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Kyungpook Mathematical Journal 2023; 63(3): 373-411

Published online September 30, 2023 https://doi.org/10.5666/KMJ.2023.63.3.373

Copyright © Kyungpook Mathematical Journal.

Ideal Classes and Cappell-Shaneson Homotopy 4-Spheres

Min Hoon Kim∗, Shohei Yamada

Department of Mathematics Education, Kyungpook National University, Daegu, Korea
e-mail: minhoonkim@knu.ac.kr

e-mail: fujijyu alcyone@yahoo.co.jp

Received: January 18, 2023; Revised: June 7, 2023; Accepted: July 15, 2023

Gompf proposed a conjecture on Cappell-Shaneson matrices whose affirmative answer implies that all Cappell-Shaneson homotopy 4-spheres are diffeomorphic to the standard 4-sphere. We study Gompf conjecture on Cappell-Shaneson matrices using various algebraic number theoretic techniques. We find a hidden symmetry between trace n Cappell-Shaneson matrices and trace 5-n Cappell-Shaneson matrices which was suggested by Gompf experimentally. Using this symmetry, we prove that Gompf conjecture for the trace n case is equivalent to the trace 5-n case. We confirm Gompf conjecture for the special cases that 64trace69 and corresponding Cappell-Shaneson homotopy 4-spheres are diffeomorphic to the standard 4-sphere. We also give a new infinite family of Cappell-Shaneson spheres which are diffeomorphic to the standard 4-sphere.

Keywords: Cappell-Shaneson homotopy 4-spheres, Ideal class monoids

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 ASL(3;) with det(AI)=1 and a choice of framing ε2. We say a matrix ASL(3;) is a Cappell-Shaneson matrix if det(AI)=1. For example, for any n, the following matrix An is a Cappell-Shaneson matrix

An=01001110n+1.

We first recall history on Cappell-Shaneson spheres ΣAnϵ corresponding to the family An 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 4, denoted by Q, corresponding to the matrix

010001110

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,yxyx=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 ϵ2. 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=0ab0cd10nc

for some integers c, d and n such that fn(c)0(Mod d) where fn(x)=x3nx2+(n1)x1. Note that fn(x) is the minimal polynomial of Xc,d,n and the entries a and b are determined as b=(c1)(nc1) and ad=fn(c) from the equalities det(Xc,d,n)=det(Xc,d,nI)=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([Θn]) where Θn is a root of fn(x). Via the bijection, the similarity class of Xc,d,n corresponds to the ideal class [Θnc,d]. (In particular, An=X1,1,n+2 corresponds to the identity element in C([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([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 ϵ2 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([Θ5]) 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=058023107

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. (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 ϵ2. 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 and ϵ2.

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 n3, 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 [Θn] to [Θ5n] which induces a monoid isomorphism between C([Θn]) and C([Θ5n]). 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 64n69.

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

When [Θn] is not a Dedekind domain, the current version of MAGMA cannot compute C([Θn]). (Nonetheless, using MAGMA, we can still compute a strictly smaller subset Pic([Θn]) of C([Θn]), consisting of the classes of invertible ideals.) We will observe that there are infinitely many integers n such that [Θn] is not a Dedekind domain. In fact, for each integer k, [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 [Θ27] explicitly (for details, see Section 4.3), and determine the monoid structure of C([Θ27]). The authors think that our method could also be used to study C([Θn]) for general n such that [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 3n69. In Table 2-5, we give representatives of elements of C([Θn]) for 3n69. 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. 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 0c94 and a19,37, or if 1d35. 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 0c94, or if 1d134 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 64n69. More generally, ΣXc,d,nϵ is diffeomorphic to S4 for any ϵ2 and for any integers c, d and n that satisfy fn(c)0(Mod d) and nn0(Mod d) for some 64n069. In particular, ΣXc,d,nϵ is diffeomorphic to S4 for any ϵ2 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([Θn]) for 3n69. 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([Θn]) for 64n69. 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 Mk (k),

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

Note that Mk=X2,7,49k+27, and hence ΣMkϵ is diffeomorphic to S4 for any ϵ2 by Corollary C. (This fact can be also checked by using a weaker version given in [20, Theorem 3.2].) We show that Mk is not similar An for any integers k and n.

