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CLASSIFICATION OF THE SMOOTH
CLOSED MANIFOLDS UP TO BLOWUP

By

Grigory MIKHALKIN

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the Degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

Advisor. Professor S.Akbulut

1993

ABSTRACT

CLASSIFICATION OF THE SMOOTH CLOSED MANIFOLDS UP TO BLOWUP

By

Grigory MIKHALKIN

The dissertation is devoted to the problem of classification of the smooth mani—
folds. The dissertation contains the classification of the smooth manifolds up to the

blowups.

The boundary of the tubular neighbourhood of a submanifold in a manifold is a
sphere bundle over the submanifold. The tubular neighbourhood of a submanifold
in a manifold is a mapping cylinder of the projection of the sphere bundle given by
the boundary of the tubular neighbourhood of the submanifold. The blowup of a
manifold along its submanifold is the replacing of the tubular neighbourhood of the
submanifold by the mapping cylinder of the projectivisation of this sphere bundle.
In the other words we replace the submanifold by the space of linear directions in
the tangent space of the manifold normal to the submanifold. This operation is

well-known in algebraic geometry.

The main theorem of the dissertaion is that any two smooth closed connected

manifolds of the same dimension are equivalent up to blowups.

ACKNOWLEDGEMENTS

The author would like to thank S.Akbulut, V.Kharlamov, A.Marin and O.Viro for

their attention to the work and for the pointing out of the errors in previous versions.

The author would like to thank S.Akbulut and A.Marin for the useful comments
and discussions that led to the generalisation of the proof in 4-dimensional case to

the case of an arbitrary dimension.

iii

Contents

LIST OF FIGURES v
INTRODUCTION 1
1 The statement of the theorem 2

2 Each cobordism class contains at least one topologically rational

manifold 3
3 Cobordant manifolds are blowup equivalent 8
4 Summary 17

List of References 18

iv

List of Figures

1 RP” x RP? ................................ 5
2 Cancelling of pairs of the intersection points .............. 12
3 Blowup of the zero section ........................ 13
4 Proof of Lemma 5 ............................. 15

INTRODUCTION

J .Nash in [8] proved that every smooth closed manifold is diffeomorphic to a com-
ponent of a real algebraic variety and completed his paper with a conjecture that
every smooth closed connected manifold is diffeomorphic to a rational real algebraic
nonsingular variety. If a rational real algebraic nonsingular surface is orientable then
its genus has to be less then 2 (see e.g. [6]). So this ”algebraic” Nash conjecture is not
true in dimension 2. It was observed in [2] and [4] that there is a topological version
of the Nash conjecture: Every smooth closed connected manifold can be obtained from
S" by a sequence of blowings up and blowings down. The topological Nash conjecture
obviously holds for 2-n'ianifolds since blowing up a surface has an affect of connecting
summing with RPQ. In [2] and [4] the topological Nash conjecture was proven for

3-manifolds.

The dissertation contains a proof of the topological Nash conjecture for any di-

mension.

1 The statement of the theorem

Let M be a smooth manifold and L be a proper smooth submanifold of M. By l/M( L)
we denote the normal bundle of L in M. Let L be the projectivisation of l/M(L). In
other words L is the RPd‘mM‘di"‘L‘1-bundle over L associated to VM(L). Let VM(L)
be the tautological I -bundle over L defined by the projectivisation p : L —-> L. Then
we have a natural diffeomorfism u : VM(L) — L —> VM(L) — L which can be extended

to a map 7r : VM(L) ——> 1/M(L) by the identity ”IL 2 p. Note that we may View VM(L)

as a tubular neighbourhood of L in M.

Definition. Map f : 11:] —+ M is called the blowup of a smooth manifold M along
a smooth submanifold L c M if M is the result of gluing of Mai) to M — L with
u, the projection f is defined by f|M_L = id and fIVM(L) = 7r. Smooth manifold
M is called the result of blowup of M along L, we will also denote M by B(M,L).
Submanifold L C M is called the center of blowup f. Submanifold L C M is called
the exceptional divisor of f. We say that l7 C M is the proper transform of a closed
subset V c M, if m7) = v, l7 is closed in M, and t7 —— t is dense in V1.

