数理解析研究所講究録
第 1707 巻 2010 年 29-36

29

SPECIAL GENERIC MAPS ON OPEN 4-MANIFOLDS
九州大学・大学院数理学研究院

佐伯修 (OSAMU SAEKI)

Faculty of Mathematics, Kyushu University

ABSTRACT. We characterize those smooth l-connected open 4-manifolds
with certain finite type properties which admit proper special generic maps
into 3-manifolds. As a corollary, we show that a smooth 4-manifold homeif and only
omorphic toadmits a proper special generic map into
. We also characterize those smooth 4-manifolds
if it is diffeomorphic tohomeomorphic tofor some closed orientable 3-manifold which admit$R^{3}$$R^{4}$$R^{4}$$L$$L\cross R$

proper special generic maps into$R^{3}$.

1. INTRODUCTION

A special generic map : $Marrow N$ between smooth manifolds is a smooth
map with at most definite fold singularities, which have the normal form
$f$

,

(1.1)

$(x_{1}, x_{2}, \ldots, x_{m})\mapsto(x_{1},x_{2}, \ldots, x_{n-1},x_{n}^{2}+x_{n+1}^{2}+\cdots+x_{m}^{2})$where $m=\dim M\geq\dim N=n$ . For some typical examples of special genericmaps, refer to Fig. 1. Note also that the map $R^{m}arrow R^{n}$ defined by (1.1) is itself
a proper special generic map, where a continuous map is proper if the inverse
image of a compact set is always compact. Submersions are also considered
special generic maps.
It has been known as the Reeb Theorem [19] that if a smooth connectedclosed m-dimensional manifold admits a special generic map into $R$ , then it is. In [20, 21], the author has shown that ahomeomorphic to the m-sphere
admits a special genericsmooth connected closed m-dimensional manifold$1\leq
n\leq
m$
only
if $M$ is diffeomorphic to
if andmap into
for every with
. In [23, 24] Sakuma and the author found some pairs
the standard m-sphereof homeomorphic smooth closed 4-manifolds such that one of them admits a, while the other does not. These show that special
special generic map into
generic maps are sensitive to detecting distinct differentiable structures on agiven topological manifold.
On the other hand, it has been known that a smooth m-dimensional manifold,
if and only if it is diffeomorphic to the standard
is homeomorphic to$m\neq4$
provided(see [15, 26]), while for $m=4$ , there exist uncountably many$S^{m}$$\Lambda/I$$R^{n}$$n$$S^{m}$$R^{3}$$R^{m}$$R^{m}$

2000 Mathematics Subject Classification. Primary $57N13$ ; Secondary $57R45,57R55$ ,$58K15$.

Key rnords and phmses. Proper special generic map, differentiable structure, open 4manifold, end.
This is an abridged version of [22].
The author has been partially supported by Grant-in-Aid for Scientific Research (B)(No. 19340018), Japan Society for the Promotion of Science.

30

$|$

$R^{a+b’}$$(b’\leq b)$

FIGURE 1. Examples of special generic maps
(for example, see [4, 6, 8, 27]). In
distinct differentiable structures onfact, it is known that most open 4-manifolds admit infinitely (and very often,
uncountably) many distinct differentiable structures [1, 3, 5, 7].
In this paper, we characterize those smooth l-connected open 4-manifolds of“finite type” which admit proper special generic maps into 3-manifolds, using
the solution to the Poincar\’e Conjecture in dimension three (see [16, 17, 18] or[14], for example). Here, an open 4-manifold is of finite type if its homology isfinitely generated and it has only finitely many ends, whose associated fundamental groups are stable and finitely presentable. As a corollary, we show thatis diffeomorphic to the standard
a smooth 4-manifold homeomorphic to.
if and only if it admits a proper special generic map into$M$
is homeomorphic toFurthermore, we show that if a smooth 4-manifold$L\cross R$ for some connected closed orientable 3-manifold
and if $M$ admits a, then $M$ is diffeomorphic to $L\cross R$ and theproper special generic map into
3-manifold admits a special generic map into .
All these results claim that among the (uncountably or infinitely) many distinct differentiable structures on a certain open topological 4-manifold, there
is at most one smooth structure that allows the existence of a proper special
generic map into a 3-manifold.
Throughout the paper, manifolds and maps between them are differentiable”
denotes a diffeounless otherwise indicated. The symbolof class
morphism between smooth manifolds or an appropriate isomorphism between
algebraic objects.
The author would like to express his sincere gratitude to Kazuhiro Sakuma
for stimulating discussions and invaluable comments.
$R^{4}$$R^{4}$$R^{4}$$R^{3}$$L$$R^{3}$$R^{2}$$L$$\cong$$C^{\infty}$