Corollary D. For any integers k and ϵ2, Cappell-Shaneson sphere ΣMkϵ corresponding to Mk is diffeomorphic to S4. For any integers k and n, Mk is not similar to An.

Recall that Akbulut [7] showed that the infinite family of Cappell-Shaneson matrices An give Cappell-Shaneson spheres ΣAnϵ are diffeomorphic to S4 for any ϵ2. Since Mk is not similar to An 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([Θ49k+27]) is not a group for any integer k, and discuss the structure of C([Θ27]). In Section 5, we use MAGMA software to find representatives of elements in Pic([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;) be the set of 3×3 integral matrices whose determinants are 1. We say two matrices A,BSL(3;) are similar if there is a matrix CSL(3;) such that A=CBC1.

Definition 2.1. A matrix ASL(3;) is a Cappell-Shaneson matrix if AISL(3;).

For a Cappell-Shaneson matrix ASL(3;), Cappell and Shaneson [14] constructed two homotopy 4-spheres ΣAε as follows. Let T3 be the 3-torus 3/3. Since ASL(3;), A induces an orientation-preserving diffeomorphism fA:T3T3. Possibly after an isotopy, we can assume that fA is the identity on a neighborhood Dy of some chosen point y∈ T3. Let WA be the mapping torus of fA, that is,

WA=T3×[0,1]/(x,0) (fA(x),1).

Since fA is the identity around the point y, we can regard Dy×S1WA. From the condition det(AI)=1, the Wang sequence applied to the fiber bundle T3WAS1, and Van Kampen theorem show that WA is a homology S1×S3 whose fundamental group π1(WA) is normally generated by [y×S1]. If we remove Dy×S1 from WA and glue S2×D2 along the boundary via a framing ϵ2, 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 WA and WB 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)=x3nx2+(n1)x1.

Remark 2.4. For ASL(3;), 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 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=0ab0cd10nc.

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,nI)=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=0ab0cd1enc

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=110010011.

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

1e00100010ab0cd1enc1e0010001=0a+ceb+de0cd10nc.

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)Δk. The above argument shows that the matrix

ΔkA=0akcbkd0cd1kc+ekd+nc

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,d0 and n, the following are equivalent.

  • (1) fn(c)0 (mod d).

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

    Xc,d,n=0ab0cd10nc

    is a standard Cappell-Shaneson matrix.

Proof. We first note that fn(c)=c3nc2+(n1)c1=c(c1)(nc1)1. Suppose that fn(c)0 (mod d). Define a,b by fn(c)=ad and b=(c1)(nc1). Consider the following matrix

A=0ab0cd10nc.

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

detA=adbc=fn(c)c(c1)(nc1)=1

and

det(AI)=(c1)(nc1)+(adb(c1))=1.

For the converse, consider the Cappell-Shaneson matrix

Xc,d,n=0ab0cd10nc.

By Remark 2.8, fn(c)=c(c1)(nc1)1=bc1=ad0(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 IJ 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 II(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=[Θ] where Θ is a root of a monic polynomial g(x)[x] which is irreducible (over ). Note that [Θ] is the number field obtained by adjoining Θ to .

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[x] is a monic, irreducible polynomial of degree 3. Let Θ be a root of g. Then there is a bijection between C[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=[Θ]. Regard A as a K-linear map A:K3K3. 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 . 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 xi lies in [Θ] by multiplying some integer. Let I be the -module generated by x1,x2 and x3. Then, I is an ideal of [Θ]. The ideal class [I]C([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)=x3nx2+(n1)x1. 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([Θn]). We will explicitly describe the bijection.

