数理解析研究所講究録 1233 巻 2001 年 102-110

102

Surgery along a projective plane in a 4-manifold
and -singularity$D_{4}$

山田裕一 (Yuichi YAMADA)
電気通信大数学教室(University of Electro-Communications, Math. Dept. )(2001 年 5/7 から 6/29 まで京大数理研に滞在長期研究員)

1Introduction
The Price surgery has been defined in [P, KSTY, Y3] as acut and paste of a4-manifoldin the 4-sphereand also in general 4-manifolds, whereis defined as atotal space of anon-0rient-bundle over aprojective
plane with normal Euler number 2(see [Ml, M2, LI, Yl]). It may be
expected to make afake pair of 4-manifolds, which means apair that arehomotopy equivalent but non-diffeomorphic to each other, but such atrial
seems not to be succeeded yet except the non-0rient
example [Al, A2](see also [KSTY]: Gluck surgery ([G1]) is realized by Price surgery).
The 4-manifold
is represented by the framed link (see [Ki, GoS]) in
Figure $1(1)$ . The boundaryis homeomorphic to the quaternion space, which is the quotient space of the unit sphereof the quaternion
field
by the quaternion group of order 8. This spaceis alsohomeomorphic to the linking 3-manifold of -singularity: $S^{5}\cap\{f^{-1}(0)\}$ ,where$S^{4}$$N_{2}$$N_{2}$$D^{2}$$N_{2}$$\partial N_{2}$$S^{3}$$Q$$\mathrm{H}\cong \mathrm{R}^{4}$$Q$$D_{4}$f:$\mathrm{C}^{3}$(x, y, z)$arrow$$\mapsto$$\mathrm{C}$$x^{2}+y^{3}+z^{3}$,and we regard
as the unit sphere (the boundary of the unit disk)in
. Throughout the paper, by the notation
we denote thecompact 4-manifold obtained ffom $D^{6}\cap\{f^{-1}(0)\}$ by resolve the singularity
minimally, which is represented by the ffamed link in Figure $1(2)$ .
$S^{5}$$D^{6}$$\mathrm{C}^{3}\cong \mathrm{R}^{6}$$D_{4}$

Mathematics Subject Classification: Primary, Secondary
Title of the author’s talk on 31 May was slightly different from that of this report$0_{2000}$$57\mathrm{Q}45$$57\mathrm{R}65,57\mathrm{N}13$

.

103

Figure 1(1) :Figure 1(2) :$N_{2}$The boundaries of the two 4-manifolds
andeach other and also to Q, thus we can define$N_{2}$

