The smooth 4-dimensional Poincaré conjecture is a central open problem in low-dimensional topology.

The smooth 4-dimensional Poincaré conjecture. Every homotopy 4-sphere is diffeomorphic to S⁴.

Cappell and Shaneson [14] constructed homotopy 4-spheres, called Cappell-Shaneson homotopy 4-spheres. These homotopy 4-spheres are the most notable, potential counterexamples of the smooth 4-dimensional Poincaré conjecture. The following folklore conjecture is a special case of the smooth 4-dimensional Poincaré conjecture and has remained open for 40 years.

Conjecture 1.1. Every Cappell-Shaneson homotopy 4-sphere is diffeomorphic to S⁴.

One of our main results, Corollary C, will give the largest known family of Cappell-Shaneson spheres that are diffeomorphic to S⁴, supporting Conjecture 1.1. To motivate our results, we recall several earlier results on Cappell-Shaneson spheres.

1.1. Historical background

Cappell-Shaneson spheres ΣAε are parametrized by a matrix A∈SL(3;ℤ) with det(A−I)=1 and a choice of framing ε∈ℤ2. We say a matrix A∈SL(3;ℤ) is a Cappell-Shaneson matrix if det(A−I)=1. For example, for any n∈ℤ, the following matrix A_n is a Cappell-Shaneson matrix

An=01001110n+1.

We first recall history on Cappell-Shaneson spheres ΣAnϵ corresponding to the family A_n which have been studied thoroughly. For more details, we refer the reader to [9, Section 14.2] where a nice discussion on ΣAnϵ is given with many handlebody diagrams. Akbulut and Kirby [10] proved that ΣA00 is diffeomorphic to S⁴ by drawing its handlebody diagram and simplifying the diagram. They claimed that ΣA00 is the double cover of the Cappell-Shaneson fake ℝℙ4, denoted by Q, corresponding to the matrix

010001−110

which was constructed in [13]. Aitchison and Rubinstein [1] pointed out that ΣA01 is indeed the double cover of Q. We remark that Q is used by Akbulut to construct several interesting fake non-orientable 4-manifolds in [2, 3], and to show that a Gluck twist can change the diffeomorphism type for a non-orientable 4-manifold in [4]. (It is unknown whether a Gluck twist can change the diffeomorphism type of an orientable 4-manifold.) In the same paper [1], Aitchison and Rubinstein proved that ΣAn0 is diffeomorphic to S⁴ for all n.

For the non-trivial framing case, Akbulut and Kirby [11] drew a handlebody diagram of ΣA01 without 3-handles. They first introduced canceling pairs of 2- and 3-handles to remove 1-handles, and turned the resulting diagram upside-down to obtain the diagram without 3-handles. We remark that similar techniques are used in [5, 6, 8]. They showed that the punctured ΣA01 can be embedded in S⁴. In particular, by topological Schönflies theorem, ΣA01 is homeomorphic to S⁴. (Of course, this fact also can be checked by using Freedman's theorem.) They also observed that the double of the punctured ΣA01 is diffeomorphic to S⁴ if a balanced presentation of the trivial group

⟨x,y∣xyx=yxy,x5=y4⟩

is Andrews-Curtis trivial. (This balanced presentation is unlikely Andrew-Curtis trivial.)

Consequently, if ΣA01 were not diffeomorphic to S⁴, then the smooth 4-dimensional Poincaré conjecture and the smooth Schönflies conjecture would be false. Also, if the double of the punctured ΣA01 were not diffeomorphic to S⁴, then the Andrews-Curtis conjecture would be false. Gompf [18] excluded this possibility by proving that ΣA01 is actually diffeomorphic to S⁴ by adding a canceling pair of 2- and 3-handles. After a lengthy handlebody calculus, Gompf [19] gave a handlebody diagram of ΣAn1 without 3-handles for each n.

Around three decades later, Freedman, Gompf, Morrison and Walker [17] tried to disprove the smooth Poincaré conjecture via the following strategy. They considered knots obtained by adding a band to the two attaching circles of 2-handles in the handlebody diagrams of ΣAn1 given in [19]. A simple, but interesting observation is that such knots are slice in a homotopy 4-ball obtained from ΣAn1 by removing a small open ball. Hence if there is a non-slice knot obtained in this way, then the smooth 4-dimensional Poincaré conjecture would be false. By choosing specific bands, they obtained explicit diagrams of such knots, and computed Rasmussen s-invariants of them to disprove the smooth 4-dimensional Poincaré conjecture, but the s-invariants of their examples are trivial.

It turns out that there is an underlying reason that their attempts cannot be successful. Akbulut [7] added a marvellous canceling pair of 2- and 3-handles to the handlebody diagram of ΣAn1 given in [19], and proved that ΣAn1 is diffeomorphic to S⁴ for any integer n.

Now we recall what was known about general Cappell-Shaneson spheres. From the construction of Cappell-Shaneson spheres, it can be easily seen that two similar Cappell-Shaneson matrices give diffeomorphic Cappell-Shaneson spheres. More precisely, if A and B are similar Cappell-Shaneson matrices, then Cappell-Shaneson spheres ΣAϵ and ΣBϵ are diffeomorphic for any ϵ∈ℤ2. Therefore it is natural to think of the set of the similarity classes of Cappell-Shaneson matrices.

Our starting point is a result of Aitchison and Rubinstein [1] which states that every Cappell-Shaneson matrix is similar to a standard Cappell-Shaneson matrix

Xc,d,n=0ab0cd10n−c

for some integers c, d and n such that fn(c)≡0(Mod d) where fn(x)=x3−nx2+(n−1)x−1. Note that f_n(x) is the minimal polynomial of Xc,d,n and the entries a and b are determined as b=(c−1)(n−c−1) and ad=fn(c) from the equalities det(Xc,d,n)=det(Xc,d,n−I)=1.