2. PRELIMINARIESLet first recall the following notion of a Stein factorization, which will play
an important role in this paper.
$11S$

Definition 2.1. Let : $Marrow N$ be a smooth map between smooth manifolds.
For two points $x^{f}\in M$ , we define $x\sim fx’$ if $f(x)=f(x’)(=y)$ , and thepoints and belong to the same connected component of $f^{-1}(y)$ . We define
$f$

$x,$

$x$

$x’$

31
$N$

FIGURE 2. Stein factorization
to be the quotient space with respect to this equivalence relation,: $Marrow W_{f}$ the quotient map. Then we see easily that there
and denote bythat makes the diagram
exists acontinuous map$W_{f}=M/\sim f$$q_{f}$$\overline{f}:W_{f}arrow N$$1lniq\iota le$$M$$arrow^{f}$$N$$\nearrow\overline{f}$$q_{f^{\backslash _{\searrow}}}$$W_{f}$

commutative. The above diagram is called the Stein factorization of[13] . Refer to Fig. 2 for an example.

$f$

(see

$)$

The Stein factorization is a very useful tool for studying topological propertiesof special generic maps. In fact, we can prove the following, which is folklore(for example, see [2, 20]).

Proposition 2.2. Let : $Marrow N$ be a proper special generic map between
smooth manifolds with $m=\dim M>\dim N=n$ . Then we have the following.
(1) The set of singular points $S(f)$ of is a regular submanifold of $M$ of
dimension $n-1$ , which is closed as a subset of $M$ .
has the structure of a smooth n-dimensional(2) The quotient space
an immersion.
manifold possibly with boundary such that :: $Marrow W_{f}$ restricted to $S(f)$ is a diffeomorphism(3) The quotient map
.
onto
$M$
is
restricted to(4) Ifis connected, then the quotient map
then the. Furthermore, if
a smooth fiber bundle over Int$S^{m-n}$.
fiber is the standard $(m-n)$ -sphere
$f$

$f$

$W_{f}$$W_{f}arrow N\dot{u}s$$\overline{f}$$q_{f}$$\partial W_{f}$$q_{f}$$M\backslash S(f)$$S(f)\neq\emptyset_{f}$$W_{f}$

See Fig. 3 for an illustrative explanation.
Using the above proposition, the author proved the following [20].
: $Marrow N$ be a proper speTheorem 2.3 (Disk bundle theorem). Let
cial generic map between smooth connected manifolds with $\dim M=m$ and$\dim N=n$ . If $m-n=1,2,3$ and, then $M$ is diffeomorphic to thewith $O(m-n+1)$ as structure group.
boundary of a $D^{m-n+1}$ -bundle over$f$

$S(f)\neq\cdot\emptyset$$W_{f}$

32$N$$\underline{f}$FIGURE 3. Proposition 2.2
In the following, we recall several notions conceming ends of manifolds. For
details, the reader is referred to Siebenmann’s thesis. [25].
Definition 2.4. Let $X$ be a Hausdorff space. Consider a collection of subsets
of $X$ with the following properties.
(i) Each
is a connected open non-empty set with compact frontier,with $G”\subset G\cap G’$ ,, then there exists(ii) If $G,$(iii).
$\epsilon$$G\in\epsilon$$\overline{G}-G$$G’\in\epsilon$$G^{u}\in\epsilon$$\bigcap_{G\in\epsilon}\overline{G}=\emptyset$

Adding to every connected open non-empty set $H\subset X$ with compact frontier, we produce a collection satisfying (i), (ii)such that $G\subset H$ for someand (iii), which we call the end of $X$ determined by .
An end of a Hausdorff space $X$ is a collection of subsets of $X$ which is
maximal with respect to the properties (i), (ii) and (iii) above.
A neighborhood of an end is any set $N\subset X$ that contains some member of. (See Fig. 4.)$\epsilon$$G\in\epsilon$$\epsilon$$\epsilon$$\epsilon$$\epsilon$