Operation :“Cut$D_{4}$

off and paste$N_{2}$$D_{4}$$D_{4}$

are homeomorphic toon” a4-manif0ld,

but the resulting 4-manifold is not well-defined because of the ambiguityof the gluing map (self-homeomorphisms on ). Thus, for agiveninan original 4-manifold $M$ , we study the set(consisting of at mostthree elements, see Section 2) of diffeomorphic class of resulting 4-manif0lds.
This operation changes some topological invariants of the ambient 4manifold: it decreases the Euler characteristic number by 4, the negative
second Betti number
by 4 and do not change the positive second Betti, thus increases the signature aof the 4-manifold by 4.
numberIn this paper, we will report two lemmas related to the operation. One
is Lemma 3.1 in \S 3, which says that acertain operation consisting of four
blowing up’s and the operation above is reduced to Price surgery. The otheris Lemma 4.1 in \S 4 on the resulting manifolds of the operation on the simple
elliptic surfaces. Before stating the results, in the next section, we will recall
some facts on Price surgery in general 4-manifolds. In \S 5, we will show somekey lemmas by “relative Kirby calculus” (see Section 5.5 in) but wedo not give the complete proof.
The author would like to express his thanks to the people at ResearchInstitute for Mathematical Sciences, Kyoto University for their hospitality
during his stay for two months$Q$$D_{4}$$\Omega_{M}(D_{4})$$\chi$$\beta_{2}^{-}$$\beta_{2}^{+}$$[\mathrm{G}\mathrm{o}\mathrm{S}]$$)$$arrow a\vee)\mathrm{f}\mathrm{f}1_{\mathrm{X}}^{*}\emptyset \mathrm{S}$$\rceil j$$\vee\supset\delta[searrow] lf[] \mathrm{f}1999\not\in\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\square \ovalbox{\tt\small REJECT} 1^{\iota}\mathrm{J}\mathrm{T}\mathit{0})ffl^{2}j_{\mathrm{L}}^{\mathrm{B}}\ovalbox{\tt\small REJECT}_{\mathrm{I}}^{\mathrm{A}}\lceil\backslash /\backslash 6^{\backslash }6^{\gamma_{X}}7\mathrm{J}\overline{\tau}\supset^{\backslash }\backslash$$-\vee \mathrm{C}^{\backslash \backslash }\emptyset\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\Phi\emptyset \mathrm{F}$$hf_{arrow\vec{-}}^{\vee}k\mathfrak{P}\mathrm{b}f_{0}^{-}.\ovalbox{\tt\small REJECT}$$\mathrm{f}^{\supset i}\mathrm{F}\not\simeq\nearrow\ovalbox{\tt\small REJECT} \mathrm{I}5\mathit{0})\backslash J\backslash \backslash \ovalbox{\tt\small REJECT} \mathrm{g}\Re$$1_{\vee}\ovalbox{\tt\small REJECT}$$T\circ$$\ovalbox{\tt\small REJECT}\%$$\mathfrak{h}$$’+^{\mathrm{o}}\backslash \mathrm{D}\sqrt[\backslash ]{}-\not\simeq$$\gamma.arrow$,$\Lambda\sim\overline{\cup}\urcorner 0$$\mathrm{f}\not\equiv\ovalbox{\tt\small REJECT},\Xi_{\backslash \backslash }\rfloor\pi\Phi\ovalbox{\tt\small REJECT}_{\mathrm{B}}\ovalbox{\tt\small REJECT}_{\mathrm{B}}^{\not\in \mathfrak{o}}\ovalbox{\tt\small REJECT} \mathrm{A}\mathrm{f}1$,$l_{arrow}^{-}*\mathrm{J}^{*}/$$\ovalbox{\tt\small REJECT} \mathrm{j}\backslash \gamma_{j}$$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}_{\mathfrak{o}}7k$$\llcorner^{\vee}C$$\mathit{0})\ovalbox{\tt\small REJECT}\grave{\mathit{1}}\hat{\mathrm{R}}’\uparrow k\#\simarrow\#\mathrm{f}4\gamma \mathrm{p}\mathrm{A}\delta[searrow] 0:?\mathrm{J}\mathrm{X}\backslash 1^{-}\mathrm{f}\mathrm{f}\mathrm{i}\emptyset\not\in \mathrm{a}\mathrm{e}x\iotaarrow.\Gamma\epsilon\frac{\Rightarrow}{\#_{\backslash }\vee}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{i}$$t_{arrow}^{arrow}\vee\supset\backslash _{J^{1}}T\ovalbox{\tt\small REJECT}\grave{\mathrm{x}}.T\iota_{\mathit{1}^{1}}f_{arrow}^{-}f^{\backslash ^{\backslash }}.’$\delta^{\grave{\grave{\mathrm{Y}}}}k\grave{\prime)}-\vee^{\backslash }\mathrm{g}^{\backslash ^{\backslash }}\backslash \iota,\backslash \ovalbox{\tt\small REJECT}$.

$|_{\vee}f_{0}^{arrow}$

3\ovalbox{\tt\small REJECT} \mathrm{b}$$f_{0}^{-}.-\vee\emptyset\ovalbox{\tt\small REJECT}\not\geq$$\mathrm{k}^{\mathrm{Y}}\mathrm{t}_{\mathrm{B}}^{4\mathrm{h}}\mathfrak{h}$$\llcorner^{\vee}\tau,\pi_{\backslash }--\Re_{\mathrm{f}}^{\tau}\mathrm{u}^{-}\mathrm{X}$

104
d)

