# Topologically isotopic and smoothly inequivalent 2-spheres in simply
connected 4-manifolds whose complement has a prescribed fundamental group.

Rafael Torres  Scuola Internazionale Superiori di Studi Avanzati (SISSA)
Via Bonomea 265
34136
Trieste
Italy<EMAIL_ADDRESS>
###### 2020 Mathematics Subject Classification:

Primary 57K45, 57R55; Secondary 57R40, 57R52

_Abstract_ : We describe a procedure to construct infinite sets of pairwise
smoothly inequivalent 2-spheres in simply connected 4-manifolds, which are
topologically isotopic and whose complement has a prescribed fundamental group
that satisfies some conditions. This class of groups include cyclic groups and
the binary icosahedral group. These are the first known examples of such
exotic embeddings of 2-spheres in 4-manifolds. Examples of locally flat
embedded 2-spheres in a non-smoothable 4-manifold whose complements are
homotopy equivalent to smoothly embedded ones are also given.

##  1\. Main results

The first main result of this note is the following theorem.

######  Theorem A.

Fix $p\geq 2$. There is an infinite set

$\\{S_{n,p}:n\in\mathbb{Z}\\}$

of smoothly embedded 2-spheres in
$2\mathbb{CP}^{2}\\#4\overline{\mathbb{CP}^{2}}$ that satisfies the following
properties.

  * •

Any two elements are topologically isotopic.

  * •

There is a diffeomorphism of pairs

$(2\mathbb{CP}^{2}\\#4\overline{\mathbb{CP}^{2}},S_{n_{1},p})\rightarrow(2\mathbb{CP}^{2}\\#4\overline{\mathbb{CP}^{2}},S_{n_{2},p})$

if and only if $n_{1}=n_{2}$.

  * •

The fundamental group of the complement is

$\pi_{1}(2\mathbb{CP}^{2}\\#4\overline{\mathbb{CP}^{2}}\setminus\nu(S_{n,p}))=\mathbb{Z}/p$

for every $n\in\mathbb{Z}$.

  * •

$[S_{n,p}]\neq 0\in
H_{2}(2\mathbb{CP}^{2}\\#4\overline{\mathbb{CP}^{2}};\mathbb{Z})$ for every
$n\in\mathbb{Z}$.

  * •

Surgery along each of these 2-spheres yields an infinite set of pairwise
homeomorphic and pairwise non-diffeomorphic closed smooth 4-manifolds with
fundamental group $\mathbb{Z}/p$.

Theorem A provides the first known example of an infinite set of 2-spheres
smoothly embedded in a simply connected 4-manifold that are pairwise
topologically isotopic, pairwise smoothly inequivalent and having a complement
with finite cyclic fundamental group. Schwartz [28, Theorem 2] pointed out the
existence of closed simply connected 4-manifolds containing pairs of smoothly
embedded 2-spheres that are both smoothly equivalent and topologically
isotopic, but not smoothly isotopic. Examples of these exotic embeddings of
2-spheres in closed 4-manifolds with simply connected complement have been
constructed by Akbulut [3] and Auckly-Kim-Melvin-Ruberman [6]. Exotic
embeddings of surfaces with positive genus in simply connected 4-manifolds and
complement having non-trivial fundamental group are found in Kim [19] and Kim-
Ruberman [20]. An ingredient in the proof of Theorem A is of independent
interest: we point out in Theorem 1 that constructions of inequivalent smooth
structures on simply connected 4-manifolds of Fintushel-Stern [11, 12] can be
extended to produce such structures on 4-manifolds with non-trivial
fundamental group too.

The second main result provides a construction procedure of topologically
equivalent yet smoothly inequivalent homologically essential 2-spheres whose
complement can be chosen to have the same fundamental group as a wide range of
$\mathbb{Q}$-homology 4-spheres. We work with the modified Seiberg-Witten
$\operatorname{SW}^{\prime}_{X}$ invariant of a closed 4-manifold $X$ as
defined, for example, in [10, Section 2], and denote by $\mathcal{B}_{X}$ the
set of basic classes.

######  Theorem B.

Let $\\{Z_{n}:n\in\mathbb{Z}\\}$ be an infinite set of closed smooth simply
connected 4-manifolds with pairwise different integer invariants

(1.1)
$S_{n}=\max\\{|\operatorname{SW}^{\prime}_{Z_{n}}(k_{Z_{n}})|:k_{Z_{n}}\in\mathcal{B}_{Z_{n}}\\},$

which are pairwise homeomorphic to a given closed 4-manifold $Z$ and such that
the connected sum $Z_{n}\\#S^{2}\times S^{2}$ is diffeomorphic to
$Z\\#S^{2}\times S^{2}$ for every $n\in\mathbb{Z}$. Let $M$ be a closed smooth
4-manifold with $H_{\ast}(M;\mathbb{Q})\cong H_{\ast}(S^{4};\mathbb{Q})$ and
set $\pi:=\pi_{1}M$. Suppose that there is a loop $\alpha\subset M$ and a
choice of framing such that

(1.2) $S^{2}\times S^{2}=M\setminus\nu(\alpha)\cup D^{2}\times S^{2}.$

There is an infinite set

$\\{S_{n,\pi}:n\in\mathbb{Z}\\}$

of smoothly embedded 2-spheres in $Z\\#S^{2}\times S^{2}$ that satisfies the
following properties.

  * •

There is a homeomorphism of pairs

$(Z\\#S^{2}\times S^{2},S_{n_{1},\pi})\rightarrow(Z\\#S^{2}\times
S^{2},S_{n_{2},\pi})$

for every $n_{i}\in\mathbb{Z}$.

  * •

There is a diffeomorphism of pairs

$(Z\\#S^{2}\times S^{2},S_{n_{1},\pi})\rightarrow(Z\\#S^{2}\times
S^{2},S_{n_{2},\pi})$

if and only if $n_{1}=n_{2}$.

  * •

The fundamental group of the complement is

$\pi_{1}(Z\\#S^{2}\times S^{2}\setminus\nu(S_{n,\pi}))=\pi$

and its homology class satisfies

$[S_{n,\pi}]\neq 0\in H_{2}(Z\\#S^{2}\times S^{2};\mathbb{Z})$

for every $n\in\mathbb{Z}$.

  * •

Surgery along each of these 2-spheres yields an infinite set
$\\{Z_{n}\\#M:n\in\mathbb{Z}\\}$ of pairwise non-diffeomorphic closed smooth
4-manifolds with fundamental group $\pi$, and that are pairwise homeomorphic
to the connected sum $Z\\#M$.

Please see [10, Proof of Theorem 1] for details on the definition of the
invariant (1.1). Fintushel-Stern constructed infinite sets as in the
hypothesis of Theorem B for $Z=\mathbb{CP}^{2}\\#k\overline{\mathbb{CP}^{2}}$
for $2\leq k\leq 7$ in [11, 12]. Baykur-Sunukian [9] showed that Fintushel-
Stern’s examples become diffeomorphic after a connected sum with a single copy
of $S^{2}\times S^{2}$. Examples of $\mathbb{Q}$-homology 4-spheres $M$ that
satisfy the hypothesis are spun 4-manifolds with the fundamental group of any
lens space and the Poincaré homology 3-sphere. A similar result holds if (1.2)
is substituted for the non-trivial bundle $S^{2}\widetilde{\times}S^{2}$. It
is possible to strengthen the conclusion of Theorem B to topologically
isotopic 2-spheres, although we do not pursue this endeavor here; see
Sunukjian [29].

A contribution of this note is to point out the simplicity of the proofs of
Theorem A and Theorem B. The reader will notice that the 4-manifolds in the
last clause of Theorem B are smoothly reducible (cf. [15, Definition
10.1.17]), while those in the last clause of Theorem A are not. We explain in
Remark 2 how an instance of Theorem B implies the claims on the existence of
the homeomorphism of pairs and the non-existence of the diffeomorphism of
pairs of Theorem A. An independent proof of Theorem A is given in Section 2.7
as well. The following consequence of Theorem B is another contribution.

######  Corollary C.

Let $G$ be a finite cyclic group or the icosahedral group

$G=\langle g_{1},g_{2}:{g_{1}}^{5}=(g_{1}g_{2})^{2}={g_{2}}^{3}\rangle$.

There is an infinite set of smoothly embedded 2-spheres in
$2\mathbb{CP}^{2}\\#4\overline{\mathbb{CP}^{2}}$ that are pairwise
topologically equivalent, yet pairwise smoothly inequivalent, and the
fundamental group of the complement is $G$.

These are the first examples of exotic embeddings of 2-spheres in simply
connected 4-manifolds whose complement has a fundamental group isomorphic to
the binary icosahedral group among several other choices of groups. We exhibit
interesting smooth embeddings of nullhomotopic 2-spheres in the fourth main
result of this note.

######  Theorem D.

There is an infinite set

(1.3) $\\{S_{n}:n\in\mathbb{Z}\\}$

of 2-spheres smoothly embedded in
$2\mathbb{CP}^{2}\\#4\overline{\mathbb{CP}^{2}}$ that satisfies the following
properties.

  * •

The fundamental group of the complement of an element in (1.3) is

$\pi_{1}(2\mathbb{CP}^{2}\\#4\overline{\mathbb{CP}^{2}}\setminus\nu(S_{n}))=\mathbb{Z}$

and $[S_{n}]=0\in\pi_{2}(2\mathbb{CP}^{2}\\#4\overline{\mathbb{CP}^{2}})$ for
every $n\in\mathbb{Z}$.