Definition. The multiblowup is a sequence of blowups
M=Mk—i...—+M0=M

Manifold M is called the result of multiblowup of M.

Definition. Two manifolds M and N of the same dimension are called blowup
equivalent (or m-equivalent [4], or topologically birationally equivalent [2]), if there
exists a sequence M = M0,M1, . . .,Mn = N such that for anyj E {1, . . . ,n} either
M,- is the result of blowup of MJ-_1 along some submanifold of MJ-_1 or Mj_1 is

the result of blowup of Mj along some submanifold of Mj. The sequence M =

 

1If V is a submanifold transverse to L then we can define the proper transform much simpler, as
the union of V — L and the projectivisation of uv(L {'1 V)

2

M0, 11/11, . . . , A1,, = N is called the blowup sequence.

Definition. We say that a pairof manifolds (M, V) is the result of blowup of a pair of
manifolds (M, V) along L C M if M is the result of blowup of M along L and V is the
proper transform of V. Similarly, two pairs of manifolds (M, V) and (N, W) are called
blowup equivalent, if there exists a sequence (M, V) = (M0, V0), (M1, V1), . . . , (Mn, Vn)
: (N, W) such that for anyj E {1, . . . ,n} either (Mj,VJ~) is the result of blowup of
(MJ-_1,V}_1) along some submanifold of M,_1 or (Mj_1, VJ-_1) is the result of blowup
of (Mi, Vj) along some submanifold of Mj.

Definition. A smooth manifold M of dimension n is called topologically rational, if

M is blowup equivalent to the standard sphere 5".

Theorem 1 If!” and N are closed smooth connected manifolds of the same dimen-

sion then M is blowup equivalent to N

In other words every closed smooth connected manifold is topologically rational.

Theorem 1 follows from Lemma 1 and Lemma 3.

2 Each cobordism class contains at least one to-
pologically rational manifold

Lemma 1 (A.Marin) Every manifold [M is cobordant to a topologically rational

manifold

Proof. The cobordism group is generated by products of non-singular hypersurfaces
RHM of bidegree (1,1) in RP” x RP",p S q (see Exercise 16.F of [7]). Lemma 2
implies that we need only to show that RHM is topologically rational. It suffices to

prove that RHM is blowup equivalent to lRPl"1 X RP“.

The hypersurface RHM is given by the equation

$oyo+$1y1+-~+$pyp=0

where [.730 : : ivp] X [yo I : yq] are standard bihomogeneous coordinates of
RP” X RPQ. It is easy to see that RHM is non-singular. Let p : RHM — RS —->

RP”"1 x RP" be the map given by the equation

p([:1:0:...:a:,,]x[y0:...:yq])=[x0:...::rp_1]x[y0:...:yq]

where [are : : :rp_1] x [yo : : yq] are standard bihomogeneous coordinates of
RP”‘1 x RP" and R5 is the subvariety of RHM given by the system of equations
$0: =xp_1 =0 (RSzRHmflm : ...:0 : 1] XRP"). In [0:...:0 : 1] XRPq
manifold RS is given by equation yp = 0, thus R5 is a non-singular subvariety of
RHM diffeomorphic to RPq‘l. We can view p geometrically. Consider RP”‘1 X RP"
as the submanifold of RP” X RP" given by the equation 33,, = 0. Let L(x.y) be the
line in RP” X {y} through (1r,y) and ([0 : : 0 : 1],y). We define p(a:,y) as the

intersection point of L(a:.y) and RP”‘1 x RP".