Consider a Cappell-Shaneson matrix with trace n,

Xc,d,n=0ab0cd10nc.

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

0ab0cd10ncx1x2x3=Θnx1Θnx2Θnx3,  ax2+bx3=Θnx1,cx2+dx3=Θnx2,x1+(nc)x3=Θnx3.

In particular, (x1,x2,x3)=((Θnn+c)(Θnc),d,Θnc) is an eigenvector of A in [Θn]3 with the eigenvalue Θn.Note that (Θnn+c)(Θnc),d,Θnc=Θnc,d. Hence the ideal class [Θnc,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([Θn]), which is defined by

Xc,d,n=0ab0cd10nc[Θnc,d]

where fn(c)0(mod d), b=(c1)(nc1) and ad-bc=1.

Remark 2.15. For k, by Proposition 2.14, Xc,d,n and Xc+kd,d,n are similar because Θnc,d=Θnckd,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=110010011.

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

0ab0cd1enc.

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 ϵ2.

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

A=0ab0cd1enc,

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 [Θnc0,d0]=[Θnc1,d1]C([Θn]),Xc,d,n GXc,d,n+kd if k.

Using Theorem 2.16 and Aitchison-Rubinstein's computation of C([Θn]) for small n, Gompf proved that Conjecture 1.2 is true for trace n if 6n9 or n=11. In Section 6.1, we will show that Conjecture 1.2 is true for trace n if 64n69.

Theorem 2.19 ([20, Theorem 3.2]). Conjecture 1.2 is true for trace n if 6n9 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)=x3nx2+(n1)x1. Consider

CS={(c,d,n)3fn(c)0(mod d)and d0}.

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=0ab0cd10nc

where b=(c1)(nc1) 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 [Θnc0,d0]=[Θnc1,d1]C([Θn]),(c,d,n) G(c,d,n+kd) if k.

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([Θn+2]) because Θn+21,1 is principal. Since (1,1,n+2) G(1,1,2), we do not have to consider the trivial element of C([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.)

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)=x3nx2+(n1)x1 for each integer n. We give a ring isomorphism between [Θn] and [Θ5n] 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 fn has the following symmetry:

Δ(fn)=n(n2)(n3)(n5)23=Δ(f5n).

On the other hand, Gompf [20][page 1672] computed the cardinality #C(On) for r108 via PARI/GP [15] and observed that #C(On)=#C(O5n) where On is the ring of integer of [Θn].

Theorem 3.1. For any integer n, let Θn be a root of fn(x)=x3nx2+(n1)x1. Then, there is a ring isomorphism φn:[Θn][Θ5n] defined by

φn(Θn)=Θ5n2+(n4)Θ5n+1.

Proof. For an aesthetic reason, we prove an equivalent statement that the ring homomorphism φ5n:[Θ5n][Θn] is an isomorphism for any integer n. The ring isomorphism φ5n will be defined as φ5n(Θ5n)=Θn2+(1n)Θn+1. Let φ¯5n:[x][Θn] be a ring homomorphism which sends x to Θn2+(1n)Θn+1. We prove that φ¯5n induces the ring homomorphism φ5n:[Θ5n][Θn] by observing that

φ¯5n(f5n(x))=f5n(Θn2+(1n)Θn+1)=0

where f5n(x)=x3(5n)x2+(4n)x1. By setting αn=φ5n(Θ5n)=Θn2(n1)Θn+1, we show f5n(αn)=0. Recall that Θn is a root of fn(x)=x3nx2+(n1)x1=0. We have

Θn(Θn1)(Θnn+1)=Θn3nΘn2+(n1)Θn=1.

The following equality will be useful.

(Θn1)αn=(Θn1)(Θn2(n1)Θn+1)    =Θn(Θn1)(Θnn+1)+Θn1=Θn.