  * •

There is a diffeomorphism of pairs

$(2\mathbb{CP}^{2}\\#4\overline{\mathbb{CP}^{2}},S_{n_{1}})\rightarrow(2\mathbb{CP}^{2}\\#4\overline{\mathbb{CP}^{2}},S_{n_{2}})$

if and only if $n_{1}=n_{2}$.

Notice that elements in (1.3) do not bound a smoothly embedded 3-ball in
$2\mathbb{CP}^{2}\\#4\overline{\mathbb{CP}^{2}}$. The smoothly inequivalent
embeddings of homotopically trivial 2-spheres of Theorem D are related to a
construction of an infinite set of closed smooth 4-manifolds with infinite
cyclic fundamental group and the homology of the connected sum
$2\mathbb{CP}^{2}\\#4\overline{\mathbb{CP}^{2}}\\#S^{1}\times S^{3}$, which is
given in Theorem 2.

While any 2-sphere in a closed simply connected 4-manifold can be assumed to
be regularly immersed, Hambleton-Kreck [16] and Lee-Wilczyński [22, 23]
completely characterized when a homology class of non-zero divisibility can be
represented by a locally flat embedded 2-sphere. The fifth and last result to
be mentioned in this introduction records the existence of a myriad of
explicit examples of locally flat embedded 2-spheres in closed simply
connected 4-manifolds whose exteriors are homotopy equivalent but not
homeomorphic.

######  Theorem E.

For every $p\geq 2$, there is a locally flat embedded 2-sphere

(1.4) $S_{p}\subset\ast\mathbb{CP}^{2}\\#\overline{\mathbb{CP}^{2}}$

whose complement has finite cyclic group $\mathbb{Z}/p$, and it is homotopy
equivalent to the complement of a smoothly embedded 2-sphere

(1.5) $S_{p}^{\prime}\subset\mathbb{CP}^{2}\\#\overline{\mathbb{CP}^{2}}.$

Theorem E is essentially derived from an existence result of non-smoothable
$\mathbb{Q}$-homology 4-spheres due to Hambleton-Kreck [17]. Other interesting
examples are found in Kasprowski-Lambert-Cole-Land-Lecuona [18].

###  1.1. Acknowledgements:

E-mail exchanges with Aru Ray, Bob Gompf, Danny Ruberman and Ian Hambleton
have increased our understanding on the subject. We are grateful to them for
patiently sharing their knowledge. We thank the anonymous referees for their
careful reading of the manuscript and for their input, which helped us improve
the manuscript. Del tega dela je nastal na Rogu in Metelkovi.

##  2\. Proofs

###  2.1. Infinitely many inequivalent smooth structures

Fintushel-Stern [12, Theorem 1] showed that there is a nullhomologous 2-torus
$T$ smoothly embedded in $\mathbb{CP}^{2}\\#3\overline{\mathbb{CP}^{2}}$ such
that performing surgeries on $T$ results in infinitely many inequivalent
smooth structures on $\mathbb{CP}^{2}\\#3\overline{\mathbb{CP}^{2}}$. We point
out that changing the coefficients of the torus surgery on $T$ introduces
homotopically non-trivial loops to the resulting 4-manifold, and their
procedure also yields infinitely many smooth structures on 4-manifolds with
prescribed cyclic fundamental group. The latter will serve as raw material to
construct the knotted 2-spheres.

We introduce terminology to state the result and follow the notation in [12,
Section 3]. Let $T\subset X$ be a smoothly embedded 2-torus with trivial
tubular neighborhood $\nu(T)=T^{2}\times D^{2}$. Let $\\{a,b\\}$ be loops in
$T$ that form a symplectic basis of $\pi_{1}T=\mathbb{Z}^{2}$, and let
$\\{S^{1}_{a},S^{1}_{b}\\}$ be loops in $\partial\nu(T)=T^{2}\times\partial
D^{2}=T^{2}\times S^{1}$ that are homologous to $a$ and $b$, respectively. The
meridian of $T$ is denoted by $\mu_{T}$ and it is any curve in the same
isotopy class of the curve $\\{x\\}\times\partial D^{2}\subset\partial\nu(T)$.
The smooth 4-manifold

(2.1) $X_{T,S^{1}_{b}}(p/n):=(X\setminus\nu(T))\cup_{\varphi}(T^{2}\times
D^{2})$

where the gluing diffeomorphism satisfies $\varphi_{\ast}([\partial
D^{2}])=n[S^{1}_{b}]+p[\mu_{T}]$ is said to be obtained by performing a
$p/n$-torus surgery to $X$ on $T$ along the curve $b$.

We first consider the case of finite cyclic fundamental group and postpone the
infinite cyclic case to the end of the section.

######  Theorem 1.

Fix $p\geq 2$. There is a smoothly embedded nullhomologous 2-torus
$T\subset\mathbb{CP}^{2}\\#3\overline{\mathbb{CP}^{2}}$ and a nullhomologous
curve in its complement
$S^{1}_{b}\subset\mathbb{CP}^{2}\\#3\overline{\mathbb{CP}^{2}}\setminus\nu(T)$
such that performing a $(p/n)$-torus surgery to
$\mathbb{CP}^{2}\\#3\overline{\mathbb{CP}^{2}}$ on $T$ along $S^{1}_{b}$
yields an infinite set

(2.2) $\\{X_{T,S^{1}_{b}}(p/n):n\in\mathbb{Z}\\}$

of pairwise non-diffeomorphic 4-manifolds such that every element is
homeomorphic to the connected sum

(2.3) $\mathbb{CP}^{2}\\#3\overline{\mathbb{CP}^{2}}\\#\Sigma_{p},$

where $\Sigma_{p}$ is a $\mathbb{Q}$-homology 4-sphere with fundamental group
$\pi_{1}\Sigma_{p}=\mathbb{Z}/p$.

###### Proof.

The only contribution in this note to the work of Fintushel-Stern [11, 12]
that provides a proof of Theorem 1 is the change in a coefficient of the torus
surgery. We then employ a homeomorphism criteria of Hambleton-Kreck to pin
down the homeomorphism class of the closed 4-manifolds that are constructed
this way. Set $X:=\mathbb{CP}^{2}\\#3\overline{\mathbb{CP}^{2}}$ so to not
overload the notation. Fintushel-Stern showed the existence of a
nullhomologous torus $T\subset X$ [12, Theorem 1.1] and the curve $b\subset T$
with framing $S^{1}_{b}\subset X\setminus\nu(T)$ as in the statement of
Theorem 1. Build $X_{T,S^{1}}(p/n)$ as in (2.1). Since $[T]=0\in
H_{2}(X;\mathbb{Z})$, we have that
$H_{1}(X_{T,S^{1}_{b}}(p/n);\mathbb{Z})=\mathbb{Z}/p$ for every
$n\in\mathbb{Z}$; in this notation, $p=0$ corresponds to $\mathbb{Z}$. We now
fix $p\geq 2$.

To see that the fundamental group of $X_{T,S^{1}_{b}}(p/n)$ is $\mathbb{Z}/p$,
we take a closer look at the constructions of Fintushel-Stern in [11, 12],
where six torus surgeries along six nullhomologous 2-tori
$\\{T_{1,i},T_{2,i}:i=1,2,3\\}$ are performed to $X$ to produce a symplectic
4-manifold $Q$ with fundamental group $\pi_{1}(Q)=\mathbb{Z}^{6}$ and that
contains six Lagrangian 2-tori $\\{L_{1,i},L_{2,i}:i=1,2,3\\}$ [11,
Proposition 7], [12, p. 77]. The complements of these 2-tori are the same,
i.e.,

(2.4)
$X\setminus\overset{3}{\underset{i=1}{\bigcup}}\big{(}\nu(T_{1,i})\sqcup\nu(T_{2,i})\big{)}=Q\setminus\overset{3}{\underset{i=1}{\bigcup}}\big{(}\nu(L_{1,i})\sqcup\nu(L_{2,i})\big{)}.$

By applying six surgeries to the symplectic 4-manifold $Q$ along the
Lagrangian 2-tori with a given choice of surgery curves [11, Theorem 2], one
obtains an infinite set of inequivalent smooth structures on $X$. The first
five surgeries are $|1/1|$-torus surgeries, while the last one is a
$1/n$-torus surgery [11, p. 1685]. In particular, this infinite set can be
obtained by performing torus surgeries to $X$ on six nullhomologous 2-tori.
For our purposes, we perform the first five surgeries verbatim as in the proof
of [11, Theorem 2], change the surgery coefficients of the sixth surgery to
perform a $p/n$-torus surgery in order to obtain an infinite set

(2.5) $\\{X_{T,S^{1}_{b}}(p/n):n\in\mathbb{Z}\\}$

for a fixed $p\geq 2$. It follows from the Seifert-van Kampen theorem that the
fundamental group is $\pi_{1}(X_{T,S^{1}_{b}}(p/n))=\mathbb{Z}/p$ [8, p. 321]
for every $n\in\mathbb{Z}$: a detailed account on the computation of the
fundamental group of the 4-manifolds obtained with such a change in the
surgery coefficient can be found in several places in the literature, for
example, [8, Section 5], [5, p. 595]. We have explained so far that six
surgeries on six nullhomologous 2-tori in $X$ as in [11, Theorem 2] produce an
infinite set (2.5) of 4-manifolds with fundamental group $\mathbb{Z}/p$.