Let f : RF —> Rap, be the blowup of RH” along as and as 6 RF be the
exceptional divisor of f. We want to find a smooth map f) : RF —-> RP”‘1 X RP"
such that p[RF_RS = p o f. By definition, the points of RS are lines tangent to RHp,q
passing through RS and normal to RS. Since R5 = RHM fl [0 : : 0 : 1] X RP" is
non-singular, these lines are transverse to [0 : : 0 : 1] X RP". For 2 6 RS passing
through ([1 : 0 : . . . : 0],y) we define 15(2) to be the intersection of RP”’1 X RP" with
the projection of 2 along [0 : . .. : 0 : 1] x RP” onto RP” X {y}. Evidently, p is a

smooth well-defined map.

We can define a partial inverse of p as a map q : RP”“1 X RP" — RT --) RF, where

 

[a mlallxk P“

 

 

Figure 1: RP” X RP”
RT is given by the equations

31;: = 0
$0310 + . . . + ftp—lyp—i = 0

in the following way. Let q(:r, y) be the intersection point Amy) of RH” and the line
L(r.y) in RP” X {y} connecting the points (2:,y) and ([0 : .. . : 0 : 1],y) if Amy) ¢ RS
(i.e. if yp 9t 0). Point Amy) is unique if L(x,y) ¢ RHM since RH,M is a hyperplane in
RP” X {y}. To define q(a:,y) in the case of Amy) E R3 (i.e. when yp = 0) we note
that if L(r.y) ¢ RHM then L(r.y) determines a. line l(x.y) in the quotient of tangent
plane to RHP,q at ([0 : :0: 1],y) by tangent plane to [0 : :0: 1] X RP” at
([0 : : 0 : 1],y). We put q(a:,y) = Q”) E RS C RF. Note that L(,_y) C RH” iff
([0 : : 0 : 1],y),($,y) 6 RH ,q, i.e. iff yp = 0 and xoyo + . . . + xp_1yp_1= 0, thus q
is a smooth well-defined map on RP”'l X RP” — RT. By construction q is the inverse

of f).

Let g : RG ——> RP”"1 x RP” be the blowup of RP”‘1 x RP” along RT and RT be

the exceptional divisor of g.
B(RHM, RS) = RF =RG = B(RP”‘l X RP”,RT)
fl \ [9
RHM D RHM — RS —p>RP”’1 x RP”

22 Q?

 

Now we want to find a diffeomorphism (j : RG —i RF extending q. Let qIRG_RT = qog.
By definition, the points of RT are lines tangent to RP”"1 X RP” passing through RT
and normal to RT. If such a line does not lie in {:r} X RP” and in RP”‘1 X {y} then
we can consider this line as a curve C of bidegree (1,1) in RP”’1 X RP”. Let D be the
surface generated in RP” x RP” by lines in RP” X {y} connecting ([0 : : 0 : 1],y)
and CO RP” x {y} for all y with C (1 RP” X {y} 7£ 0. We claim that D is a surface of
bidegree (1,1) in RP” x RP”, indeed, D is given in RP” X RP” by the same system
of equations as C in RP”‘1 X RP”. The intersection RHM H D is a curve of bidegree
(1,1) in D and, by construction, if (:13, y) 6 RT then L(x.y) C D. Therefore, RHMfl D
is reducible and consists of two intersecting lines. Define q(a:,y) as the intersection
point Amy) of these lines if A(x.y) ¢ RS and as the line [(2:41) (see the construction of
q) if Amy) 6 RS. Map 4 is a smooth injective map onto RF, therefore, RF and RG

are diffeomorphic (RC is compact) and RHM is blowup equivalent to RP”“1 X RP”

[1

Proposition 2.1 IfM is blowup equivalent to M’ and N is blowup equivalent to N’

then M X N is blowup equivalent to M’ X N’

Proof. It is easy to see that ifM = M0,M1,...,Mn = M’ and N = N0,Nl,...,Nm =
N’ are blowup sequences then M X N = M0 X N0,M1 X N0,...,Mn X N0,Mn x

N1, . . .,Mn X Nm = M’ X N’ is also a blowup sequence Cl

Lemma 2 The product oftwo topologically rational manifolds is topologically rational

Proof. Note that it is enough to prove that S1 X S" is m-equivalent to S"+1 for any
n 2 2 (the classification of surfaces implies that manifolds of dimension less then 3
are topologically rational). Once we prove this we get that S” X S” is m-equivalent to
S1 X S”‘1 X S” with the help of Proposition 2.1, by induction we get that S” X S” and
5”” are equivalent to S1 X S1 X . . . x S1 and, therefore, are equivalent to each other,
i.e. the product of S” and S” is topologically rational and Proposition 2.1 implies the

lemma.