Moreover, using a classical result of Latimer-MacDuffee and Taussky [21, 24], Aitchison and Rubinstein [1] observed that for any integer n, there are only finitely many similarity classes of trace n Cappell-Shaneson matrices. In fact, there is a bijection between the set of similarity classes of trace n Cappell-Shaneson matrices and the ideal class monoid C(ℤ[Θn]) where Θn is a root of f_n(x). Via the bijection, the similarity class of Xc,d,n corresponds to the ideal class [⟨Θn−c,d⟩]. (In particular, An=X1,1,n+2 corresponds to the identity element in C(ℤ[Θn+2]), see Remark 2.23.)

In short, to confirm Conjecture 1.1, it suffices to show that ΣXc,d,nϵ is diffeomorphic to S⁴ for finitely many pairs of such integers c, d for each integer n. (It is sufficient to check this for each representative of C(ℤ[Θn]).) However, this has been untouched mainly because finding and simplifying their handlebody diagrams of ΣXc,d,nϵ seem to be an onerous task.

In [20], Gompf proved that Cappell-Shaneson spheres ΣXc,d,nϵ and ΣXc,d,n+kdϵ are diffeomorphic for any ϵ∈ℤ2 and any integer k. It is remarkable that Gompf's proof does not involve any handlebody diagram. Nonetheless, this method is strong enough to give an alternative proof of the aforementioned result of Akbulut that ΣAn1 is actually diffeomorphic to S⁴ for any integer n. (To see this, note that An=X1,1,n+2, and hence ΣAnϵ is diffeomorphic to ΣA0ϵ which is diffeomorphic to S⁴ by [10, 18].) Using a computation of C(ℤ[Θ−5]) by Aitchison and Rubinstein, Gompf showed that more Cappell-Shaneson spheres are standard. Indeed, two Cappell-Shaneson spheres ΣX2,3,−50 and ΣX2,3,−51 corresponding to the Cappell-Shaneson matrix

X2,3,−5=0−5−802310−7

are diffeomorphic to S⁴. This result does not follow from the result of Akbulut since X2,3,−5 is not similar to A_-5.

In this context, Gompf considered an equivalence relation on the set of standard Cappell-Shaneson matrices generated by similarity and Xc,d,n GXc,d,n+kd for k∈ℤ. (We will call the equivalence relation by Gompf equivalence.) By the aforementioned result of Gompf, if two Cappell-Shaneson matrices are Gompf equivalent, then they give diffeomorphic Cappell-Shaneson spheres. Gompf conjectured the following whose affirmative answer implies Conjecture 1.1.

Conjecture 1.2 ([20, Conjecture 3.6]) Every Cappell-Shaneson matrix is Gompf equivalent to A₀.

1.2. Main results

In this paper, using various techniques in algebraic number theory, we study Conjecture 1.2 in a systematic way. For brevity of our discussion, we say Conjecture 1.1 is true for a Cappell-Shaneson matrix A if ΣAϵ is diffeomorphic to S⁴ for every ϵ∈ℤ2. Similarly, we say Conjecture 1.2 is true for trace n if every Cappell-Shaneson matrix A with trace n is Gompf equivalent to A₀.

Remark 1.3. If Conjecture 1.2 is true for trace n, then Conjecture 1.1 is true for every Cappell-Shaneson matrix with trace n. More generally, ΣXc,d,n+kdϵ is diffeomorphic to S⁴ for any k∈ℤ and ϵ∈ℤ2.

Our first result, Theorem A, shows that there is a hidden symmetry between trace n Cappell-Shaneson matrices and trace 5-n Cappell-Shaneson matrices. Theorem A implies that if Conjecture 1.2 is true for trace n≥3, then both of Conjectures 1.1 and 1.2 will be true simultaneously.

Theorem A. There is a bijection between the set of similarity classes of trace n Cappell-Shaneson matrices and the set of similarity classes of trace 5-n Cappell-Shaneson matrices. Moreover, Conjecture 1.2 is true for trace n if and only if Conjecture 1.2 is true for trace 5-n for any integer n.

To prove Theorem A, we will explicitly give a ring isomorphism from ℤ[Θn] to ℤ[Θ5−n] which induces a monoid isomorphism between C(ℤ[Θn]) and C(ℤ[Θ5−n]). This gives a bijection between the set of similarity classes of trace n Cappell-Shaneson matrices and the set of similarity classes of trace 5-n Cappell-Shaneson matrices. We will observe that the bijection is compatible with Gompf equivalence, and Theorem A will follow from the observation.

Our second result, Theorem B, shows that Conjecture 1.2 is true for trace n if |n| is small.

Theorem B. Conjecture 1.2 is true for trace n if −64≤n≤69.

To prove Theorem B, we first find representatives of elements of C(ℤ[Θn]). (Equivalently, we find a representative for each similarity class of trace n Cappell-Shaneson matrices.) When ℤ[Θn] is a Dedekind domain, this task can be done using MAGMA software (see Section 5).

When ℤ[Θn] is not a Dedekind domain, the current version of MAGMA cannot compute C(ℤ[Θn]). (Nonetheless, using MAGMA, we can still compute a strictly smaller subset Pic(ℤ[Θn]) of C(ℤ[Θn]), consisting of the classes of invertible ideals.) We will observe that there are infinitely many integers n such that ℤ[Θn] is not a Dedekind domain. In fact, for each integer k, ℤ[Θ49k+27] is not a Dedekind domain (see Proposition 4.10). Consequently, when we prove Theorem B, it is the most difficult to confirm that Conjecture 1.2 is true for trace 27. Using Dedekind-Kummer theorem, we analyze non-invertible ideals of ℤ[Θ27] explicitly (for details, see Section 4.3), and determine the monoid structure of C(ℤ[Θ27]). The authors think that our method could also be used to study C(ℤ[Θn]) for general n such that ℤ[Θn] is not a Dedekind domain.