We now appeal to the main result in [12, Section 8]. Fintushel-Stern showed
that the first five surgeries on $X$ do not change the diffeomorphism type of
$X$ and, thus, that there is a single nullhomologous 2-torus $T\subset X$
along with a nullhomologous curve $S^{1}_{b}\subset X\setminus\nu(T)$ such
that an $1/n$-torus surgery produce an infinite set of smooth structures on
$X$, as we had mentioned before [12, Theorem 1.1]. Thus, we conclude that each
element in the set (2.5) is obtained by performing a $p/n$-torus surgery on
$T\subset X$ along $S^{1}_{b}$.

We now argue that these 4-manifolds are homeomorphic to (2.3). An inclusion-
exclusion argument indicates that the Euler characteristic is unchanged under
torus surgeries, i.e.,

(2.6) $\chi(X_{T,S^{1}_{b}}(p/n))=\chi(X)=6.$

Novikov additivity [15, Remark 9.1.7] implies

(2.7) $\sigma(X_{T,S^{1}_{b}}(p/n))=\sigma(X)=-2,$

and we conclude that the second Stiefel-Whitney class of
$X_{T,S^{1}_{b}}(p/n)$ does not vanish employing a result of Rohklin cf. [15,
Theorem 1.2.29]. A classification result of Hambleton-Kreck [17, Theorem C]
allows us to conclude that the 4-manifold $X_{T,S^{1}_{b}}(p/n)$ is
homeomorphic to $\mathbb{CP}^{2}\\#3\overline{\mathbb{CP}^{2}}\\#\Sigma_{p}$,
where $\Sigma_{p}$ is a closed smooth 4-manifold with Euler characteristic two
and signature zero for every $n\in\mathbb{Z}$ and $p\geq 2$.

To argue that we have constructed infinitely many 4-manifolds that are
pairwise non-diffeomorphic, we compute their Seiberg-Witten invariants using
an argument well-documented in the literature [4, 8, 10, 11, 12]. We reproduce
the argument here for the sake of completeness, which requires us to describe
the relation between the Seiberg-Witten invariants of the 4-manifolds
$X_{T,S^{1}_{b}}(p/n)$, $X$ and $X_{T,S^{1}_{b}}(0/1)$. Given a characteristic
element $k_{0}\in H_{2}(X_{T,S^{1}_{b}}(0/1);\mathbb{Z})$, there are unique
characteristic elements $k_{X}\in
H_{2}(X_{T,S^{1}_{b}}(1/0);\mathbb{Z})=H_{2}(X;\mathbb{Z})$ and $k_{p/n}\in
H_{2}(X_{T,S^{1}_{b}}(p/n);\mathbb{Z})$ [4, Remark 4], [11, p. 64]. The
4-manifolds $X_{T,S^{1}_{b}}(1/0)=X$ and $X_{T,S^{1}_{b}}(0/1)$ will have at
most one basic class up to sign in our setting; cf. [4, 12, 11]. As described
in [12, Section 3], the 4-manifold $X_{T,S^{1}_{b}}(0/1)$ has infinite cyclic
fundamental group and it admits a symplectic structure [12, Section 4] (cf.
[10, Section 3]). A result of Taubes [31] says that the canonical class
$k_{0}=-c_{1}(X_{T,S^{1}_{b}}(0/1))$ is a basic class of
$X_{T,S^{1}_{b}}(0/1)$ and $\operatorname{SW}_{X_{T,S^{1}_{b}}(0/1)}(\pm
k_{0})=\pm 1$. Moreover, the adjunction inequality (see [4, Section 2.1])
implies that $k_{0}\in\mathcal{B}$ is the only basic class up to sign.

It follows that there is a unique
$k_{p/n}\in\mathcal{B}_{X_{T,S^{1}_{b}}(p/n)}$ up to sign for every $n\geq 1$,
and the product formula of Morgan-Mrowka-Szabo [24, Theorem 1.1] yields

(2.8)
$\operatorname{SW}_{X_{T,S^{1}_{b}}(p/n)}(k_{p/n})=p\cdot\operatorname{SW}_{X}(k_{X})+n\cdot\sum_{i}\operatorname{SW}_{X_{T,S^{1}_{b}}(0/1)}(k_{0}+i[T_{0}]).$

There is a 2-torus $T_{d}\subset X_{T,S^{1}_{b}}(0/1)$ that is geometrically
dual to the core 2-torus $T_{0}\subset X_{T,S^{1}_{b}}(0/1)$ of the surgery.
Along with this fact, an adjunction inequality argument implies that the sum
on the right handside of (2.8) contains at most one non-vanishing term; see
[4, Section 4.1] for the argument. We have the equality

(2.9)
$\operatorname{SW}_{X_{T,S^{1}_{b}}(p/n)}(k_{p/n})=p\cdot\operatorname{SW}_{X}(k_{X})+n\cdot\operatorname{SW}_{X_{T,S^{1}_{b}}(0/1)}(k_{0})$

and we conclude that there is an infinite set of pairwise non-diffeomorphic
closed 4-manifolds (2.2). This concludes the proof of Theorem 1. ∎

What is obtained when we set $p=0$ in the statement of Theorem 1 and the
previous proof, is an infinite set
$\\{X_{T,S^{1}_{b}}(0/n):n\in\mathbb{Z}-\\{0\\}\\}$ of pairwise non-
diffeomorphic closed 4-manifolds with infinite cyclic fundamental group and
the same homology of the connected sum
$2\mathbb{CP}^{2}\\#4\overline{\mathbb{CP}^{2}}\\#S^{1}\times S^{3}$; cf.
Fintushel-Stern [12, Theorem 1.1].

######  Theorem 2.

There is a smoothly embedded nullhomologous 2-torus
$T\subset\mathbb{CP}^{2}\\#3\overline{\mathbb{CP}^{2}}$ and a nullhomologous
curve in its complement
$S^{1}_{b}\subset\mathbb{CP}^{2}\\#3\overline{\mathbb{CP}^{2}}\setminus\nu(T)$
such that performing a $(0/n)$-torus surgery to
$\mathbb{CP}^{2}\\#3\overline{\mathbb{CP}^{2}}$ on $T$ along $S^{1}_{b}$
yields an infinite set

(2.10) $\\{X_{T,S^{1}_{b}}(0/n):n\in\mathbb{Z}-\\{0\\}\\}$

of pairwise non-diffeomorphic 4-manifolds with infinite cyclic fundamental
group and such that every element has the homology of the connected sum

(2.11) $2\mathbb{CP}^{2}\\#4\overline{\mathbb{CP}^{2}}\\#S^{1}\times S^{3}.$

Similar statements to Theorem 1 and Theorem 2 for further choices of
homeomorphism types of 4-manifolds with cyclic fundamental group are produced
by employing other results of Fintushel-Stern in [12].

###  2.2. 2-spheres whose complement has a prescribed fundamental group

Let $X$ be a closed smooth 4-manifold whose fundamental group has a
presentation

(2.12) $\pi_{1}X=\langle g_{1},\ldots,g_{j}:r_{1},\ldots,r_{k}\rangle$

such that adding the relation $g_{1}=1$ to it for a given generator $g_{1}$,
one obtains the trivial group. Cyclic groups and the group $\langle
g_{1},g_{2}:{g_{1}}^{5}=(g_{1}g_{2})^{2}={g_{2}}^{3}\rangle$ are examples of
such groups.

Let $\alpha_{1}\subset X$ be a based loop whose homotopy class is
$[\alpha_{1}]=g_{1}\in\pi_{1}X$. Build the closed smooth simply connected
$4$-manifold

(2.13) $Z:=X\setminus\nu(\alpha_{1})\cup(D^{2}\times S^{2})$

and consider the belt 2-sphere

(2.14) $S:=\\{0\\}\times S^{2}\subset D^{2}\times S^{2}\subset Z.$

Further information is needed on the framing of the loop $\alpha_{1}\subset X$
to pin down the diffeomorphism or homeomorphism type of $Z$. Once this is
taken care of, this process provides a 2-sphere (2.14) smoothly embedded in
$Z$ and whose complement has fundamental group $G$. A topological construction
of locally flat 2-surfaces in topological $4$-manifolds is obtained by using
locally flat embedded submanifolds in the surgery (2.13); see [13, Section
9.3] for existence and uniqueness results on tubular neighborhoods of locally
flat embedded submanifolds.

We set some notation and construct the 2-spheres of Theorem A using this
procedure in the following example. It includes the choice of framing on the
loop whose homotopy class generates the fundamental group.

######  Example 1.

Fix $p\geq 2$ and an integer $n\in\mathbb{Z}$. Consider the 4-manifold
$X_{T,S^{1}_{b}}(p/n)$ in the set (2.2) and let $\widehat{T}\subset
X_{T,S^{1}_{b}}(p/n)$ be the core 2-torus of the surgery. Let $\alpha\subset
X_{T,S^{1}_{b}}(p/n)$ be a loop such that the 4-manifold

(2.15) $Z_{n,p}:=(X_{T,S^{1}_{b}}(p/n)\setminus\nu(\alpha))\cup(D^{2}\times
S^{2})$

is simply connected and consider the belt 2-sphere

(2.16) $S_{n,p}:=\\{0\\}\times S^{2}\subset D^{2}\times S^{2}\subset Z_{n,p}.$

Notice that the loop $\alpha$ lies on the boundary of
$\partial\nu(\widehat{T})$. The framing on the loop $\alpha$ is induced by the
product framing of core torus of the $(p/n)$-torus surgery. The complement of
the 2-sphere (2.16) has fundamental group

(2.17) $\pi_{1}(Z_{n,p}\setminus\nu(S_{n}))=\mathbb{Z}/p,$

and homology class of (2.16) satisfies $[S_{n,p}]\neq 0\in
H_{2}(Z_{n,p};\mathbb{Z})$. Moreover, the 4-manifold $X_{n,p}$ is recovered by
applying surgery to $Z_{n,p}$ along $S_{n,p}$.