Let us now show that S1 X S" is m-equivalent to 5"“. Consider the result X
of blowup of S"+1 along standard Sn'l. Manifold X is diffeomorphic to SIXS".
To see this we represent S"+1 as a join Sl * Sn‘l, then X consists of spheres {:13} at
S"—1 U {—:r} * Sn‘1,a: E S]. Thus we see that X is an Sn-bundle over RP1 m S1
diffeomorphic to the fiberwise join of the trivial Sn’2 bundle en_2 over S1 and the
non-trivial SO-bundle ,u over S1. By the same arguments, the result Y of blowup of X
along en_2 % S1 X S"‘2 is diffeomorphic to the fiberwise join of the trivial S"’3-bundle

over S1 X S1 and a non-trivial SO-bundle over S1 X S1.

Consider S1 x S” now. We can represent S1 X S“ as the fiberwise join of en_2
and the trivial SO bundle over 5‘. By the same arguments, the result of blowup of
S1 X S" along en_2 x S1 X S"“2 is diffeomorphic to the fiberwise join of the trivial

Sn‘3-bundle over S1 X S1 and a non-trivial SO-bundle over S1 x S1.

Note that all the non-trivial SO-bundle over S1 X S1 are isomorphic, every 5"-
bundle is determined by its first Stiefel-Whitney class and for any two non-zero ele-
ments 0, fl 6 H1(S1 X 3‘; Z2) there exists a self-diffeomorphism f : S1 X S1 —> S1 X S1
such that f*(fl) = 0. Thus, S1 X S" is blowup equivalent to S1 XS" and therefore to

571+] Cl

3 Cobordant manifolds are blowup equivalent

Lemma 3 If smooth connected closed manifolds M and N are cobordant then they

are blowup equivalent

Let W be a cobordism between M and N, then W admits a handlebody de-
composition. A handlebody decomposition of W determines a sequence of mani-
folds M = M0,M1,...,1Wn = N such that M,- is the result of surgery of My-“
j E {1, . . . ,n}. Let PJ-_1 E Mj_1 denote the surgery sphere (i.e. the boundary of the
core of the j-th handle) and let Q,- C MJ- denote the dual surgery sphere (i.e. the
boundary of the cocore of the j-th handle). Evidently we may assume that every M,-

is connected.

Proposition 3.1
a) Sphere Pj_1 is Z2-h0m010g0U8 to zero in M34 iff
dimH.(M.-; 22) 2 dimH.(M.--1;Z2)
b) Sphere Pin] is not Zg-homologous to zero in Mj_1 ifl

dlmH.(Mj,Z2) ‘2 dth.(Mj_1;Z2) — 2

Proof. Consider the exact sequences of pairs (Mj_1, PJ-_1) and (My, Qj)

.. —+ Hk(P,-_1;Z2) 2» _H,,(M,-_,;z2) —» Hk(M,-_1,P,_1;Zg) _.
. . —) Hk(Qj;Z2) A Hk(Mj;Z2) —) Hk(Mj,Pj;Z2) —> . . .

Hence

0 '—* Hk(Mj_1;Z2)/Zmlk -—) Hk(MJ'_1,Pj_1;Z2) —* kerik_1—> 0
0 -—i Hk(Mj;Z2)/imjk —+ Hk(Mj,QJ-;Z2) —> kerjk-1 —> 0

Note that Hk(Mj_1, Pj_1; Z2) = Hk(Mj,QJ-; Z2) and therefore

dimH.(MJ-; Z2) — dimH..(M,-_1; Z2) =
(dim(im(j.)) — dim(ker(j..))) — (dim(im(i.)) — dim(ker(i.)))