By Theorem A, to prove Theorem B, it suffices to confirm that Conjecture 1.2 is true for trace 3≤n≤69. In Table 2-5, we give representatives of elements of C(ℤ[Θn]) for 3≤n≤69. We have to show that the corresponding standard Cappell-Shaneson matrices are Gompf equivalent to A₀. Recall that Gompf equivalence is an equivalence relation on the set of standard Cappell-Shaneson matrices generated by similarity and Xc,d,n GXc,d,n+kd for k∈ℤ. Understanding when two standard Cappell-Shaneson matrices are similar is important to study Conjecture 1.2, but this seems to be a difficult question in algebraic number theory. Instead, for any given standard Cappell-Shaneson matrix, we give a MAGMA code which gives a list of Cappell-Shaneson matrices with sufficiently small entries in Section 5. Using this, we could find several non-trivial Gompf equivalences. The authors think that finding such Gompf equivalences by hands is cumbersome.

In [16, Theorem 3.1], Earle considered the following special family of Cappell-Shaneson matrices

Xc,d,c+2=0ab0cd102,

and showed that Xc,d,c+2 are Gompf equivalent to A₀ if 0≤c≤94 and a≠19,37, or if 1≤d≤35. Earle found similar Cappell-Shaneson matrices by hands. As an application of our method, using our MAGMA codes, we recover and generalize the result of Earle. Indeed, we show that the Cappell-Shaneson matrices Xc,d,c+2 are Gompf equivalent to A₀ if 0≤c≤94, or if 1≤d≤134 by removing technical conditions on the entry a, and weakening the condition on the entry d (see Theorem 7.2).

Theorem B enables us to find new Cappell-Shaneson spheres that are diffeomorphic to S⁴, which we record the result as Corollary C. By Remark 1.3, Corollary C immediately follows from Theorem B.

Corollary C. Conjecture 1.1 is true for trace n Cappell-Shaneson matrices if n is an integer such that −64≤n≤69. More generally, ΣXc,d,nϵ is diffeomorphic to S⁴ for any ϵ∈ℤ2 and for any integers c, d and n that satisfy fn(c)≡0(Mod d) and n≡n0(Mod d) for some −64≤n0≤69. In particular, ΣXc,d,nϵ is diffeomorphic to S⁴ for any ϵ∈ℤ2 if |d|≤134.

By Corollary C, to find a counterexample to Conjecture 1.1, one should start from a Cappell-Shaneson matrix whose trace is either greater than 69 or less than -64. We remark that Corollary C gives the largest known family of Cappell-Shaneson spheres which are diffeomorphic to S⁴.

Remark 1.4. In Table 2-5, we give the lists of representatives (c,d,n) of elements of C(ℤ[Θn]) for 3≤n≤69. Each tuple (c,d,n) corresponds to the standard Cappell-Shaneson matrix Xc,d,n. For example, when n=21, there are three corresponding tuples (1,1,21), (5,7,21) and (9,13,21) in Table 2. This means that every Cappell-Shaneson matrix A with tr(A)=21 is similar to exactly one of the following three matrices:

X1,1,21=0100111020,X5,7,21=043600571016,X9,13,21=0618809131012.

Note that X1,1,21=A19. Using this computation and Theorem A, we can see that there are 1314 non-trivial ideal classes of C(ℤ[Θn]) for −64≤n≤69. In particular, Corollary C gives at least 2628 Cappell-Shaneson spheres that are diffeomorphic to S⁴, and this fact is not covered by the result of Akbulut [7].

It is natural to ask whether Corollary C actually gives a new infinite family of Cappell-Shaneson matrices whose corresponding Cappell-Shaneson spheres are diffeomorphic to S⁴. Our final result, Corollary D, shows that this is the case. For this purpose, we consider the following family of Cappell-Shaneson matrices M_k (k∈ℤ),

Mk=014k+749k+240271049k+25.

Note that Mk=X2,7,49k+27, and hence ΣMkϵ is diffeomorphic to S⁴ for any ϵ∈ℤ2 by Corollary C. (This fact can be also checked by using a weaker version given in [20, Theorem 3.2].) We show that M_k is not similar A_n for any integers k and n.

Corollary D. For any integers k and ϵ∈ℤ2, Cappell-Shaneson sphere ΣMkϵ corresponding to M_k is diffeomorphic to S⁴. For any integers k and n, M_k is not similar to A_n.

Recall that Akbulut [7] showed that the infinite family of Cappell-Shaneson matrices A_n give Cappell-Shaneson spheres ΣAnϵ are diffeomorphic to S⁴ for any ϵ∈ℤ2. Since M_k is not similar to A_n for any integers k and n, Corollary D is not covered by the result of Akbulut.

Organization of the paper. In Section 2, we recall several facts on Cappell-Shaneson spheres and Cappell-Shaneson matrices, and we discuss the correspondence between ideal class monoid and the similarity classes of Cappell-Shaneson matrices. In Section 3, we prove Theorem A. In Section 4, we recall Dedekind-Kummer theorem, and show that C(ℤ[Θ49k+27]) is not a group for any integer k, and discuss the structure of C(ℤ[Θ27]). In Section 5, we use MAGMA software to find representatives of elements in Pic(ℤ[Θn]). In Section 6, we prove Theorem B and Corollary D. In Section 7, we give a generalization of the result of Earle.