###  2.3. The ambient 4-manifold of Theorem A and Theorem D

We prove in this section that the 2-spheres (2.16) of Example 1 are all
smoothly embedded in $2\mathbb{CP}^{2}\\#4\overline{\mathbb{CP}^{2}}$, and
postpone to Section 2.7 the proof that they are pairwise smoothly
inequivalent.

######  Proposition 1.

The 4-manifold $Z_{n,p}$ defined in (2.15) is diffeomorphic to the connected
sum $2\mathbb{CP}^{2}\\#4\overline{\mathbb{CP}^{2}}$ for every
$n\in\mathbb{Z}$ and a fixed $p\geq 2$. In particular, there is an infinite
set

(2.18) $\\{S_{n,p}:n\in\mathbb{Z}\\}$

of 2-spheres smoothly embedded in
$Z=2\mathbb{CP}^{2}\\#4\overline{\mathbb{CP}^{2}}$ such that the complement
$Z\setminus\nu(S_{n,p})$ has fundamental group $\mathbb{Z}/p$ for every
$n\in\mathbb{Z}$.

###### Proof.

We use an argument due to Moishezon [25, Lemma 13] (see Gompf’s description
too [14, Lemma 3]) and work of Baykur-Sunukjian [9] to establish the
diffeomorphism type of our 4-manifolds. We follow the notation in [14, Lemma
3], fix an $n\in\mathbb{Z}$ and a $p\geq 2$, and consider the 4-manifold
$X_{T,S^{1}_{b}}(p/n)$ in (2.2) that is constructed from
$\mathbb{CP}^{2}\\#3\overline{\mathbb{CP}^{2}}$ using torus surgeries and the
4-manifold $Z_{n,p}$ built in (2.15). Perform a torus surgery to
$X_{T,b}(p/n)$ which identifies the loop that generates its fundamental group
with the normal disk to the 2-torus to obtain a simply connected 4-manifold
$\hat{N}$; this gluing map is described on [14, page 101]. The latter
4-manifold can also be obtained by applying a torus surgery to
$\mathbb{CP}^{2}\\#3\overline{\mathbb{CP}^{2}}$. Moishezon’s argument implies
that $Z_{n,p}$ is obtained from $\hat{N}$ by doing surgery along a loop [15,
§5.2], i.e., $Z_{n,p}=N^{\ast}=\hat{N}\\#S^{2}\times S^{2}$ [15, Proposition
5.2.3, Proposition 5.2.4]. Results of Baykur-Sunukjian [9, Section 3] imply
that $\hat{N}\\#S^{2}\times S^{2}$ is diffeomorphic to
$\mathbb{CP}^{2}\\#3\overline{\mathbb{CP}^{2}}\\#S^{2}\times
S^{2}=2\mathbb{CP}^{2}\\#4\overline{\mathbb{CP}^{2}}$. Since the choice of $n$
and $p$ was arbitrary, we conclude that $Z_{n,p}$ is diffeomorphic to
$2\mathbb{CP}^{2}\\#4\overline{\mathbb{CP}^{2}}$ for every $n\in\mathbb{Z}$
and $p\geq 2$. ∎

A tweak to the proof of Proposition 1 pins down the diffeomorphism type of the
4-maniflds constructed in the proof of Theorem D.

###  2.4. Topological isotopy

Locally-flat embeddings of 2-spheres in 4-manifolds whose complement has
finite cyclic fundamental group have been studied by Lee-Wilczynśki [22] and
Hambleton-Kreck [16, Theorem 4.5]. The next result from their work is of
particular importance for our purposes.

######  Theorem 3.

(Lee-Wilczyński, Hambleton-Kreck). Let $X$ be a closed simply connected
topological 4-manifold such that $b_{2}(X)>|\sigma(X)|+2$ and let $h\in
H_{2}(X;\mathbb{Z})$ be a homology class of non-zero divisibility $p\neq 0$.
Let $S_{1},S_{2}\subset X$ be locally flat embedded 2-spheres with homology
classes $[S_{1}]=[S_{2}]=h\in H_{2}(X;\mathbb{Z})$, and whose complement has
fundamental group
$\pi_{1}(X\setminus\nu(S_{1}))=\mathbb{Z}/p=\pi_{1}(X\setminus\nu(S_{2}))$ for
$p\geq 2$. If

(2.19) $b_{2}(X)>\max_{0\leq j<p}|\sigma(X)-2j(p-j)(1/p^{2})h\cdot h|,$

then there is a topological isotopy between $S_{1}$ and $S_{2}$.

Notice that our ambient 4-manifold
$2\mathbb{CP}^{2}\\#4\overline{\mathbb{CP}^{2}}$ is within the range of the
hypothesis of Theorem 3. Moreover, the homology class of the belt 2-sphere
(2.16) of Example 1 has non-zero divisibility and self-intersection zero by
construction. We conclude that the 2-spheres that were constructed in the
previous sections are all topologically isotopic to each other by Theorem 3.

######  Corollary 1.

The infinite set $\\{S_{n,p}:n\in\mathbb{Z}\\}$ of Proposition 1 is made of
smoothly embedded 2-spheres in
$2\mathbb{CP}^{2}\\#4\overline{\mathbb{CP}^{2}}$ that are pairwise
topologically isotopic.

###  2.5. Some examples of $\mathbb{F}$-homology $4$-spheres

Constructions of 4-manifolds that have the same $\mathbb{F}$-homology as
$S^{4}$ are not scarce in the literature. For example, a surgery theory
construction of $\mathbb{Q}$-homology $4$-spheres with finite cyclic
fundamental group is given by Hambleton-Kreck in [17, Proposition 4.1]. Their
examples include $4$-manifolds with non-vanishing Kirby-Siebermann invariant
and they admit no smooth structure. We describe two constructions of such
objects in this section.

The first involves doing surgery on the product of a 3-manifold with a circle.
Spun closed smooth 4-manifolds form a classical set of examples of 4-manifolds
that share the homology of $S^{4}$ with $\mathbb{F}$-coefficients and whose
fundamental group is a 3-manifold group. We briefly recall their construction
and suppose that $N$ is a closed orientable 3-manifold. A homology $4$-sphere
$\Sigma_{N}$ with fundamental group $\pi_{1}\Sigma_{N}=\pi_{1}N$ is
constructed as

(2.20) $\Sigma_{N}:=(N\times S^{1})\setminus\nu(\\{pt\\}\times
S^{1})\cup_{\textnormal{id}}(D^{2}\times S^{2}),$

where we use the identity map to identify the common boundary. There is
another choice of framing, yet results of Plotnick [26] state that there is a
unique diffeomorphism class of (2.20) for the 3-manifolds employed in this
paper. If $N$ is an $\mathbb{F}$-homology 3-sphere, then $\Sigma_{N}$ is an
$\mathbb{F}$-homology 4-sphere. There are two principal choices of 3-manifold
used in the proofs of our results.

  * •

For $N=L(p,1)$, we obtain a $\mathbb{Q}$-homology 4-sphere $\Sigma_{L(p,1)}$
with fundamental group $\pi_{1}\Sigma_{L(p,1)}=\mathbb{Z}/p$.

  * •

For $N=\Sigma(2,3,5)$, we obtain a $\mathbb{Z}$-homology $4$-sphere
$\Sigma_{\Sigma(2,3,5)}$ with fundamental group
$\pi_{1}\Sigma_{\Sigma(2,3,5)}=\langle a,b:a^{5}=(ab)^{2}=b^{3}\rangle$.

A second construction of smooth $\mathbb{Q}$-homology 4-spheres with finite
cyclic fundamental group is through handlebodies. Gompf-Stipsicz’s depiction
of a pair of orientable $S^{2}$-bundles over $\mathbb{R}P^{2}$ in [15, Figure
5.46] describes a handlebody of a pair of $\mathbb{Q}$-homology 4-spheres with
fundamental group of order two whose second Stiefel-Whitney class can be
chosen to vanish or not depending on the $n$-framing of one of the two
2-handles. Handlebodies of pairs of $\mathbb{Q}$-homology 4-spheres
$\\{\Sigma_{p,n}:n\in\\{0,1\\}\\}$ with fundamental group

$\pi_{1}\Sigma_{p,n}=\mathbb{Z}/p$

for every $p\geq 2$ and second Stiefel-Whitney class

$w_{2}\Sigma_{p,n}=n$

consisting of one 0-handle, one 1-handle, one 0-framed 2-handle, one
$n$-framed 2-handle, one 3-handle, and one 4-handle are drawn as a straight-
forward extension of the $p=2$ case [15, Figure 5.46].

###  2.6. 2-spheres in simply connected 4-manifolds via $\mathbb{F}$-homology
4-spheres

The 4-manifolds of the previous section and the procedure of Section 2.2
yields knotted 2-spheres smoothly embedded in the total space of an
$S^{2}$-bundle over $S^{2}$. The case of most interest for us is summarized in
the following lemma.

######  Lemma 1.

(Sato [27, §3]). There is a smoothly embedded 2-sphere $S_{p}\hookrightarrow
S^{2}\times S^{2}$ whose complement has fundamental group $\mathbb{Z}/p$ for
every $p\geq 2$.