Since Θn10, the following shows that f5n(αn)=0:

(Θn1)3f5n(αn)=(Θn1)3(αn3(5n)αn2+(4n)αn1)       =Θn3(5n)Θn2(Θn1)+(4n)Θn(Θn1)2(Θn1)3       =Θn3(Θn1)3Θn(Θn1)((5n)Θn(4n)(Θn1))       =3Θn(Θn1)+1Θn(Θn1)(Θnn+4)       =1Θn(Θn1)(Θn(n1))=0.

Therefore, we have a ring homomorphism φ5n:[Θ5n][Θn] such that φ5n(Θ5n)=Θn2+(1n)Θn+1. Now we prove that φ5nφn is the identity map on [Θn] by showing that φ5nφn(Θn)=Θn. To simplify the proof, we give two elementary observations. Since f5n(αn)=0, αn(αn1)(αn+n4)=1. Note that (Θn1)(αn1)=(Θn1)αnΘn+1=1.

φ5nφn(Θn)=φ5n(Θ5n2+(n4)Θ5n+1)     =αn2+(n4)αn+1     =(Θn1)(αn1)(αn2+(n4)αn+1)     =(Θn1)((αn1)αn(αn+n4)+αn1)     =(Θn1)(1+αn1)     =(Θn1)αn     =Θn.

By substituting n by 5-n, φ5nφn is also the identity. Hence, φ5n is a ring isomorphism and this completes the proof.

Remark. By tensoring to the ring isomorphism φn:[Θn][Θ5n] given in Theorem 3.1, we obtain a field isomorphism from [Θn] to [Θ5n]. From this field isomorphism, we can see that their ring of integers On and O5n are also isomorphic and Δ(fn)=Δ(f5n) for any integer n.

Corollary 3.2. There exists a monoid isomorphism ψn:C([Θn])C([Θ5n]) for any integer n. Furthermore, for any integers c, d with fn(c)0 fn(c)0(modd), ψn sends [Θnc,d] to [Θ5npn(c),d] where pn(x)=x2+(1n)x+1.

Proof. Define a monoid homomorphism ψn:C([Θn])C([Θ5n]) 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(Θnc,d)=Θ5npn(c),d and this clearly implies the desired statement.

Claim 3.3. Θnc,d=pn(Θn)pn(c),d

Proof of Claim. The following calculation shows that pn(Θn)pn(c),dΘnc,d:

pn(Θn)pn(c)=Θn2+(1n)Θn+1c2(1n)c1      =(Θnc)(Θn+cn+1)Θnc,d.

Now we prove Θnc,dpn(Θn)pn(c),d. Note that

fn(x)+1=x3nx2+(n1)x=(x1)(x2+(1n)x)=(x1)(pn(x)1).

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

Θnc=Θn1(c1)  =(Θn1)(fn(c)+1)(Θn1)fn(c)(c1)(fn(Θn)+1)  =(Θn1)(c1)(pn(c)1)(Θn1)fn(c)(Θn1)(c1)(pn(Θn)1)  =(Θn1)(c1)(pn(c)pn(Θn))(Θn1)fn(c).

From the assumption fn(c)0 (mod d), fn(c)pn(Θn)pn(c),d. This shows that Θnc,dpn(Θn)pn(c),d and hence the claim follows.

Note that pn(Θn)=Θn2+(1n)Θn+1=φ5n(Θn). By the claim,

φn(Θnc,d)=φn(φ5n(Θn)pn(c),d)=Θnpn(c),d.

Here the last equality follows from the fact that φnφ5n 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=0ab0cd10ncA*=0a*b*0c*d*105nc*

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

Proof. Since every Cappell-Shaneson matrix is similar to a standard Cappell-Shaneson matrix, the bijection AA* 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([Θn]) (respectively, C([Θ5n])) by Proposition 2.14. On the other hand, we have a monoid isomorphism ψn:C([Θn])C([Θ5n]) 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 [Θnc,d]. By Corollary 3.2, ψn sends the ideal class [Θnc,d] to the ideal class [Θ5npn(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=A7.)