Acknowledgement. The authors would like to thank Tetsuya Abe, Selman Akbulut, Jae Choon Cha, Hisaaki Endo, Robert Gompf, Mark Powell and Motoo Tange for their encouragements and helpful discussions. The first author would like to thank Jung Won Lee for helping him to use MAGMA software. The first author was partially supported by the POSCO TJ Park Science Fellowship.

In this section, we collect several facts on Cappell-Shaneson spheres and matrices following [1, Appendix] and [20].

2.1. Cappell-Shaneson spheres and matrices

Let SL(3;ℤ) be the set of 3×3 integral matrices whose determinants are 1. We say two matrices A,B∈SL(3;ℤ) are similar if there is a matrix C∈SL(3;ℤ) such that A=CBC−1.

Definition 2.1. A matrix A∈SL(3;ℤ) is a Cappell-Shaneson matrix if A−I∈SL(3;ℤ).

For a Cappell-Shaneson matrix A∈SL(3;ℤ), Cappell and Shaneson [14] constructed two homotopy 4-spheres ΣAε as follows. Let T³ be the 3-torus ℝ3/ℤ3. Since A∈SL(3;ℤ), A induces an orientation-preserving diffeomorphism fA:T3→T3. Possibly after an isotopy, we can assume that f_A is the identity on a neighborhood D_y of some chosen point y∈ T³. Let W_A be the mapping torus of f_A, that is,

WA=T3×[0,1]/(x,0) (fA(x),1).

Since f_A is the identity around the point y, we can regard Dy×S1⊂WA. From the condition det(A−I)=1, the Wang sequence applied to the fiber bundle T3↪WA→S1, and Van Kampen theorem show that W_A is a homology S1×S3 whose fundamental group π1(WA) is normally generated by [y×S1]. If we remove Dy×S1 from W_A and glue S2×D2 along the boundary via a framing ϵ∈ℤ2, then we obtain a homotopy 4-sphere ΣAϵ.

Definition 2.2 [Cappell-Shaneson spheres] For a Cappell-Shaneson matrix A, two homotopy 4-spheres ΣA0 and ΣA1 are called Cappell-Shaneson spheres corresponding to A.

Remark 2.3. From the construction of Cappell-Shaneson homotopy 4-spheres, if A and B are similar Cappell-Shaneson matrices, then W_A and W_B are diffeomorphic, and hence ΣAε and ΣBε are diffeomorphic.

By Remark 2.3, to study Cappell-Shaneson spheres up to diffeomorphism, it is natural to consider the similarity classes of Cappell-Shaneson matrices. In [1, Appendix], the similarity classes of Cappell-Shaneson matrices in terms of ideal classes are systematically studied using a result of Latimer-MacDuffee and Taussky [21, 24] which we recall in below.

Let A be a Cappell-Shaneson matrix with trace n. The characteristic polynomial of A is

fn(x)=x3−nx2+(n−1)x−1.

Remark 2.4. For A∈SL(3;ℤ), A is a Cappell-Shaneson matrix with trace n if and only if the characteristic polynomial of A is fn(x). Note that fn(x) is irreducible over ℤ for all n (for example, see [1, Lemma A4]).

Definition 2.5 ([1, 20]). We say a Cappell-Shaneson matrix is called standard if it is of the form

Xc,d,n=0ab0cd10n−c.

By the following theorem of Aitchison and Rubinstein and Remark 2.3, we restrict our attention to Cappell-Shaneson spheres ΣAε which correspond to standard Cappell-Shaneson matrices A since we are interested in their differentiable structures.

Theorem 2.6 (Aitchison and Rubinstein [1]) Every Cappell-Shaneson matrix is similar to a standard Cappell-Shaneson matrix.

Remark 2.7. Technically, Aitchison and Rubinstein [1] proved that every Cappell-Shaneson matrix is similar to the transpose of a standard Cappell-Shaneson matrix. However, this is clearly an equivalent statement.

Remark 2.8. Since Xc,d,n is a Cappell-Shaneson matrix, det(Xc,d,n−I)=1 and det(Xc,d,n)=1. From these conditions, b=(c-1)(n-c-1) and ad-bc=1, that is, Xc,d,n is uniquely determined by c,d and n.

Remark 2.9. Gompf [20] considered slightly general matrices of the form

A=0ab0cd1en−c

which Gompf called A is a Cappell-Shaneson matrix in the standard form. Gompf proved that ΔkA and AΔk are Cappell-Shanseon matrices and the corresponding Cappell-Shaneson homotopy spheres ΣΔkAε and ΣAΔkε are diffeomorphic to ΣAε for ε=0,1 where

Δ=1−10010011.

Observe that A is equivalent to Xc,d,n as follows. (Note that the values of c, d and n are preserved.)

1e00100010ab0cd1en−c1−e0010001=0a+ceb+de0cd10n−c.

We observe that both ΔkA and AΔk are similar to Xc,d,n+kd as follows. Note that ΔkA and AΔk are similar because ΔkA=Δk(AΔk)Δ−k. The above argument shows that the matrix

ΔkA=0a−kcb−kd0cd1kc+ekd+n−c

is similar to Xc,d,n+kd.

We end this subsection by giving a simple, algebraic characterization of standard Cappell-Shaneson matrices which will be used frequently.

Proposition 2.10. For integers c,d≠0 and n, the following are equivalent.

• (1) fn(c)≡0 (mod d).

• (2) There exist integers a and b such that

  Xc,d,n=0ab0cd10n−c

  is a standard Cappell-Shaneson matrix.