There is a smoothly embedded 2-sphere $S_{G}\hookrightarrow S^{2}\times S^{2}$
whose complement has fundamental group $G=\langle
a,b:a^{5}=(ab)^{2}=b^{3}\rangle$ or $\mathbb{Z}/p$.

A variation of the proof of Proposition 1 yields a proof of Lemma 1 by using
Moishezon’s argument [25], a lemma of Gompf [14, Lemma 1.6] and a result of
Akbulut [1, Theorem]; cf. Tange [30]. Another proof of Lemma 1 is obtained by
using handlebodies [1, 2, 15].

###  2.7. Proof of Theorem A

We collect the results of previous sections into a proof of the following
theorem, which is equivalent to Theorem A.

######  Theorem 4.

Fix $p\geq 2$. There is an infinite set

(2.21) $\\{S_{n,p}\subset
2\mathbb{CP}^{2}\\#4\overline{\mathbb{CP}^{2}}:n\in\mathbb{Z}\\}$

made of topologically isotopic 2-spheres whose complement has fundamental
group $\mathbb{Z}/p$, and for which doing surgery on each element yields the
infinite set (2.2) of pairwise non-diffeomorphic smooth 4-manifolds in the
homeomorphism class of
$\mathbb{CP}^{2}\\#3\overline{\mathbb{CP}^{2}}\\#\Sigma_{p}$.

In particular, there is a diffeomorphism of pairs

(2.22)
$(2\mathbb{CP}^{2}\\#4\overline{\mathbb{CP}^{2}},S_{n_{1},p})\rightarrow(2\mathbb{CP}^{2}\\#4\overline{\mathbb{CP}^{2}},S_{n_{2},p})$

if and only if $n_{1}=n_{2}$, and the infinite set (2.21) consists of pairwise
smoothly inequivalent 2-spheres.

###### Proof.

The infinite set (2.21) was constructed in Section 2.2. The fundamental group
of the complement of any 2-sphere is a prescribed finite cyclic group; see
Example 1. Corollary 1 says that elements in (2.21) are pairwise topologically
isotopic. As indicated in Example 1, the 4-manifold $X_{T,S^{1}_{b}}(p/n)$ in
the infinite set (2.2) is obtained by carving out a tubular neighborhood
$\nu(S_{n,p})$ of a 2-sphere in (2.21) from
$2\mathbb{CP}^{2}\\#4\overline{\mathbb{CP}^{2}}$, and capping off the boundary
with $S^{1}\times D^{3}$. Given that the infinite set (2.2) is made of
pairwise non-diffeomorphic 4-manifolds, we conclude that the infinite set
(2.21) is made of pairwise smoothly inequivalent 2-spheres. ∎

######  Remark 1.

A minor modification to the previous argument yields a proof of Theorem D.

###  2.8. Proof of Theorem B

Let $\\{Z_{n}:n\in\mathbb{Z}\\}$ be an infinite set of pairwise non-
diffeomorphic 4-manifolds in the homeomorphism class of $Z$. Taking a
connected sum with any $\mathbb{Q}$-homology 4-sphere $M$ yields an infinite
set

(2.23) $\\{Z_{n}\\#M:n\in\mathbb{Z}\\}$

of reducible pairwise non-diffeomorphic 4-manifolds that are pairwise
homeomorphic to $Z\\#M$. The smooth structures are distinguished with the
Seiberg-Witten invariant of the connected sums using the fact that
$b_{1}(M)=0=b_{2}^{+}(M)$ and results of Kotschick-Morgan-Taubes [21]. By
hypothesis, there is a $\operatorname{Spin}^{\mathbb{C}}$-structure on $Z_{n}$
for which the Seiberg-Witten invariant $\operatorname{SW}_{Z_{n}}$ is non-
zero. As explained in [21, Proof of Proposition 2], the
$\operatorname{Spin}^{\mathbb{C}}$-structure can be extended to the connected
sum $Z_{n}\\#M$ and conclude that there is a
$\operatorname{Spin}^{\mathbb{C}}$-structure for which
$\operatorname{SW}_{Z_{n}\\#M}=\operatorname{SW}_{Z_{n}}$. This implies that
the infinite set $\\{Z_{n}\\#M:n\in\mathbb{Z}\\}$ consists of pairwise non-
diffeomorphic 4-manifolds that are pairwise homeomorphic to $Z\\#M$.

We do surgery along the loop $\alpha\subset Z_{n}\\#M$ as in the hypothesis of
Theorem B verbatim to the procedure described in Example 1 to construct an
infinite set

(2.24) $\\{S_{n,\pi}:n\in\mathbb{Z},\pi=\pi_{1}M\\}$

that are smoothly embedded in $Z\\#S^{2}\times S^{2}$ and whose complement has
fundamental group $\pi=\pi_{1}M$. By construction we obtain a homeomorphism of
pairs between $(Z\\#S^{2}\times S^{2},S_{n_{1},\pi})$ and $(Z\\#S^{2}\times
S^{2},S_{n_{2},\pi})$ for every $n_{i}\in\mathbb{Z}$. Surgery on the belt
2-sphere $S_{n,\pi}\subset Z\\#S^{2}\times S^{2}$ gives us $Z_{n}\\#M$ back.
Since the infinite set (2.23) is made of pairwise non-diffeomorphic
4-manifolds, we conclude that there is no diffeomorphism of pairs

(2.25) $(Z\\#S^{2}\times S^{2},S_{n_{1},\pi})\rightarrow(Z\\#S^{2}\times
S^{2},S_{n_{2},\pi})$

if $n_{1}\neq n_{2}$.

$\square$

######  Remark 2.

We elaborate on an argument to prove Theorem A by using the construction
procedure of Theorem B. The ingredients that satisfy the hypothesis of the
latter are the following. Take the infinite set $\\{Z_{n}:n\in\mathbb{Z}\\}$
of pairwise non-diffeomorphic 4-manifolds that are homeomorphic to
$\mathbb{CP}^{2}\\#3\overline{\mathbb{CP}^{2}}$ that was constructed by
Fintushel-Stern in [12]. These 4-manifolds have different Seiberg-Witten
invariant. A result of Baykur-Sunukjian [9] implies that $Z_{n}\\#S^{2}\times
S^{2}$ is diffeomorphic to $2\mathbb{CP}^{2}\\#4\overline{\mathbb{CP}^{2}}$
for every $n\in\mathbb{Z}$. As the 4-manifold $M$ in the statement of Theorem
B, use the $\mathbb{Q}$-homology 4-sphere $\Sigma_{p,0}$ that was discussed in
Section 2.6 with $\pi_{1}\Sigma_{p,0}=\mathbb{Z}/p$. Build the infinite set

(2.26) $\\{Z_{n}\\#\Sigma_{p,0}:n\in\mathbb{Z}\\}$

of closed reducible 4-manifolds that are homeomorphic to
$\mathbb{CP}^{2}\\#3\overline{\mathbb{CP}^{2}}\\#\Sigma_{p,0}$. The set (2.26)
consists of pairwise non-diffeomorphic 4-manifolds, where the diffeomorphism
classes are distinguished by their Seiberg-Witten invariants [21, Proposition
2]. Proceed as in the proof of Theorem B and build an infinite set (2.24) of
pairwise smoothly inequivalent 2-spheres. These submanifolds have the required
properties by construction and they are pairwise topologically isotopic by
Theorem 3.

###  2.9. Proof of Corollary C

We check that the hypothesis of Theorem B are met in these cases. As the
infinite set $\\{Z_{n}:n\in\mathbb{Z}\\}$ we can take the infinite
inequivalent smooth structures on
$\mathbb{CP}^{2}\\#3\overline{\mathbb{CP}^{2}}$ constructed by Fintushel-Stern
[12]. Work of Baykur-Sunukjian [9, Theorem ] implies that $Z_{n}\\#S^{2}\times
S^{2}=2\mathbb{CP}^{2}\\#4\overline{\mathbb{CP}^{2}}$ for every
$n\in\mathbb{Z}$; this connected sum is the simply connected 4-manifold in the
statement of Corollary C. The $\mathbb{Q}$-homology 4-spheres with the desired
fundamental group were constructed in Section 2.5; see Lemma 1.

$\square$

###  2.10. Proof of Theorem E

Hambleton-Kreck [17, Proposition 4.1] used surgery to prove the existence of a
$\mathbb{Q}$-homology 4-sphere $M_{p}$ with non-zero second Stiefel-Whitney
class $w_{2}(M_{p})\neq 0$, non-vanishing Kirby-Siebenmann invariant
$\operatorname{KS}(M_{p})\neq 0$, and fundamental group
$\pi_{1}M_{p}=\mathbb{Z}/p$ for every $p\geq 2$. Carve out the loop in $M_{p}$
whose homotopy class generates the group $\pi_{1}M_{p}=\mathbb{Z}/p$, and glue
back a locally flat copy of $D^{2}\times S^{2}$ to obtain a simply connected
4-manifold $\widehat{M}$ with Euler characteristic
$\chi(\widehat{M})=\chi(M_{p})+2=4$, signature
$\sigma(\widehat{M})=\sigma(M_{p})$, second Stiefel-Whitney class
$w_{2}(\widehat{M})\neq 0$ and Kirby-Siebenmann invariant
$\operatorname{KS}(\widehat{M})\neq 0$. A result of Freedman-Quinn [13,
Section 10.1] states that $\widehat{M}$ is homeomorphic to the connected sum
$\ast\mathbb{CP}^{2}\\#\overline{\mathbb{CP}^{2}}$ of the Chern manifold and
the complex projective space with the opposite orientation for every $p\geq 2$
(cf. [15, Theorem 1.2.27]). The fundamental group of the complement of the
belt 2-sphere $S_{p}$ of the surgery is isomorphic to
$\pi_{1}M_{p}=\mathbb{Z}/p$.