A=010011106,B=058023107.

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*=07251280173107.

By Proposition 2.14, the ideal classes correspond to A* and B* are [1,Θ108] and [3,Θ1017], respectively. Note that [1,Θ108=[1,Θ101] and [3,Θ1017]=[3,Θ102]. 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. By Theorem 3.4, A*=Xc*,d,5n and B*=Xc*,d,5nkd, 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,5n)*=Xp5n(pn(c)),d,n where pn(c)=c2+(1n)c+1. Note that

p5n(pn(c))=pn(c)2+(n4)pn(c)+1    =(c2+(1n)c+1)2+(n4)(c2+(1n)c+1)+1    =c+fn(c)(cn+2)c(modd)

since fn(c)0(Mod d). Since p5n(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.

In this section, we use several techniques from algebraic number theory. We will recall Dedekind-Kummer theorem, and show C([Θ49k+27]) is not a group for any integer k. We will also determine the structure of the ideal class monoid C([Θ27]). We first collect some definitions following [23].

Definition 4.1 (Number rings and orders) A number field K is a finite degree field extension of the field of rational numbers. A number ring is an integral domain R for which the field of fractions K is a number field. For a number field K with degree n, a subring R of the number field K is called an order if R is a free -module of rank n.

Example 4.2 ([Θn] is an order). Let α be a root of some monic, irreducible polynomial f[x] of degree n. Then [α] is a number field of degree n. The ring [α] obtained by adjoining to has a free -basis 1,α,,αn1 and hence [α] is an order in the number field [α]. We are principally interested in the orders of the form [Θn] where Θn is a root of the monic, irreducible polynomial fn(x)=x3nx2+(n1)x1.

Definition 4.3 (Ring of integers) Let K be a number field. An element x in K is an integral element if x is a root of monic, irreducible polynomial with integer coefficients. The set of integral elements in K is called the ring of integer of K and denoted by OK.

We recall elementary facts on orders discussed in [23].

Theorem 4.4 ([23, Sections 6-7]). A number ring R⊂ K is an order in K if and only if R is of finite index in OK. In particular, OK is the maximal order in K. For an order RK, the following conditions are equivalent.

  • (1) R is integrally closed.

  • (2) R is the maximal order OK.

  • (3) R is a Dedekind domain.

  • (4) Every ideal of R is invertible.

  • (5) C(R) is a group.

4.1. Dedekind-Kummer theorem

As in Section 2.2, for two ideals I and J in [Θn], we say I and J are equivalent (and denoted by IJ) if αI=βJ for some non-zero α,β[Θn]. By the definition of C([Θn]), IJ if and only if [I]=[J]C([Θn]). By Proposition 2.14, every ideal of [Θn] is equivalent to Θnc,d for some c,d such that fn(c)0(Mod d). By Proposition 4.10, we know that there are infinitely many n such that C([Θn]) is not a group. For those n, there is a non-invertible ideal Θnc,d of [Θn]. Therefore we want to determine when the ideal Θnc,d such that fn(c)0 (mod d) is invertible. For this purpose, we can assume that d is prime power by the following remark.

Remark 4.5. Suppose that p and q are relatively prime integers. Then Θnc is a linear combination of p(Θnc) and q(Θnc). It follows that

Θnc,pΘnc,q=(Θnc)2,p(Θnc),q(Θnc),pq=Θnc,pq.

More generally, consider the prime factorization d=p1e1pmem. Then

Θnc,d=Θnc,p1e1Θnc,p2e2Θnc,pmem.

Note that fn(c)0 (mod piei) since fn(c)0 (mod d).

Following [23, Theorem 8.2], we recall Dedekind-Kummer theorem, which can be used to determine when the ideal of the form Θnc,p such that p is prime and fn(c)0(mod p) is invertible.