Price surgery

Ve recall notations and facts on Price surgery from [KSTY, Y3].
(1) We denote by
the total space of anon-0rientablebundle overaprojective plane with normal Euler number 2, which is acompact
oriented 4-manifold with aboundary, and which is described by the
Kirby diagram in Figure 1(1). Note thathas ahandlebody decomposition with one 0-handle, one 1-handle and one 2-handle.
$D^{2}$$N_{2}$$N_{2}$(2) The boundary
is diffeomorphic to the quaternion space , which
admits aSeifert fibered structure whose Seifert invariants in the senseof\S 5.2 are given by $\{-1; (0_{1},0);(2,1), (2,1), (2,1)\}$ . We call thethree singular fibers, , .
$\partial N_{2}$$[\mathrm{O},$

$Q$

$]$

$c_{-1}$$c_{0}$$c_{1}$(3) In [P], Price has investigated the self-diffeomorphisms of the quaternion space and has shown that the mapping class group(thegroup of isotopy classes of orientation preserving self-diffeomorphisms)
is isomorphic to , the symmetric group on three letters {-1, 0, 1}.
For each element in @3, there is aself-diffeomorphismof which$Q$$\mathrm{M}(Q)$$\mathrm{S}_{3}$$\sigma$$f_{\sigma}$

preserves the Seifert fibered structure and satisfies
map
represents the class of.

$Q$$f_{\sigma}(c_{i})=c_{\sigma(i)}$. Each$\mathrm{M}(Q)$$f_{\sigma}$(4) Price has also shown that there is aself-diffeomorphism $g(g_{1}$ in ,p.116]) of
whose order is two inand that can extend
over
as aself-diffeomorphism. (In fact, is abundle isomorphism“-,,: $N_{2}arrow N_{2}$ which maps each vector to) Thus, for agiven
oriented 4-manifold $E$ whose boundary, we have at most only
three 4-manifolds up to diffeomorphism
obtained by gluing
to $E$ along the boundary, where we use the compositions of a
fixed orientation reversing map fromtoand an orientation
preserving self-diffeomorphism on as the gluing map. The three 4manifolds correspond to the classes of in the right coset $\mathrm{M}(Q)/\{1, g\}$ ,which consists of three elements$[\mathrm{P}$$Q=\partial N_{2}$$\mathrm{M}(Q)$$N_{2}$$g$$\vec{v}$$-\vec{v}.$$\mathrm{i}\mathrm{s}-Q$$E \bigcup_{i\mathrm{o}\varphi}N_{2}$$N_{2}$$i$$\varphi$$\partial N_{2}$$Q$$\varphi$$\partial E$

105
3Equivalence of two operations
Let $M$ be aclosed oriented 4-manifold and $K$ asmoothly embedded 2-sphere
in $M$ whose normal bundle is trivial. We define two operations Aand on$M$ along $K$ .
$\mathrm{B}$Operation :Taking apairwise connected sum of $(M, K)$ with the (positive) standard projective plane(see [PR], [LI], [Y1]), we have anembedded projective plane $(M, K\# P\circ)$ in $M$ whose normal Euler number 2.
The tubular neighborhood $N(K\# P_{0})$ is diffeomorphic to. Let
be the set of diffeomorphic class of 4-manifolds obtained by pasting
to the$M$exterior
along the boundary. The original manifold
itselfand the Gluck surgery
of $M$ along $K$ , by Theorem 4.1 in [KSTY],. By (4) in Section 2, $\Pi_{M}(K\# P_{0})$ consists
are contained in the setof at most three elements.
$\mathrm{A}$$(S^{4}, P_{0})$$N_{2}$$\Gamma \mathrm{I}_{M}(K\# P_{0})$$N_{2}$$M\backslash \mathrm{i}\mathrm{n}\mathrm{t}N(K\beta P_{0})$$\Sigma_{M}(K)$$\square _{M}(K\# P_{0})$Operation :This operation consists of five steps, see Figure 2: (1) Blow
up at apoint in K. (2) Blow up at the intersection point of the proper
lift of $K$ and the exceptional curve. (3) Blow up at apoint on the newest
exceptional curves. (4) Blow up at apoint on the newest exceptional curves
again. After this step, we have ain the ambient 4-manifold.
(5) Do the operation “Cutoff and pasteon” the 4-manifold By$\Omega_{M}(D_{4}(K))$ , we denote the set of the of diffeomorphic class of the resulting
4-manif0lds.
$\mathrm{B}$$M\# 4\overline{\mathrm{C}P^{2}}$$D_{4}$$D_{4}$$N_{2}$$01\mathrm{C}|\prec$$|..’.\cdot\circ 0$