Proof. We first note that fn(c)=c3−nc2+(n−1)c−1=−c(c−1)(n−c−1)−1. Suppose that fn(c)≡0 (mod d). Define a,b∈ℤ by fn(c)=−ad and b=(c−1)(n−c−1). Consider the following matrix

A=0ab0cd10n−c.

Note that A is a Cappell-Shaneson matrix (and hence is equal to Xc,d,n) since

detA=ad−bc=−fn(c)−c(c−1)(n−c−1)=1

and

det(A−I)=−(c−1)(n−c−1)+(ad−b(c−1))=1.

For the converse, consider the Cappell-Shaneson matrix

Xc,d,n=0ab0cd10n−c.

By Remark 2.8, fn(c)=−c(c−1)(n−c−1)−1=−bc−1=−ad≡0(mod d).

2.2. Latimer-MacDuffee-Taussky correspondence

In this subsection, we recall a classical result due to Latimer-MacDuffee and Taussky [21, 24]. For more details, see Newman's book [22].

Let R be an integral domain and I(R) be the set of nonzero ideals of R. Define an equivalence relation ≈ on I(R) by I≈J if and only if there exist non-zero elements α,β such that αI=βJ. Each equivalence class is called an ideal class and the ideal class of I∈I(R) is denoted by [I]. The set of all ideal classes is called the ideal class monoid of R denoted by C(R). The multiplication is given by the multiplication of ideals: [I]⋅[J]=[IJ]. The identity element is the class of principal ideals. An ideal I of R is called invertible if there exists an ideal J of R such that IJ is a principal ideal. The subset of C(R) which consists of the ideal classes of invertible ideals of R is an abelian group, called the Picard group of R and denoted by Pic(R).

Remark 2.11. We remark that the monoid C(R) is not a group in general. In fact, the following are equivalent for an integral domain R:

• (1) R is a Dedekind domain.

• (2) Every ideal of R is invertible.

• (3) C(R) is a group.

• (4) C(R)=Pic(R).

Example 2.12. If R is the ring of integers of an algebraic number field, then R is a Dedekind domain and hence C(R) is a group.

We are mainly interested in the special case that R=ℤ[Θ] where Θ is a root of a monic polynomial g(x)∈ℤ[x] which is irreducible (over ℤ). Note that ℚ[Θ] is the number field obtained by adjoining Θ to ℚ.

We recall a classical result due to Latimer-MacDuffee [21] and Taussky [24]. For simplicity and our purposes, we spell out the degree 3 case only. For more details and generalizations, we refer the reader to [22].

Theorem 2.13 (Latimer-MacDuffee [21], Taussky [24]). Suppose g∈ℤ[x] is a monic, irreducible polynomial of degree 3. Let Θ be a root of g. Then there is a bijection between Cℤ[Θ] and the set of similarity classes of matrices whose characteristic polynomials are g.

We describe an explicit description of the bijection. Let A be a 3×3 matrix whose characteristic polynomial is g and let K=ℚ[Θ]. Regard A as a K-linear map A:K3→K3. Then Θ is an eigenvalue of A and there exists a corresponding eigenvector in K³. In addition, the eigenvalues of A are distinct, because g is irreducible over ℚ. It follows that any two eigenvectors of A corresponding to Θ are proportional. Let x=(x1,x2,x3) be an eigenvector of A corresponding to Θ. We may assume that each x_i lies in ℤ[Θ] by multiplying some integer. Let I be the ℤ-module generated by x1,x2 and x₃. Then, I is an ideal of ℤ[Θ]. The ideal class [I]∈C(ℤ[Θ]) is independent of the choice of an eigenvector (x1,x2,x3), and called the ideal class which corresponds to A.

2.3. The ideal class which corresponds to a Cappell-Shaneson matrix

Aitchison and Rubinstein [1] applied Theorem 2.13 to Cappell-Shaneson matrices which we recall in below for the reader's convenience. Let Θn be a root of fn(x)=x3−nx2+(n−1)x−1. Recall that the set of Cappell-Shaneson matrices with trace n is exactly the set of 3×3 integral matrices A whose characteristic polynomial is fn(x). Since fn(x) is irreducible, Theorem 2.13 gives a bijection between the set of similarity classes of Cappell-Shaneson matrices with trace n and C(ℤ[Θn]). We will explicitly describe the bijection.

Consider a Cappell-Shaneson matrix with trace n,

Xc,d,n=0ab0cd10n−c.

We find an eigenvector x=(x1,x2,x3)∈ ℤ[Θn]3 of Xc,d,n corresponding to Θn.

0ab0cd10n−cx1x2x3=Θnx1Θnx2Θnx3,  ax2+bx3=Θnx1,cx2+dx3=Θnx2,x1+(n−c)x3=Θnx3.

In particular, (x1,x2,x3)=((Θn−n+c)(Θn−c),d,Θn−c) is an eigenvector of A in ℤ[Θn]3 with the eigenvalue Θn.Note that ⟨(Θn−n+c)(Θn−c),d,Θn−c⟩=⟨Θn−c,d⟩. Hence the ideal class [⟨Θn−c,d⟩] corresponds to the standard Cappell-Shaneson matrix Xc,d,n by Theorem 2.13.

Proposition 2.14 ([1, page 44]) here is a one-to-one correspondence between the set of similarity classes of Cappell-Shaneson matrices with trace n and C(ℤ[Θn]), which is defined by

Xc,d,n=0ab0cd10n−c↦[⟨Θn−c,d⟩]

where fn(c)≡0(mod d), b=(c−1)(n−c−1) and ad-bc=1.

Remark 2.15. For k∈ℤ, by Proposition 2.14, Xc,d,n and Xc+kd,d,n are similar because ⟨Θn−c,d⟩=⟨Θn−c−kd,d⟩.