If Pj_1 is Zg-homologous to zero in MJ-_1 then dim(im(i.)) = 1,dim(ker(i,.)) = 1, if
not then dim(im(i.)) : 2,dim(ker(i,..)) = 0. If (21- is Zz-homologous to zero in Mj
then dim(im(j.)) = 1, dim(ker(j.)) = 1, if not then dim(im(j..)) 2: 2,dim(ker(j..)) =
0. Thus, it is sufficient to prove that either Pj_1 is Zg-homologous to zero in Mj_1

or Q,- is Zg-homologous to zero in M]- (or both). Consider the boundary 8U z

Sd‘mPJ-l‘1 X Sd‘mQJ'l of the tubular neighbourhood U of PJ-_1 in Mj_1. Note that

0U = 0(Mj_1 — U). Therefore, if k : 8U —> MJ-_1 — U is the inclusion map then
dim(lcer(k,,)) = $dimH.(8U;Z2) = 2

It follows that either Sd’m’DJ-l‘l X {pt} is Zg-homologous to zero in Adj--1 — U and
then Pj_1 is Zg-homologous to zero in Mj_1, or {pt} X S‘file‘l is Zg-homologous to

zero in MJ-_1 — U and then Q,- is Zg-homologous to zero in Mj Cl

Remark 3.2 (A.Marin) It is not always true that either Pj_1 is not Zg-homologous
to zero in MJ-_1 or Q,- is not Zg-homologous to zero in M], if dim.P,-_1 : dimQJ- then
Sd‘im’DJ-l‘1 x {pt} may be Zg-homologous to {pt} X S""""QJ‘1 in Mj_1 — U (consider,

for instance the standard (+1)—surgery of S3).

Definition. We say that the j-th handle of W is odd, if Pj_1 is Zg-homologous
to zero in MJ-_1 and the Zg—self-linking number of PJ-_1 equipped with the surgery
trivialization of the normal bundle in M j_1 is not equal to zero (evidently, dimW =

dimM + 1 = 2dimPJ-_1 + 2 in this case).

The following proposition implies that the case when both PJ-_1 is Zg-homologous

to zero in Mj_1 and Qj is Zg-homologous to zero in Mj is exactly the case of odd

handle.

10

Proposition 3.3 The j-th handle ofW is odd iff

dimH,.(Mj_1;Z2) = dimH.(MJ-; Z2)

Proof. By definition, the j-th handle is odd iff Sd'ImPJ-l‘1 X {pt} is Zg-homologous to
{pt} X Sd‘in‘l, therefore, the proposition follows from the proof of Proposition 3.1

[1

Proposition 3.4 The number of odd handles of any handlebody decomposition ofW
is congruent modulo 2 to x(W) + MW

Proof. Proposition 3.1 and Proposition 3.3 follow that this number is congruent
modulo 2 to dimH.(W; Z2) — %(dimH..(M; Z2) + dimH..(N; Z2) that is equivalent to

the proposition D

Proposition 3.5 If two manifolds M and N are cobordant then there exists a cobor-

dism W between M and N and a handlebody decomposition ofW containing no odd
handles

Proof. If dimW is odd then W may not contain odd handles. Suppose that dimW =
2k is even. If the jth handle of W is odd then we attach RP2" to the tubular
neighbourhood of Mj in W to get the new cobordism W’ m W#RP2’°. Manifold
RP“ admits a standard handlebody decomposition containing exactly one handle of
index It. By Propositon 3.4 this is an odd handle (since x(RP2’° = 1). The handlebody
decomposition of W and the standard handlebody decomposition of RP“ induce a
handlebody decomposition of W’. By switching the order of the jth handle and the
handles of RPZ" of index less than k we can make the two odd handles adjanced,
these operations do not change the oddness of the handles. Then we slide ode of

two adjanced via another. After this both of them become even, the first one is

11

even because the self—intersection number is even, the second one is even because its

surgery sphere is not Zg-homologous to zero after the attaching of the new handle Cl