Figure 2
Lemma 3.1 Two operations Aand along $K$ on $M$ are equivalent, $i.e.$ ,it holds that $\Pi_{M}(K\# P_{0})=\Omega_{M}(D_{4}(K))$ as sets$\mathrm{B}$106
4Operation on elliptic surfaces
Let $E(n)$ be the simply connected elliptic surface (with section) whose Euler. $E(n)$ is the fiber sum of copies
characteristic is $12n$ , (of $E(1)$ . $E(2)\cong$ “the $K3$ surface”, .. .). In [BGo], using amethod “relative), adecomposition of $E(n)$ as a
Kirby diagram” (see Section 5.5 inisunion of $n+1$ pieces $N_{n}\cup W_{n}\cup W_{n-1}\cup\cdots W_{1}$ has been shown, whereis the -plumbing. Each $Wj(j\geq 2)$ is athe nuclei of $E(n)$ ([Go]) andcobordism represented by the relative Kirby diagram in Figure 3(modified
from Figure 27 in [BGo] , which clearly contains one -plumbing. An. Thus we can do the operation “Cutoffplumbing contains an obviouson” $E(n)$ at most times. To study the resulting 4-manif0lds,
and paste, see Lemma 5.2.
. For
we do the operation on$E(1)\cong \mathrm{C}P^{2}\# 9\overline{\mathrm{C}P^{2}}$$n$$[\mathrm{G}\mathrm{o}\mathrm{S}]$$N_{n}$$E_{8}$$W_{1}$$E_{8}$$)$$E_{8^{-}}$$D_{4}$$D_{4}$$N_{2}$$n$$W_{j}$$W_{1}$

off andLemma 4.1 The resulting 4-manifold of the operation “Cutand ispasteon” $W_{j}(j\geq 2)$ does not depend on the gluing map of, wherediffeomorphic tois the 4-manifold represented by the
relative Kirby diagram in Figure 4.
$D_{4}$$\partial N_{2}$$N_{2}$$\gamma \mathrm{V}j\# 4\overline{\mathrm{C}P^{2}}$Figure 3:$W_{j}$$\mathrm{W}_{j}$Figure 4:$\mathrm{W}_{j}$is also acobordism ffom the Seifert homology 3to $\Sigma(2,3,6j-1)$ for $j\geq 2$ . We conjecture that all
the resulting 4-manifolds, the (non-trivial) union of possible Wj’s and Wj’s
capped by
and their “logarithmic transformation” (as a4-manifold, not.
are all diffeomorphic to
as acomplex surface) in
Note that$W_{j}$

is, thus$\gamma\eta_{j}$$\mathrm{s}\mathrm{p}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}-\Sigma(2,3,6(j-1)-1)$$N_{n}$$N_{n}$$\beta_{2}^{+}(\mathrm{C}P^{2})\#\beta_{2}^{-}(\overline{\mathrm{C}P^{2}})$

5Key of the proof
We show some key lemmas for Lemma 3.1 and give aproof of Lemma 4.1.
They are shown by (ordinary) Kirby calculus and relative Kirby calculus).
(see Section 5.5 in$[\mathrm{G}\mathrm{o}\mathrm{S}]$107
Lemma 5.1 See the Kirby calculus from the diagram (A) to (B) of 3manifolds in Figure 5. It corresponds to ahomeomorphism $\varphi kom$ theboundary. Calculating the curvesoftowith 0-framing in(A) during the process of the Kirby calculus, we get the curves ’s with. Thus (under some conditions)framings ( ) in (B). They are’s inthe local change $kom(A)$ to (B) in aKirby diagram of a4-manifold $M$
corresponds to (one of) the operation “Cutoff and pasteon” $M$ .
$\partial D_{4}$$D_{4}$$c_{i}’ s$$\partial N_{2}$$c_{i}$$\varphi(c_{i})$$\cdot$$\partial N_{2}$$D_{4}$$N_{2}$$c_{-1}$Figure 5