2.4. Gompf equivalences and a reformulation of Gompf conjecture

In [20], Gompf introduced a certain equivalence relation (which we call Gompf equivalences) between standard Cappell-Shaneson matrices which preserve the diffeomorphism types of the corresponding Cappell-Shaneson homotopy 4-spheres. We recall Gompf equivalences and give a reformulation of Conjecture 1.2 in Conjecture 2.20.

As in Remark 2.9, let Δ be the following matrix,

Δ=1−10010011.

Theorem 2.16 ([20, page 1673]). Let A be a Cappell-Shaneson matrix given by

0ab0cd1en−c.

Then, AΔk and ΔkA are also Cappell-Shaneson matrices and corresponding Cappell-Shaneson spheres ΣAΔkε and ΣΔkAε are diffeomorphic to ΣAε for every integer k and ϵ∈ℤ2.

Remark 2.17. In Remark 2.9, we remarked that if A is a Cappell-Shaneson matrix given by

A=0ab0cd1en−c,

then A is similar to Xc,d,n and ΔkA and AΔk are similar to Xc,d,n+kd. We know that similar Cappell-Shaneson matrices give diffeomorphic homotopy 4-spheres by Remark 2.3. Therefore, the content of Theorem 2.16 is that two standard Cappell-Shaneson matrices Xc,d,n and Xc,d,n+kd give diffeomorphic homotopy 4-spheres for any integer k.

Definition 2.18 (Gompf equivalence). Define an equivalence relation ~, called Gompf equivalence, on the set of standard Cappell-Shaneson matrices generated by S and G where

Xc0,d0,n SXc1,d1,n  if [⟨Θn−c0,d0⟩]=[⟨Θn−c1,d1⟩]∈C(ℤ[Θn]),Xc,d,n GXc,d,n+kd  if k∈ℤ.

Using Theorem 2.16 and Aitchison-Rubinstein's computation of C(ℤ[Θn]) for small n, Gompf proved that Conjecture 1.2 is true for trace n if −6≤n≤9 or n=11. In Section 6.1, we will show that Conjecture 1.2 is true for trace n if −64≤n≤69.

Theorem 2.19 ([20, Theorem 3.2]). Conjecture 1.2 is true for trace n if −6≤n≤9 or n=11.

We end this preliminary section by giving a reformulation of Conjecture 1.2. This reformulation will be convenient to give the proof of Theorem B given in Section 6. Let Θn be a root of a polynomial fn(x)=x3−nx2+(n−1)x−1. Consider

CS={(c,d,n)∈ℤ3∣fn(c)≡0(mod d)and d≠0}.

By Proposition 2.10, there is a bijection between CS and the set of standard Cappell-Shaneson matrices such that the tuple (c,d,n)∈CS corresponds to the standard Cappell-Shaneson matrix

Xc,d,n=0ab0cd10n−c

where b=(c−1)(n−c−1) and ad-bc=1. (In particular, a and b are determined by c, d and n.)

We define an equivalence relation ~ on CS generated by S and G where

(c0,d0,n) S(c1,d1,n)  if [⟨Θn−c0,d0⟩]=[⟨Θn−c1,d1⟩]∈C(ℤ[Θn]),(c,d,n) G(c,d,n+kd)  if k∈ℤ.

Conjecture 2.20. For every (c,d,n)∈CS, (c,d,n) (1,1,2).

Definition 2.21. For an integer n, we say Conjecture 2.20 is true for trace n if for any integers c and d such that (c,d,n)∈CS, (c,d,n) (1,1,2).

Remark 2.22. By Proposition 2.14, (c0,d0,n) S(c1,d1,n) if and only if Xc0,d0,n and Xc1,d1,n are similar. The second relation G corresponds to the equivalence relation Xc,d,n GXc,d,n+kd. It is clear that Conjecture 1.2 for trace n is equivalent to Conjecture 2.20 since A0=X1,1,2.

Remark 2.23. The pair (1,1,n+2)∈CS corresponds to the trivial element of the ideal class monoid C(ℤ[Θn+2]) because ⟨Θn+2−1,1⟩ is principal. Since (1,1,n+2) G(1,1,2), we do not have to consider the trivial element of C(ℤ[Θn+2]). (In fact, X1,1,n+2=An and, as mentioned in the introduction, it has been known that ΣAnε is diffeomorphic to S⁴ for ε=0,1 and n∈ℤ.)

In this section, we prove Theorem A which says that Conjecture 1.2 for the trace n case is equivalent to the trace 5-n case. Throughout this section, let Θn be a root of fn(x)=x3−nx2+(n−1)x−1 for each integer n. We give a ring isomorphism between ℤ[Θn] and ℤ[Θ5−n] which will induce a bijection between corresponding ideal class monoids which is compatible with Gompf equivalence.

Theorem 3.1 is inspired by some evidences which are given in work of Aitchison-Rubinstein [1] and that of Gompf [20]. Aitchison and Rubinstein \cite[page 43]{Aitchison-Rubinstein:1984-1} observed that the discriminant Δ(fn) of the polynomial f_n has the following symmetry:

Δ(fn)=n(n−2)(n−3)(n−5)−23=Δ(f5−n).

On the other hand, Gompf [20][page 1672] computed the cardinality #C(On) for r≤108 via PARI/GP [15] and observed that #C(On)=#C(O5−n) where On is the ring of integer of ℚ[Θn].

Theorem 3.1. For any integer n, let Θn be a root of fn(x)=x3−nx2+(n−1)x−1. Then, there is a ring isomorphism φn:ℤ[Θn]→ℤ[Θ5−n] defined by

φn(Θn)=Θ5−n2+(n−4)Θ5−n+1.