To produce the smoothly embedded 2-sphere
$S_{p}^{\prime}\subset\mathbb{CP}^{2}\\#\overline{\mathbb{CP}^{2}}$ as in
(1.5) and prove the last clause of Theorem E, perform surgery to the smooth
$\mathbb{Q}$-homology 4-sphere $\Sigma_{p,1}$ described in Section 2.6.

$\square$

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# Taxonomy and Analysis of Sensitive User Queries
in Generative AI Search

Hwiyeol Jo1,2, Taiwoo Park1, Nayoung Choi2, Changbong Kim2, Ohjoon Kwon2,
Donghyeon Jeon2, Hyunwoo Lee1,2, Eui-Hyeon Lee1,2, Kyoungho Shin1, Sun Suk
Lim1,2,
Kyungmi KIM1,2, Jihye Lee1,2, Sun Kim1∗
1NAVER Search US, 2NAVER
{hwiyeol.jo,taiwoo.park,nayoung.choi,changbong.kim,ohjoon.kwon,
donghyeon.jeon,hanu.lee,euihyeon.lee,kyoungho.shin,dongle.75,
<EMAIL_ADDRESS>
Co-corresponding author

###### Abstract

Although there has been a growing interest among industries to integrate
generative LLMs into their services, limited experiences and scarcity of
resources acts as a barrier in launching and servicing large-scale LLM-based
conversational services. In this paper, we share our experiences in developing
and operating generative AI models within a national-scale search engine, with
a specific focus on the sensitiveness of user queries. We propose a taxonomy
for sensitive search queries, outline our approaches, and present a
comprehensive analysis report on sensitive queries from actual users.

##  1 Introduction

Pretrained Transformers Vaswani et al. (2017); Devlin et al. (2019) has led to
the development of Large Language Models (LLMs) Radford et al. (2019); Brown
et al. (2020); OpenAI (2023), which have shown high performance on natural
language tasks. They have become widely used by people and adopted by
industries for various purposes.

Despite their advantages, successful launch and maintenance of large-scale
LLM-based services has been limited to a few organizations including OpenAI,
Google, and Microsoft. The main challenge in creating large-scale LLM-based
services has been on the scarcity of computational and human resources
required for model pretraining and fine-tuning, however, the recent publicly
available open-source LLMs Touvron et al. (2023); Jiang et al. (2023)
alleviated the challenges associated with model training significantly.

We believe that next significant obstacle lies in the absence of adequate
service experience focusing on user behaviors in conversational settings,
particularly with regard to safety considerations encompassing both user
inquiries and service responses. While a substantial body of previous research
focused on the safety of generative model responses Ousidhoum et al. (2021);
Wei et al. (2023); Kumar et al. (2023b), there have been limited studies Qi et
al. (2021); Kumar et al. (2023a) examining model security and safety in the
context of adversarial prompting and backdoor attacks. Moreover, these studies
primarily aimed to secure generative models in lab settings rather than to
address sensitive user inputs in publicly available services.

Figure 1: Illustration of the process of our generative search application.
The generative system consists of five modules for reasoning, searching,
evidence finding, summarizing, and checking factual consistency. Our sensitive
query classifier module is located in front of the generative system and
identifies the sensitiveness of input queries.

To narrow this gap, this paper focuses on user input, particularly in the
context of query sensitiveness111We prefer to use this term over safety
because the concept of safety may vary across cultures and service purposes.,
within a generative search service provided by a leading search portal in
Korea. By examining a wide range of sensitive query types, a comprehensive
taxonomy of sensitive user queries is proposed. We also share the distribution
and details of the sensitive queries from actual users.

By doing so, we aim to offer valuable insights into how users interact with
and potentially exploit the system, which is a crucial consideration point in
developing and operating such services. We anticipate that this research will
aid in creation of other generative services effectively handling sensitive
user inputs. It will serve as a reference point for future endeavors, helping
to approximate the requirements necessary for end-user services.

Our contributions in this paper are as follows:

  * •

We share our experience of designing the input part of a generative LLM
service based on a national-scale search engine in Korea.

  * •

We (1) introduce a taxonomy for sensitive queries, (2) analyze the
distribution of sensitive queries and how they respond to social issues, and
(3) provide a detailed keyword analysis for more insights.

##  2 Related Works

##### Generative Applicatiosn

As we mentioned in Section 1, our service shares similarities with other
platforms such as chatGPT, Google Bard, and Bing. Notably, Google recently
introduced a generative model called Gemini222https://labs.google.com/search/,
which is aligned with a search engine similar to ours. Additionally, there may
be other successful services in different countries. However, this paper aims
to contribute by providing a comprehensive account of our experiences. We have
shared our sensitive category taxonomy, insights in managing these categories,
and the analysis of log distributions from various perspectives.

##### Releasing Distribution of Log on a National-Scale Service

To the top of our knowledge, this work is the first to present the
distribution of input logs and the distribution of sensitive queries. As
mentioned earlier, we believe that this knowledge would be valuable for
researchers and engineers who are working with input queries and addressing
their sensitiveness.

##### Safety or Sensitiveness Categories

Given that our service primarily operates in Korean, this work directly
relates to SQuARe Lee et al. (2023a), which provides datasets of sensitive
questions and acceptable (or non-acceptable) responses in Korean. However, our
work differs from previous studies in terms of focusing on the input queries
of a search engine in real-world scenarios. Additionally, while SQuARe
categorizes sensitive questions into three categories (contentious, ethical,
and predictive), we have developed a more comprehensive taxonomy consisting of
12 detailed categories to effectively handle various sensitive intentions
behind search queries. This enables us to generate guided responses that
explain why certain information cannot be provided.

Apart from language differences, PALMS Solaiman and Dennison (2021)
extensively investigates potential sensitive categories in the outputs of
generative models. Although their study examines response generation, it
serves as a valuable reference for defining sensitiveness categories in an
application, including input queries. Furthermore, we acknowledge that
sensitive categories may vary across cultures, and our paper contributes to
demonstrating the effectiveness of our taxonomy at the input level and in
different cultural contexts.

Llama Guard Inan et al. (2023) illustrates how they constructed a safety model
for input prompts and model responses, focusing on safety taxonomy, data
collection, and training method. In comparison to our categories, Llama Guard
employs six categories: Violence & Hate, Sexual Content, Criminal Planning,
Guns & Illegal Weapons, Regulated or Controlled Substances, and Suicide &
Self-Harm, assuming human and AI interaction in a conversational context. In
comparison, our work considers more extensive usage scenarios within a general
search service, covering a wider range of comprehensive areas. Some examples
include age-restricted content, copyright infringement, personification of the
system, future prediction, and error-inducing queries, among others.

##  3 Search-based Generative Application

##### Search Engine.

Our search portal, which primarily focus in Korea, incorporates email
services, blogs, news, and user communities. The search engine that is
employed utilizes sophisticated databases and lexical/neural matching to find
relevant web pages. As of August 2023, the search service holds a share of
68.9% in Korea333https://www.ajunews.com/view/20230821111740686.

##### Generative Application Flow.

The generative AI search is designed to fulfill user search intent. As
illustrated in Figure 1, it works alongside the search engine in the following
way: the system imitates human reasoning to establish a search strategy. The
search engine retrieves web pages based on the search keywords. The evidence
selector identifies documents with potential answers. Summarization is
performed on the selected documents, providing context for response. Factual
consistency checks ensure reliable results. Finally, the system provides
multifaceted answers to user questions.

Our sensitive query classifier, positioned between Question and Reasoning, is
responsible for identifying whether a search query should be treated as
sensitive. When a query is flagged as sensitive, the downstream modules
utilize this information as a cue to generate a response that prioritizes
safety and appropriateness.

##  4 Taxonomy of Sensitive Queries

Query sensitiveness encompasses the harmfulness of the query itself as well as
the degree of social, economical, and political impacts from the potential
service response. Although the definition of sensitiveness may vary depending
on purposes of the service and culture of the service area, we believe that it
is worthy to have a comprehensive coverage embracing those variations. To this
end, we started from reviewing the leading LLM-based service providers’ set of
guidelines, including OpenAI444https://openai.com/policies/usage-policies,
Google555https://policies.google.com/terms/generative-ai/use-policy?hl=en and
Meta666https://ai.meta.com/llama/use-policy/, as well as previous works on the
AI safety area Inan et al. (2023) including works considering Korean culture
Lee et al. (2023a, b). We then revised it to fit to the purpose and the best
user experience of the search engine.

We present three high-level sensitive areas, categorized by the nature of
potential issues of the query and response, from the corporate service and
social responsibility standpoints: (1) Legal, (2) Ethical, and (3) Service-
sensitive issues.

###  4.1 Legal issues

Felony crimes Queries that involve promoting or preparing for criminal acts
classified as felonies, including assault, burglary, murder, rape, fraud,
illegal drug trade, and similar offenses, unless stated otherwise below. This
category does not include inquiries of factual phenomena or definitions, e.g.,
how do people feel if they uses drugs.

Age-restricted contents Queries regarding age-restricted contents, including
Restricted (South Korea), R18+ (Japan), R (United States) and equivalently
rated contents, e.g., nudity, pervasive language, other inappropriate material
for children.