Of course, another Kirby calculus from the diagram (A) to (B) corresponds to another homeomorphism from. To prove Lemma 4.1
tocompletely, we need every (six or three) calculus from the diagram (A) to(B) for each element of the mapping class groupof order six, but inthis paper, we omit the other calculus.
Now we use Lemma 5.1 to study the resulting 4-manifold of the operation.
“Cut
off and pasteon” the obvious
in the -plumbing$\partial D_{4}$$\partial N_{2}$$\mathrm{M}(Q)$$D_{4}$$N_{2}$$D_{4}$$E_{8}$$W_{1}$, whereLemma 5.2 The resulting 4-manifold is diffeomorphic tois the 4-manifold represented by the final Kirby diagram (-l-framed
left-hand trefoil) in Figure 6.
$\mathrm{W}_{1}\# 3\overline{\mathrm{C}P^{2}}$$\mathrm{W}_{1}$108
Proof. See the Kirby calculus in Figure 6.

$\square$$\overline{\mathrm{O}}\overline{\mathrm{c}}\backslash (\hat{y_{\vee}\sim\overline{\mathrm{o}}^{2}}\cap(_{J}\backslash ^{\gamma}\backslash \mathrm{c}^{\overline{\gamma}^{2}}\overline{\mathrm{O}}^{\mathit{2}}22^{-}2- 2-\mathit{2}$Figure 6

Lemma 4.1 is shown by application of such method.
Note that the action ofonis obvious. Thus we can calculate every resulting 4-manifold of the operation onin the -plumbingfor each choice of the gluing map in. For asmoothly embedded 2sphere $K$ in , we can also study the resulting 4-manifold of the operation
cut theand paste anof aprojective planein( $N_{2}\cong-X(P_{0})$ , see [PR, ,instead of, L2, Yl, Y2]) by the method”“circle with adot and with asymbol $K$ in Kirby diagrams introduced inAppendix of [KSTY]. They are all diffeomorphic to. Note that
the Gluck surgery
along any $K$ in.
satisfies that$\mathrm{M}(Q)(\cong@_{3})$$\partial D_{4}$$D_{4}$$E_{8}$$\mathrm{M}(Q)$$S^{4}$$D_{4}$$S^{4}$$P_{0}\# K$$\mathrm{e}\mathrm{x}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{o}\mathrm{r}-X(P\circ\# K)$$N_{2}$$\mathrm{P}$$\mathrm{L}\mathrm{I}$$\mathrm{W}_{1}\# 3\overline{\mathrm{C}P^{2}}$$S^{4}$$\Sigma(K)$$\Sigma(K)\#\overline{\mathrm{C}P^{2}}\cong\overline{\mathrm{C}P^{2}}$Outline of the proof of Lemma S. :See the Kirby calculus in Figure 2
again. It describes the process of operationnear the 2-sphere $K$ , but
we have not done the final step yet. Doing the change in Lemma 5.1 tothe final diagram, we finish the operationand get the first diagram in$\mathrm{I}$$\mathrm{B}$$\mathrm{B}$Figure 7(The dotted circle corresponds to ameridian to $K$ in M. The thincircle corresponds to the boundary of a -core of the 2-handle . Once
ignore them). The diagram describes a4-manifold obtained by attaching a. All we have to do is to verify that
2-handle to, where we use the notation
for the piece since we switch
the core and the -core. See Figure $4(1)$ and the proof of Theorem 4.1 in[KSTY] for the goal$h$$\mathrm{c}\mathrm{o}$$h$$N_{2}$$(M\backslash \mathrm{i}\mathrm{n}\mathrm{t}N(K))\cup h^{[perp]}\cong$$h^{[perp]}$$M\backslash \mathrm{i}\mathrm{n}\mathrm{t}N(K\# P_{0})$$\mathrm{c}\mathrm{o}$

$h$

109

Figure 7
By the calculus in Figure 7, we have the attachinig circle ofinand the ffaming: it is the thin circle in the diagram. (If one careorientation of the diagram, it would be better take the mirror image.) We
have the lemma. Cl$h^{[perp]}$$\partial(M\backslash \mathrm{i}\mathrm{n}\mathrm{t}N(K))\cong$$S^{1}\cross S^{2}$

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YAMADA Yuichi
yyyamada@matha. e–one. uec.ac.jp
Dept. of Systems Engineering
The Univ. of ElectrO-Communications
1-5-1,Chofugaoka, Chofu,
Tokyo, 182-8585, JAPA