Proof. For an aesthetic reason, we prove an equivalent statement that the ring homomorphism φ5−n:ℤ[Θ5−n]→ℤ[Θn] is an isomorphism for any integer n. The ring isomorphism φ5−n will be defined as φ5−n(Θ5−n)=Θn2+(1−n)Θn+1. Let φ¯5−n:ℤ[x]→ℤ[Θn] be a ring homomorphism which sends x to Θn2+(1−n)Θn+1. We prove that φ¯5−n induces the ring homomorphism φ5−n:ℤ[Θ5−n]→ℤ[Θn] by observing that

φ¯5−n(f5−n(x))=f5−n(Θn2+(1−n)Θn+1)=0

where f5−n(x)=x3−(5−n)x2+(4−n)x−1. By setting αn=φ5−n(Θ5−n)=Θn2−(n−1)Θn+1, we show f5−n(αn)=0. Recall that Θn is a root of fn(x)=x3−nx2+(n−1)x−1=0. We have

Θn(Θn−1)(Θn−n+1)=Θn3−nΘn2+(n−1)Θn=1.

The following equality will be useful.

(Θn−1)αn=(Θn−1)(Θn2−(n−1)Θn+1)    =Θn(Θn−1)(Θn−n+1)+Θn−1=Θn.

Since Θn−1≠0, the following shows that f5−n(αn)=0:

(Θn−1)3f5−n(αn)=(Θn−1)3(αn3−(5−n)αn2+(4−n)αn−1)       =Θn3−(5−n)Θn2(Θn−1)+(4−n)Θn(Θn−1)2−(Θn−1)3       =Θn3−(Θn−1)3−Θn(Θn−1)((5−n)Θn−(4−n)(Θn−1))       =3Θn(Θn−1)+1−Θn(Θn−1)(Θn−n+4)       =1−Θn(Θn−1)(Θn−(n−1))=0.

Therefore, we have a ring homomorphism φ5−n:ℤ[Θ5−n]→ℤ[Θn] such that φ5−n(Θ5−n)=Θn2+(1−n)Θn+1. Now we prove that φ5−n∘φn is the identity map on ℤ[Θn] by showing that φ5−n∘φn(Θn)=Θn. To simplify the proof, we give two elementary observations. Since f5−n(αn)=0, αn(αn−1)(αn+n−4)=1. Note that (Θn−1)(αn−1)=(Θn−1)αn−Θn+1=1.

φ5−n∘φn(Θn)=φ5−n(Θ5−n2+(n−4)Θ5−n+1)     =αn2+(n−4)αn+1     =(Θn−1)(αn−1)(αn2+(n−4)αn+1)     =(Θn−1)((αn−1)αn(αn+n−4)+αn−1)     =(Θn−1)(1+αn−1)     =(Θn−1)αn     =Θn.

By substituting n by 5-n, φ5−n∘φn is also the identity. Hence, φ5−n is a ring isomorphism and this completes the proof.

Remark. By tensoring ℚ to the ring isomorphism φn:ℤ[Θn]→ℤ[Θ5−n] given in Theorem 3.1, we obtain a field isomorphism from ℚ[Θn] to ℚ[Θ5−n]. From this field isomorphism, we can see that their ring of integers On and O5−n are also isomorphic and Δ(fn)=Δ(f5−n) for any integer n.

Corollary 3.2. There exists a monoid isomorphism ψn:C(ℤ[Θn])→C(ℤ[Θ5−n]) for any integer n. Furthermore, for any integers c, d with fn(c)≡0 fn(c)≡0 (mod d), ψn sends [⟨Θn−c,d⟩] to [⟨Θ5−n−pn(c),d⟩] where pn(x)=x2+(1−n)x+1.

Proof. Define a monoid homomorphism ψn:C(ℤ[Θn])→C(ℤ[Θ5−n]) by [I]↦[φn(I)] where φn is the ring homomorphism given in Theorem 3.1. Since φn is a ring isomorphism for any integer n by Theorem 3.1, ψn is also a monoid isomorphism for any integer n. To give an explicit formula of ψn, we prove that φn(⟨Θn−c,d⟩)=⟨Θ5−n−pn(c),d⟩ and this clearly implies the desired statement.

Claim 3.3. ⟨Θn−c,d⟩=⟨pn(Θn)−pn(c),d⟩

Proof of Claim. The following calculation shows that ⟨pn(Θn)−pn(c),d⟩⊂⟨Θn−c,d⟩:

pn(Θn)−pn(c)=Θn2+(1−n)Θn+1−c2−(1−n)c−1      =(Θn−c)(Θn+c−n+1)∈⟨Θn−c,d⟩.

Now we prove ⟨Θn−c,d⟩⊂⟨pn(Θn)−pn(c),d⟩. Note that

fn(x)+1=x3−nx2+(n−1)x=(x−1)(x2+(1−n)x)=(x−1)(pn(x)−1).

Using the above equation on fn(x)+1 and the fact that fn(Θn)=0, we observe that

Θn−c=Θn−1−(c−1)  =(Θn−1)(fn(c)+1)−(Θn−1)fn(c)−(c−1)(fn(Θn)+1)  =(Θn−1)(c−1)(pn(c)−1)−(Θn−1)fn(c)−(Θn−1)(c−1)(pn(Θn)−1)  =(Θn−1)(c−1)(pn(c)−pn(Θn))−(Θn−1)fn(c).

From the assumption fn(c)≡0 (mod d), fn(c)∈⟨pn(Θn)−pn(c),d⟩. This shows that ⟨Θn−c,d⟩⊂⟨pn(Θn)−pn(c),d⟩ and hence the claim follows.