Without loss of generality we may assume that the handlebody decomposition of
W consists of a single handle (otherwise we proceed with induction). By Proposition
3.5 we may assume that the handle is not odd. Suppose that the handle is of index
p, S”“1 m P C M is the surgery sphere (the boundary of the core of the handle
of W) and S”‘1 a: Q C N is the dual surgery sphere (the boundary of the cocore
of the handle of 1V). In particular, dim(M) = p + q — 1. By Proposition 3.1 and
Proposition 3.3 we may assume that P is Zg-homologous to zero in M and Q is not
Zg—homologous to zero in N (if by chance P is not Zg-homologous to zero in M and

Q is Zg-homologous to zero we turn W upside down).

Lemma 4 There exists a manifold N such that N — Q is the result of multiblowup of
N — Q and a smooth p-dimensional submanifold W C N transversal to Q and such

that W O Q = {q}, where q is a point.

Proof. By the Nash theorem [8] we may assume that N is a non-singular algebraic
variety. Since Q is not Zg-homologous to zero in N, there exists fl E Hp(N; Z2) such
that [3.[Q] = 1 E Z2 . By a theorem of Thom (Théoréme 111.2 of [9]) fl is representable
by smooth map f : B —+ N, where B is a smooth p-manifold and f.([B]) = 3. By the
Akbulut-King normalization theorem (Theorem 2.8.3 of [3]) ,8 is representable by a
subalgebraic set of dimension p in N (i.e. there exists a non-singular component Z of
an algebraic variety and a degree 1 rational map F : Z ——> N such that F..([Z]) = fl).
Denote by W the Zariski closure of F (Z ) Since F (Z ) is subalgebraic (and, therefore,
semialgebraic by the Tarski-Seidenberg theorem), dimW = dimZ : p. By changing
Q with a small isotopy we can assume that Q intersects W transversally in non-

singular points of W.

12

 

 

Q

Figure 2: Cancelling of pairs of the intersection points

By the Hironaka theorem [5] there exists a multiblowup of N with centers not
intersecting the transforms of Q such that the proper transform of W is a smooth
submanifold of the result N of the multiblowup. Therefore, the proper transform T of
F(Z) is a smooth submanifold of N intersecting Q C N transversally at odd number
of points (the fact that T is a submanifold without the boundary follows from the
local version of Corollary 2.3.3 of [3]). Let q1,. . .,q2k+1 be the points of T O Q. Let
71, . . . ,yk be the disjoint paths in Q connecting ql to q2, q3 to q.,, ..., q2k_1 to (12k.
Then, the tubular neighbourhood S of 71 U. . .U'yk such that W = TUUS— intS may

be smoothened to a p-dimensional submanifold of N intersecting Q transversally at

q = 92k+1 D

Proposition 3.6 There exists a manifold M’ such that M’- P is the result of multi-

blowup ofM — P and P = (9V for a submanifold V C M’

Proof. Since M — P = N -— Q, the multiblowup from Lemma 4 produces the required
multiblowup of M, the complement of the small disk neighbourhood of q in W gives
V such that P = 8V D

Now we may assume that P = 8V C M.

Definition. Let P m S”"1 be the submanifold of M equipped with a trivialization r

13

 

 

2|

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2‘ V i=B(V.Zi)

 

 

 

Figure 3: Blowup of the zero section

of uM(P). Let us denote by erM, P) the result of surgery of M along P. If P = 8V
for some submanifold V we say that the surgery trivialization T is compatible with V if
there exists a trivialization rv of VM(V) such that Tvlp and the natural trivialization

of z/V(P) form r.

Proposition 3.7 There exists a manifold M’ such that M’— P is the result of multi-
blowup ofM— P, P = 8V’ and the surgery trivialization ofz/Mi(P) is compatible with
V.