Theorem 4.6 (Dedekind-Kummer [23, Theorem 8.2]). Let p be a prime integer and α be a root of a monic, irreducible polynomial f(x)[x]. Let f¯p[x] be a polynomial such that f¯f(mod p). Let the factorization of f¯ in p[x] be i=1lg¯iei. Let gi[x] be a polynomial such that gi g¯i(mod p). If ri[x] is the remainder of f upon division by gi in [x], that is, f=giqi+ri, then the ideal pi=p,gi(α)[α] is prime and pi is invertible if and only if at least one of the following conditions holds.

  • (1) ei=1.

  • (2) p2 does not divide ri[x].

By applying Dedekind-Kummer theorem to the case that α=Θn and f=fn(x), we obtain the following proposition which gives a simple, but complete characterization when ideals of the form Θnc,p such that p is prime and fn(c)0 (mod p) are invertible. This will be useful in our analysis of the structure of C([Θ27]).

Proposition 4.7. Suppose that integers c, n and p satisfy fn(c)0 (mod p). If p is prime, then Θnc,p is a prime ideal of [Θn]. The ideal Θnc,p is invertible if and only if at least one of the following conditions holds.

  • (1) c is a simple root of fn(x) modulo p.

  • (2) p2 does not divide fn(c).

Proof of Proposition 4.7. Recall fn(x)=x3nx2+(n1)x1 is a monic, irreducible polynomial with a root Θn. If fn(c)0(mod p), then x-c is a factor of fn¯ in p[x]. On the other hand, we can write fn(x)=(xc)q(x)+fn(c). By applying Theorem 4.6 for p=p,Θnc where g(x)=xc and r(x)=fn(c), we obtain the conclusion.

Proposition 4.8. Suppose that p is a prime integer and an integer c satisfies fn(c)0(mod pk) for some positive integer k.

  • If Θnc,p is invertible, then Θnc,pk is invertible.

  • If fn(c)0(mod pk+1), then Θnc,pk is invertible.

Proof. (1) Denote I=Θnc,pk and p=Θnc,p. Assume that p is invertible. We first observe that I=p where I is the radical of I. Let α be an element in p. We can write α=xp+y(Θnc) for some x,y[Θn]. Then αk=(xp+y(Θnc))kI. This shows that pI. Recall that I is the intersection of all prime ideals which contain I. By Proposition4.7, p is a prime ideal which contains I. It follows that Ip. Since we are assuming p is invertible, by Lemma 4.9 below, we conclude that I is invertible.

(2) From the hypothesis, we can write fn(c)=pkq where q is relatively prime to p. Then, Θnc,pkΘnc,q=Θnc,pkq by Remark4.5. Since fn(Θn)=0, we have pkq=fn(c)fn(Θn)Θnc. It follows that Θnc,pkq=Θnc which is a principal ideal. This completes the proof.

Lemma 4.9. Let R be a number ring and p be an invertible prime ideal of R. If I is an ideal of R such that I=p, then I=pk for some k. In particular, I is an invertible ideal.

Proof. By [23, page 213], every ideal of R is finitely generated, and every prime ideal of R is maximal. In particular, p is maximal. By [12, Proposition 4.2], I is p-primary. That is, I=p, and if xyI, then either xI or ynI for some n>0.

Let K be the quotient field of R. Consider the localization Rp={rsKrR,sp} of R at p and the canonical homomorphism fp:RRp. Since p is an invertible prime ideal, every ideal of Rp is a power of pRp by [23]. In short, Rp is a discrete valuation ring.

Let Ip be the extension of I. That is, Ip is the ideal of Rp generated by fp(I). Since Rp is a discrete valuation ring, Ip=(pRp)k for some k. Then

I=fp1(Ip)=fp1((pRp)k)=(fp1(pRp))k=pk.

We remark that the first equality uses the fact that I is p-primary (see [12, Proposition 3.11(2) and Lemma 4.4(3)]). This completes the proof.