Definition 2.5. Let be an end of a topological manifold $X$ . The fundamentalgroup is stable at if there exists a sequence of path connected neighborhoods
points and base paths, withsuch that (withofchosen) the sequence$\epsilon$$\pi_{1}$$\epsilon,$$X_{1}\supset X_{2}\supset\cdots$$\epsilon$$ba_{\iota}se$$\cap\overline{X}_{i}=\emptyset$$\pi_{1}(X_{1})arrow^{f_{1}}\pi_{1}(X_{2})arrow^{f_{2}}$...

induced by the inclusions induces isomorphisms${\rm Im}(f_{1})arrow^{\simeq\underline}{\rm Im}(f_{2})arrow^{\simeq\underline}$

The following lemma is proved in [25].

.. .

33
I
$\bullet$$\bullet$

FIGURE 4. Ends of a manifold
is any path connected
then for any choice of baseis stable at andLemma 2.6. If
sequence of neighborhoods of such
points and base paths, the inverse sequence$Y_{1}\supset Y_{2}\supset\cdots$$\epsilon$$\pi_{1}$$that\cap\overline{Y}_{i}=\emptyset_{f}$$\epsilon$$\mathcal{G}$:$\pi_{1}(Y_{1})arrow^{g_{1}}\pi_{1}(Y_{2})arrow^{g_{2}}$induced by the inclusions is stable, $i.e$ . there exis$f_{\wedge}s$. ..

a subsequence$\pi_{1}(Y_{i_{1}})arrow^{h_{1}}\pi_{1}(Y_{i_{2}})arrow^{h_{2}}$...

inducing isomorphisms${\rm Im}(h_{1})arrow^{\simeq\underline}{\rm Im}(h_{2})arrow^{\simeq\underline}$$\tau i)here$each$h_{j}$

is a suitable composition of$g_{i}s$... ,.

to be theis stable at an end , we define
Definition 2.7. Whenas above. According to [25],for some fixed systemprojective limit
is well defined
to isomorphism.
$\pi_{1}(\epsilon)$$\epsilon$$\pi_{1}$$\mathcal{G}$$\lim_{arrow}\mathcal{G}$$\pi_{1}(\epsilon)$$11p$

Let us introduce the following definition.

Definition 2.8. An open manifold $M$ is of finite type if(i) $M$ has finitely many ends,(ii) for each end
is stable at$\epsilon,$$\pi_{1}$$\epsilon$

with$\pi_{1}(\epsilon)$

being finitely presentable,and(iii) $H_{*}(M;Z_{2})$ is finitely generated.

We will need the following result due to Husch-Price [11, 12].
Lemma 2.9 (Husch-Price, 1970). Let $W$ be an open orientable 3-manifoldof finite type. Then there exists a compact orientable 3-manifold and anembedding $h:Warrow W$ such that (Int $W$ ) Int $W$ .
$\overline{W}$

–

$h$

$=$

34
3. OPEN 4-MANIFOLDS THAT ADMIT SPECIAL GENERIC MAPS
In the following, a manifold is open if it has no boundary and each of its
component is non-compact, while a manifold is closed if it has no boundary
and is compact.
Theorem 3.1. Let $M$ be a smooth l-connected open 4-manifold of finite type.
Then there exists a proper special generic map : $Marrow N$ into a smooth 3manifold $N$ with
if and only if $M$ is diffeomorphic to the connected
sum of a finite number of copies of the following 4-manifolds:(1)(2) the interior of the boundary connected sum of a finite number of copies,of(3) the total space of a 2-plane bundle over ,(4) the total space of an -bundle over ,where at least one manifold of the form (1), (2) or (3) should appear in theconnected sum.
$f$