Proof. Let 61,...,€q be the trivialization of VM(P). Let {1 be a generic section
of VM(V) extending e]. The zero set of f; is a smooth submanifold Z1 of V. Let
M1 = B(M, Z1) and let V1 C M1 be the proper transform of V, V1 z B(V, Z1). Then

61 extends to a non-vanishing section {1 of UM, (V1) (see [1] for explicit proof of this).

We now proceed by induction. Suppose now that there exists Mk such that Mk —P

is the result of multiblowup of M — P and €1,. . . , 6k extend to non-vanishing linearly
independent sections $1,. . . ,{k of l/Mk(Vk).
Let ff,“ be a generic section of VMk(Vk) such that {1, . . . ,Ehéffl are linearly in-

dependent. Let Zk+1 C Vk be the zero set of (2+1. Let Mk = B(Mk,Zk+1), let V},
be the proper transform of V1,, and let E be the exceptional divisor of B(Mk, Zk+1).
It is easy to see that 6k+1 extends to a non-vanishing section Ek+1 of VMk(Vk) and
sections 61, . . . , ck extend to sections 51,. . . ,5}, of VMk(l_/k) transverse to V}, and van-

ishing exactly on E (1 V1,. The projectivisation F of the orthogonal complement of

14

{1, . . . ,{k in VMk(Zk+1) is contained in E and contains Efl Vk. Let Mk“ 2 B(Mk, F).
Sections 61,. . . , en+1 now extend to some non-vanishing linearly independent sections
(produced by 6—1, . . . , Ek+1)- since F is tangent to €k+1- Induction completes the proof

of the lemma D

The following lemma is the main tool that enables us to blow down submanifolds

inside the ambient manifold.

Lemma 5 Let i7 be a submanifold of M. IN = B(V, K) and VM(V) is trivial then
there exists a manifold AI’ containing submanifold V and such that (M’, V) is blowup

equivalent to (M, V)

Addendum 6 Ifa trivialization T of VM(V)|6V extends to a trivialization of VM(V)

then the induced by r trivialization ofz/M,(V)|3v extends to a trivialization of VM,(V)

Proof of Lemma 5. Let If C V C M be the exceptional divisor of B(V,]i). Let
M = B(M,]i). The proper transform of I7 in M is diffeomorphic to V (since the
center of the blowup is of codimension 1 in V), we shall denote the proper transform
of V in M still by V. The bundle VM(V) is isomorphic to the tensor product of
the trivial vector bundle and the 1-dimensional vector bundle 771? over V dual to If
(cf. proof of Proposition 3.7). By the adjunction formula, the normal bundle 11,7(Ii')
is isomorphic to Tlli'lk- Hence uM(Ii’) = (e”‘1 (8) VV(Ii’)) 69 147(1?) = e” (8) 11,-,(Ii').
Therefore, since dim(up(1~{)) = 1, the projectivisation E of the normal bundle of If
in M is diffeomorphic to If X RP”‘1, where the diffeomorphism is induced by the
trivialization of VM(V), and the section F given by the normal bundle of If in V

extends to some trivialization of E R: If X RP”‘1.

Let M = B (M, 1?). The proper transform V of V in M is diffeomorphic to V (If

is of codimension 1 in V). Therefore, the normal bundle of F in V is diffeomorphic to

15

V=B(V,K)

 

 

 

 

 

 

 

Figure 4: Proof of Lemma 5

the normal bundle of the exceptional divisor of blowup. But F z 11' X {x} for some
x 6 RP”‘1 and, therefore, for any y 6 RP”‘1 the restriction of the normal bundle
of E in NI to If X {y} is diffeomorphic to the normal bundle of the exceptional
divisor of a blowup. In other words the tubular neighbourhood of If X RP”“ in
M is diffeomorphic to the I-bundle (ft x ape-1);] 3 (Kid) x ape-1 which is
canonical over each fiber of the map it x ape-1 —» K x RPq-l. It follows that
M = B(M’,K x RP”'1) for some manifold M’ so that the exceptional divisor of