4.2 The case n=49k+27

In this subsection, we prove that there are infinitely many integers n such that [Θn] is not a Dedekind domain. Hence, to study general Cappell-Shaneson spheres, we need to understand equivalence classes of non-invertible ideals of [Θn] for those n. For general n, finding an explicit formula for #C([Θn]) (and its representatives) seems to be a difficult problem in algebraic number theory. Because of these subtleties, proving Conjecture 1.2 is difficult.

Proposition 4.10. For any integer k, the ideal Θ49k+272,7 is not an invertible ideal in [Θ49k+27], and hence C([Θ49k+27]) is not a group. Consequently, [Θ49k+27] is not a Dedekind domain for any integer k.

Proof. We first observe that

f49k+27(x)=x3(49k+27)x2+(49k+26)x1(x2)3(mod 7).

It is straightforward to check that f49k+27(2)=49(2k+1). Therefore, 2 is not a simple root of f49k+27(x)0(mod 7), and 49 divides f49k+27(2). By Proposition 4.7, Θ49k+272,7 is not an invertible ideal of [Θ49k+27] for any integer k.

Recall from Theorem 4.4 that C(R) is not a group if and only if R is not integrally closed. We give another proof of the fact that C([Θ49k+27]) is not a group by directly showing that [Θ49k+27] is not integrally closed for any integer k.

Proposition 4.11. For any integer k, [Θ49k+27] is not integrally closed, and hence C([Θ49k+27]) is not a group for any integer k. Equivalently, [Θ49k+27] is not a Dedekind domain for any integer k.

Proof. Fix an integer k and it suffices to prove that [Θ49k+27] is a proper subset of O49k+27 where O49k+27 is the ring of integer of [Θ49k+27]. Set ηk=17(Θ49k+272)2 [Θ49k+27]. We will show that ηk is an integral element or equivalently ηkO49k+27. Let

gk(x)=x3(343k2+336k+83)x2+(245k2+238k+58)x(28k2+28k+7),u(x)=17(x2)2.

Then gk(ηk)=gk(u(Θ49k+27))=0 since

343gk(u(x))=f49k+27(x)(x3+(49k+15)x2(343k+142)x+588k+265).

The last equality can be easily checked by expanding terms in both sides.

Now we prove that gk is irreducible over for any integer k. Suppose that the cubic, monic polynomial gk is reducible over . Then gk is reducible over 2 so it has a solution in 2. Since gk(0) and gk(1) are odd, gk does not have a solution in 2. It follows that gk is irreducible over . That is, gk is the minimal polynomial of ηk, and hence ηkO49k+27. Since ηk [Θ49k+27], this completes the proof that [Θ49k+27] is not equal to O49k+27.

4.3. The computation of the ideal class monoid when n = 27

In the previous section, we showed that C([Θ49k+27]) is not a group for any integer k. Among 3n75, n=27 is the only case that C([Θn]) is not a group, but a monoid (this can be checked either using MAGMA or PARI/GP). Nonetheless, MAGMA can still compute the Picard group Pic([Θ27]) consists of the ideal classes of invertible ideals in [Θ27] (see Section 5.3).

The goal of this subsection is to prove Theorem 4.17 where we determine the monoid structure of C([Θ27]). The key ingredients are Proposition 4.12 and the computation of Pic([Θ27]).

Proposition 4.12. Let I be a non-zero ideal of [Θ27]. Then exactly one of the following holds.

  • (1) I is invertible.

  • (2) I is equivalent to Θ272,7J for some invertible ideal J.

Proof. Let I be a non-zero ideal of [Θ27]. By Proposition 2.14, I is equivalent to Θ27c,d where f27(c)0(mod d). Consider the prime factorization d=p1e1p2e2pmem. By Remark 4.5,

Θ27c,d=Θ27c,p1e1Θ27c,p2e2Θ27c,pmem.

Since the product of invertible ideals is invertible, it suffices to determine which ideals Θ27c,piei are not invertible. (Note that f27(c)0 (mod piei) for any i.)