Note that pn(Θn)=Θn2+(1−n)Θn+1=φ5−n(Θn). By the claim,

φn(⟨Θn−c,d⟩)=φn(⟨φ5−n(Θn)−pn(c),d⟩)=⟨Θn−pn(c),d⟩.

Here the last equality follows from the fact that φn∘φ5−n is the identity map. This completes the proof.

Theorem 3.4. There is a bijection between the set of similarity classes of Cappell-Shaneson matrices with trace n and the set of similarity classes of Cappell-Shaneson matrices with trace 5-n, which is explicitly defined by

A=0ab0cd10n−c↦A*=0a*b*0c*d*105−n−c*

where c*=pn(c)=c2+(1−n)c+1, d^*=d. In particular, Xc,d,n*=Xpn(c),d,5−n*.

Proof. Since every Cappell-Shaneson matrix is similar to a standard Cappell-Shaneson matrix, the bijection A↦A* gives the ones which represent all the similarity classes of Cappell-Shaneson matrices with trace 5-n.

Recall that there is a bijection between the similarity classes of Cappell-Shaneson matrices with trace n (respectively, trace 5-n) with the ideal class monoid C(ℤ[Θn]) (respectively, C(ℤ[Θ5−n])) by Proposition 2.14. On the other hand, we have a monoid isomorphism ψn:C(ℤ[Θn])→C(ℤ[Θ5−n]) by Corollary 3.2. The composition of these three bijections gives a bijection between the set of similarity classes of Cappell-Shaneson matrices with trace n and the set of similarity classes of Cappell-Shaneson matrices with trace 5-n. It remains to show is that the aforementioned bijection actually sends a standard Cappell-Shaneson matrix A to a standard Cappell-Shaneson matrix A^*.

By Proposition 2.14, the ideal class correspond to the standard Cappell-Shaneson matrix A is [⟨Θn−c,d⟩]. By Corollary 3.2, ψn sends the ideal class [⟨Θn−c,d⟩] to the ideal class [⟨Θ5−n−pn(c),d⟩], which is the ideal class corresponding to the standard Cappell-Shaneson matrix A^* by Proposition 2.14. This completes the proof.

Example 3.5. As an illustration, we explicitly describe the bijection given in Theorem 3.4 for the case that trace n=-5. Aitchison and Rubinstein [1] showed that there are only two similarity classes of Cappell-Shaneson matrices with trace -5, which are represented by as follows. (Note that A=A−7.)

A=01001110−6,B=0−5−802310−7.

By Theorem 3.4, it follows that there are only two similarity classes of Cappell-Shaneson matrices with trace 10, which are represented by

A*=0577081102,B*=0−725−128017310−7.

By Proposition 2.14, the ideal classes correspond to A^* and B^* are [⟨1,Θ10−8⟩] and [⟨3,Θ10−17⟩], respectively. Note that [⟨1,Θ10−8⟩=[⟨1,Θ10−1⟩] and [⟨3,Θ10−17⟩]=[⟨3,Θ10−2⟩]. We obtain similarity relations by Proposition 2.14:

A* 010011109,B* 057023108.

To complete the proof of Theorem A, we prove two lemmas which illustrate that the bijection given in Theorem 3.4 behaves nicely with Gompf equivalence.

Lemma 3.6. Suppose that A and B are two standard Cappell-Shaneson matrices such that A and B are Gompf equivalent. Then A^* and B^* are also Gompf equivalent.

Proof. Since Gompf equivalence is generated by S and G, we can assume without loss of generality that either A SB or A GB holds. If A SB, then A* SB* by Theorem 3.4. Now we assume that A GB, that is, A=Xc,d,n and B=Xc,d,n+kd for some c,d,k and n∈ℤ. By Theorem 3.4, A*=Xc*,d,5−n and B*=Xc*,d,5−n−kd, and hence A* GB*. (Note that d^*=d.) This completes the proof.

Lemma 3.7. Let A be a standard Cappell-Shaneson matrix. Then A is similar to (A*)*.

Proof. Since A is a standard Cappell-Shaneson matrix, A=Xc,d,n for some c,d and n with fn(c)≡0(Mod d).Then (A*)*=(Xpn(c),d,5−n)*=Xp5−n(pn(c)),d,n where pn(c)=c2+(1−n)c+1. Note that

p5−n(pn(c))=pn(c)2+(n−4)pn(c)+1     =(c2+(1−n)c+1)2+(n−4)(c2+(1−n)c+1)+1     =c+fn(c)(c−n+2)≡c( mod d)

since fn(c)≡0(Mod d). Since p5−n(pn(c))≡c(mod d), (A*)* is similar to A by Remark 2.15.

Now we prove Theorem A.

Proof of Theorem A. We have already seen that there is a bijection between the set of similarity classes of trace n Cappell-Shaneson matrices and the set of similarity classes of trace 5-n Cappell-Shaneson matrices in Theorem 3.4.

Assume that Conjecture 1.2 is true for trace n for some integer n. Let X be a standard Cappell-Shaneson matrix with trace 5-n. (Recall that every Cappell-Shaneson matrix is similar to a standard Cappell-Shaneson matrix.) Then X^* given in Theorem 3.4 is a standard Cappell-Shaneson matrix with trace n. Since we are assuming that Conjecture 1.2 is true for trace n, X* is Gompf equivalent to A₀. By Lemma3.7 , X is similar to (X*)*. By Lemma 3.6, (X*)* is Gompf equivalent to (A0)*. As in Example 3.5, A0* is similar to A₁, which is Gompf equivalent to A₀ by Remark 2.23. Therefore X is Gompf equivalent to A₀. This shows that Conjecture 1.2 is true for trace 5-n if Conjecture 1.2 is true for trace n. This completes the proof.