$S(f)\neq\emptyset$$R_{f}^{4}$$S^{2}\cross D^{2}$$S^{2}$$S^{2}$$S^{2}$: $Marrow N$ be a proper special generic map into a
Sketch of proof. Let
in the Stein
3-manifold $N$ . Then we can prove that the quotient spacefactorization of is an open 3-manifold of finite type. Since $M$ is l-connected,. By the solution to the Poincar\’e Conjecture together with the Husch–
so is, whereorPrice Lemma (Lemma 2.9), we see thatis a compact surface (possibly with boumdary) contained in the boundary.
On the other hand, $M$ is diffeomorphic to the boundary of a -bundle overby the Disk bundle theorem, Theorem 2.3. Then we easily get the desiredconclusion.
Conversely, it is easy to construct explicitly a proper special generic map into
a 3-manifold for each 4-manifold in the list.
$f$$W_{f}$$f$$W_{f}$$W_{f}\cong D^{3}\backslash F$$\mathfrak{h}^{k}(S^{2}\cross[0,1])\backslash F$$F$$D^{2}$$W_{f}$$\square$Remark 3.2. Every 4-manifold as in Theorem 3.1 admits infinitely many (or uncountably many) distinct smooth structures. Theorem 3.1 implies that among
them there is exactly one structure that allows the existence of a proper special
generic map into a 3-manifold.

In particular, we have the following.

. ThenCorollary 3.3. Let $M$ be a smooth 4-manifold homeomorphic to$MarrowR^{3}$
there exisbs a proper special generic map :if and only if $M$ is.
diffeomorphic to the standard$R^{4}$$f$$R^{4}$We also have the followingl.
Theorem 3.4. Let be a smooth connected closed orientable 3-manifold.
smooth 4-manifold $M$ homeomorp $hic$ to $L\cross R$ admits a proper special genericmap into
if and only if $M$ is diffeomorphic to $L\cross R$ and is a smooth.
closed 3-manifold that admits a special generic map into
$A$

$L$

$L$

$R^{3}$$R^{2}$

lTheorem 3.4 was first conjectured by Kazuhiro Sakuma to whom the author would liketo express his sincere gratitude.

35
Sketch of proof. Suppose $M$ is homeomorphic to $L\cross R$ and let : $Marrow N$
be a proper special generic map into a 3-manifold $N$ . Then one can show that
$i=1,2$ , for someis of finite type and has exactly two ends
induce isomorphisms
surfaces . Furthermore, the inclusions
together with-manifolds
of fundamental groups. By the standard theory of
the solution to the Poincar\’e Conjecture and the Husch-Price Lemma, we see(for example, see [10]). Since $M$ is homeomorphic
that
R. Therefore, $M$ is diffeomorphic to $L’\cross R$
to $L\cross R$ , we see thatis free. Therefore, $L’\cong L\cong$
for some 3-manifold . Note that, and hence there exists a special generic map $g:Larrow R^{2}$ by a resultof Burlet-de Rham [2].
Conversely, if admits a special generic map $g:Larrow R^{2}$ , then$f$$F_{i}\cross[0, \infty),$$W_{f}$$F_{i}\cross\{0\}arrow W_{f}$$F_{i}$$W_{f}\cong(F_{1}\cross R)\#(\#^{k}D^{3})$$W_{f}\cong F_{1}\cross$$L’$$\pi_{1}(L’)\cong\pi_{1}(L)$$\#^{\ell}(S^{1}\cross S^{2})$$L$$g\cross id_{R}:L\cross Rarrow R^{2}\cross R$

is a proper special generic map, where$id_{R}$denotes the identity map of R.

$\square$Conjecture 3.5. Let $M$ be a topologica14-manifold. Then there exists at mostone smooth structure on $M$ that allows the existence of a proper special generic.
map into
Remark 3.6. In the above conjecture, the propemess of the special generic map
is essential. Let : $Marrow N$ be a special generic map of an open 4-manifold andassume that $M$ ‘ is homeomorphic to $M$ . Then there exists a “formal solution“
over $M$ ‘ on thejet level for the open differential relation corresponding to special
generic maps. Therefore, $M’$ admits a special generic map by the Gromov hprinciple for open manifolds [9]. Note that even if is proper, the reslllting
may not be proper.
special generic map on
Compare this with the following: if a smooth 4-manifold $M$ is homeomorphic, then there exists a proper special generic map : $Marrow R^{4}$ . In thetodense h-principle holds and the properness can beequidimensional case, thepreserved (see [9]).
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$Cla_{n}qsi\theta ing$

$R^{4}$$R^{2}$$Actualit6_{\backslash }s$$R^{3}$$R^{3}$FACULTY OF MATHEMATICS, KYUSHU UNIVERSITY, MOTOOKA 744, NISHI-KU,
FUKUOKA 819-0395, JAPANE-mail address:$s$aeki math. kyushu-u. ac. jp