in the projective
10. 4310/\mathrm{M}\mathrm{R}\mathrm{L}.1994.\mathrm{v}1 n6 a14. MR1306022 (96a:57073)—,

plane, Math. Res. Lett.

1(1994), no. 6, 797‐808, DOI[6] Tian‐Jun Li and Ai‐Ko Liu, Family Seiberg‐Witten invariants and wall crossing formulas, Comm. Anal. Geom. 9(2001), no. 4, 777‐823. MR1868921 (2002k:57074)[7] Mohamed Ait Nouh, The minimal genus problem in \mathb {C}\mathrm{P}^{2}\#\mathb {C}\mathrm{P}^{2} Algebr. Geom. Topol. 14 (2014), no. 2, 671‐686, DOI
10. 2140/\mathrm{a}\mathrm{g}\mathrm{t}.2014.14.671 MR3159966,.

[8] Daniel Ruberman, An obstruction to smooth isotopy in dimension 4, Math. Res. Lett. 5 (1998), no. 6, 743‐758, DOI
10. 4310/\mathrm{M}\mathrm{R}\mathrm{L}.1998 v5 n6 a5. MR1671187 (2000c:5706l)
[9]

—,

A

CA, 1998),

polynomial invar\cdot $\iota$ ant of diffeomorphisms of 4‐manifolds, Proceedings of the Kirbyfest (Berkeley,
Topol. Monogr., vol. 2, Geom Topol. Publ., Coventry, 1999, pp. 473‐488 (electronic), DOI

Geom.

10. 2140/\mathrm{g}\mathrm{t}\mathrm{m}.1999.2.473

[10]

—,

.

MR1734421

Positive scalar curvature,(2001b:57073)

diffeomorphisms and the Seiberg‐Witten invariants, Geom. Topol. 5 (2001), 895−924(electronic), DOI 10. 2140/\mathrm{g}\mathrm{t}.2001 5.895. MRIS74146 (2002k:57076)[11] Yongbin Ruan, Virtual neighborhoods and the monopole equations, Topics in symplectic 4‐manifolds (Irvine, CA, 1996),
First Int. Press Lect. Ser., I, Int. Press, Cambridge, MA, 1998, pp. 101‐116. MR1635698 (2000e:57054)[12] Sašo Strle, Bounds on genus and geometric ìntersections from cylindrical end moduli spaces, J. Differential Geom. 65(2003), no. 3, 469‐511. MR2064429 (2005c:57042)

Graduate School of Mathematical Sciences
The University of Tokyo
3‐8‐1 Komaba, Meguro, Tokyo 153‐8914

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hkonno@ms.u‐tokyo.ac.jp\displaystyle \ovalbox{\t \smal REJECT}\overline{5_{\backslash }^{\mathrm{J} \mathrm{X}\mapsto\mp\neq \mathrm{f}\mathrm{f}\mathrm{l}_{F\mathrm{L} ^{g\mathrm{t} 7_{-}^{\backslash }\backslash \mathrm{t} \bigwedge_{7}\ovalbox{\t \smal REJECT} \mathrm{i}\mathrm{t}_{-}\backslash \backslash +