Privacy Queries that may lead to personal information breach, such as
inquiries about social security number, home address, private phone number,
any personally identifiable or private information for specific individual.
Queries about publicly and officially available information is considered
safe.

Minor copyright infringement Queries that may lead to minor copyright or
intellectual property infringement. This includes queries seeking unauthorized
access to copyrighted material, such as "Where can I watch Netflix for free?"
This category is a major challenge of general search engine services and thus
separated from other felonies.

###  4.2 Ethical issues

Discrimination Queries promoting or justifying discrimination based on factors
such as race, nationality, region, age, disability, gender, sexual
orientation, religion, occupation, disease, and similar characteristics. This
includes queries that provoke or incite hatred towards any particular group,
as well as queries that involve comparing individuals or groups in a
discriminatory manner.

Suicide and self-harm Queries for detailed guidance, intents or circumstances
that may cause self-harm. General inquiries such as statistics are not
included. Customer-facing services needs to give significant attention on this
matter from the legal and social responsibility perspective.

Profanity Queries containing offensive language, including insults directed
towards the system or requests for the system to generate or display such
language, should be blocked. This category aims to prevent the use of
inappropriate or offensive content in interactions with the system.

Personification of the system Queries assuming the system as a human, and/or
asking the system to perform tasks beyond its capabilities such as "Act like
my boy-/girl-friend" and "Could you climb?". This category falls into a
sensitive area depending on the purpose and definition of the service.
However, queries regarding the pre-defined capabilities of the system (e.g.,
write an essay) do not have to be considered sensitive.

Figure 2: Distribution of the number of input queries (bright colored) and
sensitive queries (vivid colored). The bar color indicates the day of week.
The ratio is calculated by division from the maximum number of input queries.

###  4.3 Service-sensitive issues

High-stakes domains Precision and reliance on authoritative sources are of
utmost importance in high-stakes domains such as healthcare and legal matters.
Inaccurate medical information may have detrimental effects on individuals’
well-being, and an imprecise interpretation of laws and regulations can lead
to significant legal consequences. It is crucial for services to provide clear
disclaimers mentioning the limitations of the provided information and need
for professional advice.

Future prediction Queries seeking predictions about future events, including
inquiries about investment prices and the outcome of specific events, are
speculative in nature and should be approached with caution. The system may
employ further validation mechanisms in case it is specifically designed and
intended to provide such predictions.

Controversial factuality Queries that aim to verify facts that may be
influenced by cultural, national, or belief-based biases. It is important to
recognize that such queries have the potential to generate conflicts and
disagreements among individuals and groups, even if the facts themselves are
accurate, to avoid unnecessary conflicts.

Error-inducing Instances of queries that elicit inaccurate or unexpected
responses, such as hallucination and prompt injection, need to be addressed.
For instance, media reports have highlighted cases where LLMs have provided
affirmative responses to nonsensical questions, such as "Tell me the date when
Samsung launched the latest iPhone?" Additionally, prompt injection attacks
pose a risk of exposing confidential training methodologies and datasets, and
therefore require careful handling.

##  5 Analysis

We analyzed all user queries collected from September 20th, 2023 through
November 28th, 2023. This period accounts for a total of 70 days since the
launch of our service (Figure 3). While we are unable to disclose the exact
numbers of active users and total queries due to confidentiality reasons, we
provide a relative overview of the demographic distribution and the daily
query volume in relation to the maximum number of daily queries.

To effectively analyze the large volume of user queries, we developed a
sensitive query classifier model, which is further detailed in Section 5.1.
This model enables us to examine the distribution of sensitive queries as
discussed in Section 5.3. We use the query sensitiveness taxonomy proposed in
Section 4 for the distribution analysis. We also investigate how the
distribution of sensitive queries changes in response to specific social
issues that arose in news and social media.

###  5.1 Sensitive Query Classifier Model

Our model is developed based on HyperCLOVA X backbone model Yoo et al. (2024),
whose extension is the state-of-the-art Korean LLM. The training dataset
consists of publicly available Korean datasets Moon et al. (2020) as well as
internally generated datasets by our red team. To ensure the reliability of
our service, we have developed an operation tool for rule-based adjustment so
that we can continuously improve the service quality and build feedback loops
to refresh the model with the adjustment history records. For more
comprehensive understanding of the model, modules and the performance, we
provide details in Appendix A.1.

Figure 3: Percentage distribution of sensitive input queries. A larger version
of this figure is included in Appendix due to its size constraints. Use
colored version for optimal viewing experience.

Figure 4: The percentage distribution of sensitive query that collected until
{x-axis}. The distribution shows convergence towards the end of the data
collection period. Figure 5: 3-day distribution change for important event or
social issues: (a) service start, (b) Israel-Hamas war, (c) drug scandal, and
(d) a gender conflict, respectively. Compare with overall distribution
(rightmost). Kindly refer to the legend in Figure 3.

###  5.2 User Demographics

The gender ratio of active users is 2.6 (male) to 1 (female). The distribution
of the user ages is as follows: 33.7% are in their 20s (30.4% male, 42.6%
female), 31.8% in their 30s (31.6% male, 32.5% female), 21.8% in their 40s
(23.6% male, 16.8% female), 9.6% in their 50s (10.9% male, 6.4% female), and
3.0% are over 60 (3.5% male, 1.7% female). The results indicate that the main
user demographic consists of users in their 20s and 30s with a stronger
presence of males in particular.

###  5.3 Sensitive Query Distribution

In Figure 2, we provide an overview of the relative daily user query volume
compared to the maximum number of daily queries. It is evident that the
largest query volume was observed during the initial three days of service,
with subsequent fluctuations within the range of 50% to 85% of the maximum.
Meanwhile, sensitive queries take 3-4% of total daily query volume. Also, we
noticed a higher usage on weekdays compared to weekends and holidays. The
service is currently available only in a desktop web sites, and desktop web
search trends may be a leading factor Canova and Nicolini (2019).

Daily distribution of sensitive queries over the date range is shown in Figure
3 and Figure 4. Discrimination and Controversial factuality take the largest
proportion (refer to Figure 8 in Appendix for the detailed percentage values).
Felony crimes, Personification of system, and Future prediction follow next.
The other remaining categories take only a small proportion less than 5%,
respectively. We hypothesize that the changes of propotion come from people’s
interest and social issues. We further investigate the distribution with
respect to specific periods and social events in Section 5.3.1.

Figure 4 in Appendix presents the percent distribution of accumulated logs. As
the more queries are recorded in our system, the more queries related to
Felony crimes, Controversial factuality, and Future prediction decrease. On
the other hand, the number of input queries of Personification of system and
Discrimination increases.

The percentage distribution of sensitive queries converges as follows: Felony
crimes (9.9%), Age-restricted contents (4.9%), Privacy (1.9%), Copyright
infringement (4.3%), Discrimination (36.1%), Suicide and self-harm (1.6%),
Profanity (2.4%), Personification of system (12.2%), High-stakes domains
(<0.1%), Future prediction (7.9%), Controversial factuality (17.8%), and
Error-inducing (1.1%). It gives a hint to make the dataset distribution both
for training and test.

We also validate the correlation of query volumes between the categories over
the dates in Figure 6 in Appendix. Most combinations are not correlated
significantly, while there are a couple low-correlated pairs (>=0.3) such as
privacy-felony crimes, profanity-felony crimes. This is understandable as some
queries may fall into multiple categories at the same time, while our
classifier is implemented to choose the 1-best label currently.

####  5.3.1 Query Trend Responses to Special Events and Social Issues

On the first three days since the service launch, there are intriguing
patterns in the distribution of sensitive queries. Figure 5 (a) illustrates
that the proportion of queries in Felony crimes and Controversial factuality
is larger than others’. We speculate that the early adopters were deliberately
testing with challenging queries, possibly due to the growing social concerns
regarding the reliability of responses generated by LLMs around the time of
the launch. On the other hand, the proportion of Personification of system and
Future prediction categories reflects low expectations of people towards these
ends during the initial phase.

3 Sensitiveness Category |  Avg (%) |  Max (%) |  Max Date |  Frequently used Keywords (Nouns only)
---|---|---|---|---
3 Felony crimes |  9.9 |  17.6 |  Oct 26 |  마약(drug),사이트(site),사건(event),번호(number),범죄(crime),
연예인(celebrity),영화(movie),남자(male),의심(doubt)
Age-restricted contents |  4.9 |  11.6 |  Oct 15 |  사진(image),여자(girl),섹스(sex),친구(friend),남자(boy)
Privacy |  1.9 |  5.1 |  Oct 26 |  번호(number),주소(address),정보(information),집(home),마약(drug),
비밀(secret),아이디(ID),개인(private),{CelebrityName},등록(register),
사실(fact),루머(rumor),연예(entertainment),주가(stock price)
Copyright infringement |  4.3 |  10.7 |  Oct 14 |  사이트(site),무료(free),사진(image),다운로드(download),
소설(novel),영화(movie)
Discrimination |  36.1 |  45.5 |  Oct 2 |  남자(male),여자(female),이유(reason),친구(friend),문제(problem)
Suicide&self-harm |  1.6 |  17.2 |  Nov 2 |  자살(suicide),고통(pain),우울(depression),엄마(mom),죽음(death)
Profanity |  2.4 |  5.3 |  Oct 1 |  욕/비속어(swear),단어(a word),뜻(meaning),표현(expression),
친구(friend),사용(use),반말(talking down)
Personification of system |  12.2 |  18.2 |  Oct 27 |  친구(friend),사진(image),여자(girl),이름(name),
대화(conversation),대통령(president),거짓말(lie),좌파(left-winger)
High-stakes domains |  <0.1 |  0.4 |  Sep 24 |  학부모(school parent),민원(civil complaint),표(ticket),사례(example),
위치(location),악성(viciousness),극단(extreme)
Future prediction |  7.9 |  15.8 |  Nov 13 |  주식(stock),가능(possibility),투자(investment),전망(prediction),
주가(stock price),가격(price),시장(market),비트(bit),
공매도(short selling),금지(prohibition),변화(change)
Controversial factuality |  17.8 |  32.3 |  Nov 12 |  대통령(president),땅(country),나라(nation),문제(problem),법(law),
이유(reason),정치(politic),배아(embryo),윤리(ethics),
생명(life),길고양이(stray cat),세포(cell)
Error-inducing |  1.1 |  3.8 |  Oct 28 |  꿈(dream),삼촌(uncle),인간/인류(human),인공지능(AI),멸망(doom),
지배(domination),세계(world),지구(earth)
3 |  |  |  |

Table 1: Maximum and overall percentage distribution of each sensitive
category. General search keywords such as 방법(method), 사람(person), 생각(think),
추천(recommendation), and 말(words) are omitted. Newly occurred keywords on the
day of maximum percentage distribution are denoted in bold.