B(M’, K X RP”'1) is E and V is the proper transform of submanifold V C M’ D

ProofofAddendum 6. We need to prove that 61,. . . , eq_1 extend to a trivialization
of VM,(V’). We see that 61,. . ., eq_1 extend to a trivialization of VM(V), since to get
M we blow M up twice along Ii’ C V and by assumption 6], . . .,eq_1 extend to a

~ .—

trivialization of VM(V). Let {1,. . . , {94 be the trivialization of VM(V) induced by the

16

~

trivialization of VM(V). Then {1, . . . ,§q_1 extend £1,” .,eq_1. Since we defined the
diffeomorphism E z Ii' X RP”‘1 using {1, . . . , 5,4, the fibers of the blowup M -> M’
are tangent to (1,. . . ,{q_1 (i.e. the trivialization of VE(F) given by the restriction
(E, F) —i (K x RP”‘1, K X {x}) of the blowup is equivalent to {1, . . . ,§q_1). It follows
that {1, . . . ,§q_1 is induced by some trivialization $1,...,wq_1 of the normal bundle

of V’ in M’. Therefore, 1f)1,...,1/}q_1 extend €1,...,€q_1 C1

Proof of Lemma 3 and Theorem 1. By Proposition 3.6 we need only to show that if N
is the result of surgery XT(M, (9V), where V is a submanifold of M, then N is blowup
equivalent to M. Note that V is of positive codimension in M since N is connected.

By Proposition 3.7 we may assume that r is compatible with V.

We prove this by induction. Since dimV < dimM we may assume that there
exists a blowup sequence between V and D” (0V z S”‘1). If V m D” then N z
M#SlD X S”‘1 (T is compatible with V) and, by Lemma 2, N is blowup equivalent
to M. We want to construct, using the blowup sequence V = V0,. ..,Vl = D”, a
sequence M = M0, . . .,M1 such that V, is a submanifold of M,, (M,,V,-) is blowup
equivalent to (M,_1,V,-_1) and r is compatible with V,, j E {1,...,l} (note that
P = 3V 2 81/1,: = 014). This suffices for the proof since this implies that
N = XT(M,8V) is blowup equivalent to XT(M1,8V1) z .M[#S” X 5”“.

We construct such a sequence by induction. If V,-l = B(V,-,K,) then we apply
Lemma 5 to (M,_1, V,-l) and get M,-. By Addendum 6 7' is compatible with V,. If
V, = B(V,-_1,L,-_1) then let M,’ = B(M,--1,L,-_1), then V,- is the proper transform of
V,-l. By Proposition 3.7 there exists the result (M,, 17,-) of multiblowup of (MJ’, V,)
such that r is compatible with V,. Then there is the multiblowup V, : Wk —->

. -—> W0 = V,. Now we apply Lemma 5 consequently k times to construct M,- so

that (M,,V,-) is blowup equivalent to (M,,V,~) and, therefore, to (M,_1,V,-_1). By

17

Addendum 6 the surgery trivialization r is compatible with V,- D

4 Summary

Theorem 1 is proven with the help of the classification of the smooth manifolds up
to cobordism due to R.Thom. We prove first that every smooth closed connected
manifold is cobordant to a topologically rational manifold and then we prove that

two cobordant smooth closed connected manifolds are blowup equivalent.

 

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[1] S.Akbulut, H.King. Submanifolds and homology of nonsingular algebraic varieties.
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[3] S.Akbulut, H.King. Topology of Real Algebraic Sets. Mathematical Sciences Re-

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[4] R.Benedetti, A.Marin. Déchirures de variéts de dimension trois et la conjecture de

Nash rationalité en dimension trois. Comment. Math. Helv., 67 (1992), 514-545

[5] H.Hironaka. Resolutions of singularities of an algebraic variety over a field of

characteristic zero. Ann. of Math., 79 (1964), 109-326.

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