EXOTIC CODIMENSION-1 SUBMANIFOLDS IN 4-MANIFOLDS AND STABILIZATIONS HOKUTO KONNO ANUBHAV MUKHERJEE AND MASAKI TANIGUCHI

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EXOTIC CODIMENSION-1 SUBMANIFOLDS IN 4-MANIFOLDS

AND STABILIZATIONS

HOKUTO KONNO, ANUBHAV MUKHERJEE, AND MASAKI TANIGUCHI

Abstract. In a small simply-connected closed 4-manifold, we construct inﬁn-

itely many pairs of exotic codimension-1 submanifolds with diﬀeomorphic com-

plements that remain exotic after any number of stabilizations by S2×S2. We

also give new constructions of exotic embeddings of 3-spheres in 4-manifolds

with diﬀeomorphic complements.

1. Introduction

It is one of the key topics of the study in 4-manifold topology to understand

exotic phenomena, i.e. those properties that are true in the topological category

but not in the smooth category. The main three exotic phenomena that have been

getting lots of attention in recent times are existence of exotic smooth structures

in 4-manifolds, existence of exotic surfaces in 4-manifolds and existence of exotic

diﬀeomorphisms in 4-manifolds. Motivated from the 4-dimensional smooth Schoen-

ﬂies conjecture, we recently started studying exotic codimension-1 submanifolds in

4-manifolds [16]. Here let us ﬁrst clarify the term “exotic codimension-1 submani-

folds” of a 4-manifold and “exotic embeddings of 3-manifolds” into a 4-manifold in

this paper:

Deﬁnition 1.1. We say that two codimension-1 smooth submanifolds Y1and Y2

in a smooth oriented 4-manifold X(resp. two smooth embeddings i1, i2:Y→X)

are exotic if

(i) there is a topological ambient isotopy Ht:X×[0,1] →Xsuch that H0= Id

and H1(Y1) = Y2(resp. H1◦i1=i2),

(ii) there is no such smooth isotopy,

(iii) the complements of Y1and Y2are diﬀeomorphic, i.e there exists a diﬀeo-

morphism f:X→Xsuch that f(Y1) = Y2(resp. f◦i1=i2).

Exotic codimension-1 submanifolds are closely related to interesting topics of

diﬀeomorphisms of a 4-manifold. For instance, the existence of exotic codimension-

1 submanifolds leads to a counterexample to the π0-version of the Smale conjecture

in the following sense: If there is a pair of exotic codimension-1 submanifolds in S4

whose diﬀeomorphism type is Yso that the mapping class group of Yis trivial,

then it is easy to provide a non-trivial element of the kernel of

π0(Diﬀ(S4)) →π0(Homeo(S4)).

This gives a way to disprove the 4-dimensional Smale conjecture diﬀerent from the

one by Watanabe [42].

Remark 1.2.If the mapping class group of Yis trivial, then exotic embeddings

yield exotic submanifolds, and vice versa.

arXiv:2210.05029v2 [math.GT] 29 Nov 2022

2 HOKUTO KONNO, ANUBHAV MUKHERJEE, AND MASAKI TANIGUCHI

Remark 1.3.The notion obtained by dropping the condition (iii) of Deﬁnition 1.1

might be called “weakly exotic submanifolds/embeddings”. It is relatively easy

to construct such exotic codimension-1 submanifolds using (exotic) corks, as de-

scribed in [16, Introduction]. Corks yield also examples of this weaker version of

exotic embeddings. Indeed, let (W, τ) be a cork, where τis a diﬀeomorphism of

∂W . Then it is easy to see that the double W∪∂W (−W) admits weakly exotic

embeddings of ∂W by considering two embeddings of ∂W that correspond to Id∂W

and τ. However, this cork example does not give exotic embeddings in the sense of

Deﬁnition 1.1.

We shall study the existence of exotic codimension-1 submanifolds, but also

consider behavior of them under stabilizations, i.e. connected sum with S2×S2’s, or

more general 4-manifolds. In the context of exotica (in the category of orientable 4-

manifolds), one remarkable discovery by Wall [41] in 1960’s says that exotic smooth

structures disappear under suﬃciently many stabilizations by S2×S2. For surfaces

in a 4-manifold and diﬀeomorphisms of a 4-manifold, analogous results were later

established by Perron [34], and Quinn [35] (combined with a result of Kreck [19]).

One can similarly expect that such a principle should be true for exotic 3-manifolds

in 4-manifolds. Such a principle holds for previously known examples of exotic

embeddings of 3-manifolds since those examples were constructed as images of

exotic diﬀeomorphisms, and those diﬀeomorphisms are smoothly isotopic to the

identity after stabilizations, as described.

In order to state our ﬁrst result, we use the Heegaard Floer tau-invariant [33] τ:

C → Z, where Cdenotes the knot concordance group. Let sign : Z\ {0}→{1,−1}

be the sign function. The ﬁrst main theorem of this paper is:

Theorem 1.4. Let Kbe a knot in S3with τ(K)6= 0 and let n > 0. If the mapping

class group of the 3-manifold S3

sign(τ(K))/n(K)is trivial, then

Xn:= (#2S2×S2, n is even

#3(CP2#(−CP2)), n is odd

contains exotic codimension-1 submanifolds Y1and Y2whose diﬀeomorphism types

are S3

sign(τ(K))/n(K), and which survives after a connected sum by any connected

4-manifold M, where Mis attached to a point away from Y1∪Y2. In particluar,

we can choose M= #mS2×S2for any m > 0.

Example 1.5.Many examples of knots satisfying the assumptions of Theorem 1.4

can be found as follows: Take a hyperbolic knot Kwhich has trivial symmetry

group, i.e. the isometry group of the hyperbolic knot complement is trivial. For

large n, the mapping class group of S3

1/n(K) becomes trivial. (Here, we used

the generalized Smale conjecture for hyperbolic 3-manifolds proven by Gabai [12]

together with Thurston’s hyperbolic Dehn ﬁlling theorem [39].) For example, using

Snappy [7], we can ensure that one may take a hyperbolic knot K= 10149 with

trivial symmetry group and with τ(K) = 2.

The authors do not know whether there exists a pair of codimension-1 subman-

ifolds in smaller 4-manifolds than #2S2×S2and #3(CP2#(−CP2)). So we can

ask the following question:

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EXOTICCODIMENSION-1SUBMANIFOLDSIN4-MANIFOLDSANDSTABILIZATIONSHOKUTOKONNO,ANUBHAVMUKHERJEE,ANDMASAKITANIGUCHIAbstract.Inasmallsimply-connectedclosed4-manifold,weconstructinn-itelymanypairsofexoticcodimension-1submanifoldswithdieomorphiccom-plementsthatremainexoticafteranynumberofstabilizationsbyS2...

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EXOTIC CODIMENSION-1 SUBMANIFOLDS IN 4-MANIFOLDS AND STABILIZATIONS HOKUTO KONNO ANUBHAV MUKHERJEE AND MASAKI TANIGUCHI

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