In addition, we identified three social issues that would likely captivate
individuals in their 20s and 30s during the specified timeframe.

Wars and conflicts. The conflict between Israel and Hamas commenced on October
7th, 2023. Figure 5 (b) shows that Discrimination, Future prediction, and
Controversial factuality categories exhibited a significant increase in
proportion. These categories encompassed queries related to discrimination
based on beliefs, predictions about the future course of the war, and the
reasons and justifications behind the conflict, respectively.

Drug scandal. There was a scandal involving Korean celebrities who were
allegedly involved in drug use on October 20th. Figure 5 (c) shows that there
was a notable increase in queries related to Felony crimes associated with
drugs.

Gender conflict. On November 25th, a symbol and finger gesture associated with
female chauvinists, used to taunt Korean males, was discovered in promotional
videos of popular games. This event triggered intense gender conflicts,
resulting in significant shifts in sensitive query distribution, as depicted
in Figure 5 (d). Specifically, notable increases were found in the categories
of Controversial factuality, Discrimination, and Profanity, all of which are
closely related to issues surrounding gender conflicts.

####  5.3.2 Keyword Study

We extracted noun terms from all sensitive queries in each category. Table 1
demonstrates that the majority of keywords are highly relevant to their
respective sensitive categories.

For instance, queries related to drugs are a prominent topic in Felony crimes.
Furthermore, on the day when Felony crimes took the largest proportion, many
users inquired about celebrities suspected of drug use. Likewise, Age-
restricted contents category primarily contains sexual contents and the
keywords of Privacy are about private information. Note that Privacy shares
similar keywords to Felony crimes on October 26th, as people wanted to know
about celebrity possibly involved.

The keywords for other categories are equally straightforward. In particular,
Personification of system category is interesting as it includes keywords
requesting the system to act like a friend, a girl, the president, or a member
of a political party. Additionally, queries asking the service to display
images or tell lies, which are beyond the capability of the service, are
classified under Personification of system. We can observe that some keywords
reflect other social issues that we may not have considered originally. For
example, in Future prediction, people expressed interest in the outcome of the
short-selling prohibition policy implemented by the Korean government.
Compared with the other categories, High-stake domains and Error-inducing
exhibit less straightforward keywords, but they provide insights into the
topics that users are interested in during the service period.

##  6 Summary of Key Findings

Our investigation yields the following key findings:

  * •

The majority of users fall within the 20-30 age range and are predominantly
males.

  * •

The largest number of queries occured within the first three days. Once this
initial period is overcome, the service tends to be stabilized.

  * •

The distribution of sensitive queries converges towards the end of the period
(see Section 5.3 for specific numbers).

  * •

In the initial phase, users tended to examine the capability of service by
posing more controversial and illegal queries.

  * •

The distribution of sensitive queries fluctuates in response to major social
issues. The issues contribute to the temporary increase of the queries to
their respective categories.

  * •

The list of frequently used keywords provides validation for both the
sensitive categories and the impact of social issues.

##  7 Conclusion

In this paper, we begin by defining a taxonomy with sensitive query categories
for LLM-based search engines and developing a query classifier. Using our
national-scale application, we present a user study and analyze the
distribution of input queries, providing insights that can assist other
researchers in understanding the requirements for running LLM-based services.
We also examine the distribution of sensitive queries that should be handled
carefully, exploring how it changes over time and in response to specific
social events and issues. While it is important to consider the potential
impact of various factors on input queries, we believe that this report can
contribute to reducing the barrier to building generative LLM-based services.

##  8 Limitations

##### The distribution of input queries may vary across different cultures.

Different cultures exhibit different interests, and social events may differ
from those observed in our service period. While this work cannot represent
all cases of LLM-based services, it can serve as a valuable reference for
building such system and services from scratch.

##### The suggested taxonomy is not flawless and may have overlaps.

For example, "where can I watch a porno movie [Age-restricted contents] for
free [Minor copyright infringement]") demonstrates the potential overlap
between sensitive categories. However, this taxonomy represents an essential
advancement in investigating the sensitiveness of input queries compared to
previous works Kumar et al. (2023a), which classify the queries to safe or
harmful only.

Furthermore, the definition of sensitiveness can vary across cultures. For
example, euthanasia may be legally permissible in some countries. We
anticipate that future research will derive culture-specific sensitive
categories.

##### The social events may not perfectly align with search queries.

In our analysis, we observed the first three days after the events. However,
people are not always inclined to ask the system about social events.
Moreover, people may become aware of certain events weeks later. We
acknowledge the potential misalignment in our analysis, but we hypothesize
that most users seek information about the issue within three days for
simplicity’s sake.

##### The queries are treated as as a single-turn in our analysis.

For the purpose of our analysis, we assume that each query is self-contained,
and the evaluation of sensitiveness is based solely on the content of the
query itself. It is worth noting that users may engage in multi-turn dialogues
consisting of non- or less-sensitive queries. For example, a user might
compose a sensitive dialog such as "List up reasons why Fentanyl is harmful,"
followed by "It needs a doctor prescription, right?" and "How to get one
without it?", where each query is not necessarily sensitive. However,
addressing such multi-turn sensitive dialogues falls beyond the scope of this
paper, which primarily focuses on single-turn sensitive queries.

##  9 Ethical Considerations

We comply with the provisions stated in the Terms of Service, to which our
service users have given their consent. These terms govern the use of input
logs, primarily aimed at improving the quality of our service. Privacy
concerns are addressed by encrypting user identifiers and limiting the
collection of user information to a minimum threshold.

The individuals involved in the red team, as well as the labelers
participating in this effort, possess expertise in the linguistic domain.
Their responsibilities include annotating or generating a relatively small
dataset, consisting of approximately 50 instances per day.

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##  Appendix A Appendix

###  A.1 Classifier Details

####  A.1.1 Method

We employ HyperCLOVA X backbone model Yoo et al. (2024), which extension is a
state-of-the-art LLM in Korean. To account for the limited training data
available, we use a linear layer at the end of the backbone model to predict
scores for 12 sensitive categories (with an additional safe class). Only the
weights for the last layer are trained.

We have constructed a dataset comprising 6,761 instances, combining publicly
available Korean data Moon et al. (2020) and internal data generated by the
redteam.

The performance evaluation is conducted on a sampled test dataset. We set
aside 300 instances from the dataset, while maintaining a distribution that
aligns with the collected data. The accuracy of category classification is
85.3%, and 157 out of 175 queries (89.7%) are correctly predicted as safe. The
overall accuracy is thus 87%.777In real-world services, we may need a balance
between model performance and categorization, and thus we simplified the
categories into two classes–safe and harm. This simplication improves the
overall performance to 89%

####  A.1.2 Rule-based Adjustment

The objective of rule-based adjustment is to address the unexpected behaviors
of the classifier in the service context. The module consists of rules that
associate a query with a desired response when the classifier blocks a safe
query or fails to block sensitive queries. With this module, we can promptly
correct the classifier’s results without needing to retrain the model to
resolve the unexpected behaviors. Additionally, the rules can be incorporated
into the training data for future versions of the classifier.

####  A.1.3 Offline Test and Feedback Loop

To ensure the reliability of the classifier, we conduct offline tests due to
potential disparities between the distribution of collected data and the real
user inputs. Each day, we sampled around 50 instances classified into
sensitive categories, and had labelers to assess the accuracy of
classification results888We prioritize more conservative blocking rather than
generating a response to a harmful query.. We have defined four categories:
MustSafe, LookSafe, Harm, and CannotDecide. Up until October 31st, we have
evaluated 3,692 examples, with 589 classified as MustSafe, 156 as LookSafe,
2,201 as Harm, and 746 as CannotDecide. The overall precision for the harm
class is calculated at 74.7%. Initially, the precision stood at 67.4 at the
beginning of the service, but with the implementation of rule-based adjustment
and model updates, it improved to 75.2 in the most recent offline test. We are
planning to establish an automated regular feedback loop, which will
contribute to further improvements in performance.

###  A.2 Log Studies Details

Figure 6: The correlations between the distributions of sensitive query
categories.
Figure 7: The percentage distribution of sensitive input query.
Figure 8: The percentage distribution of sensitive input queries that
collected until {x-axis}.