Recall that Aitchison and Rubinstein [1, page 43] computed the discriminant

Δ(f27)=2725242223=356377

which has the prime factorization 731039. If a prime p does not divide the discriminant Δ(f27), then every root of f27(x) modulo p is a simple root. By Propositions 4.7 and 4.8, if the prime pi is not equal to 7 and 1039, then Θ27c,piei is invertible.

Consider an ideal Θ27c,1039k such that f27(c)0(mod 1039k). We show that Θ27c,1039 is invertible. By Proposition 4.8, this implies that Θ27c,1039k is invertible. Since

f27(x)=x327x2+26x1(x453)2(x160) (mod 1039),

c=1039l+453 or c=1039l+160. Since 160 is a simple root of f27(x)0(mod 1039), Θ27c,160 is invertible by Proposition 4.7. On the other hand, consider the prime factorization f27(453)=1310396473. By Proposition 4.7, this shows that Θ27c,453 is also invertible.

Now it remains to consider an ideal Θ27c,7k such that f27(c)0(mod 7k). If f27(c)0 (mod 7k+1), then Θ27c,7k is invertible by Proposition 4.8(2). If f27(c)0(mod 7k+1), then Θ27c,7kΘ272,7 by Lemma 4.13 below. This completes the proof.

Lemma 4.13. Let k be a positive integer and c be an integer such that f27(c)0 (mod 7k+1). Then Θ27c,7kΘ272,7.

Proof of Lemma 4.13. By Proposition 4.14, c2(mod 7). If k=1, then Θ27c,7=Θ272,7 since c2(mod 7).

If k≥ 2, we apply Proposition 4.15 several times to obtain the desired conclusion

Θ27c,7kΘ27c,7k1Θ27c,7=Θ272,7

The last equality follows because c2(mod 7).

Proposition 4.14. Let c be an integer such that f27(c)0(mod 7). Then c2(mod 7).

Proof. Since f27(x)=x327x2+26x1(x2)3(mod 7), the conclusion directly follows.

Proposition 4.15. If k2 and f27(c)0 (mod 7k+1), then Θ27c,7kΘ27c,7k1.

Proof. Note that

f27(x)=x327x2+26x1=(x2)37(3(x2)2+10(x2)+7).

Since Θ27 is a root of f27(x), (Θ272)3=7(3(Θ272)2+10(Θ272)+7). By Proposition 4.14, we can write c=7l+2 for some l.

It follows that

(Θ27c)(Θ272)2=(Θ272)37l(Θ272)2=7((3l)(Θ272)2+10(Θ272)+7).

In Proposition 4.16, if k2, then we will observe that

Θ27c,7k=(3l)(Θ272)2+10(Θ272)+7,7k2(Θ272)2.

This observation completes the proof since

Θ27c,7k1(Θ27c)(Θ272)2,7k1(Θ272)2    =7(3l)(Θ272)2+10(Θ272)+7,7k2(Θ272)2    (3l)(Θ272)2+10(Θ272)+7,7k2(Θ272)2    =Θ27c,7k

where we have used the equality (Θ27c)(Θ272)2=7((3l)(Θ272)2+10(Θ272)+7).

Proposition 4.16. If k2 and f27(c)0 (mod 7k+1), then Θ27c,7k=(3l)(Θ272)2+10(Θ272)+7,7k2(Θ272)2.

Proof. Recall that

f27(x)=x327x2+26x1=(x2)37(3(x2)2+10(x2)+7).

By Proposition 4.14, we can write c=7l+2 for some l. Then f27(c)=49(7l321l210l1). Since f27(c)0 (mod 7k+1) and k2,

7l321l210l10(mod 7k1).

We first prove that (3l)(Θ272)2+10(Θ272)+7,7k2(Θ272)2Θ27c,7k.

Since k2, the following computation shows that 7k2(Θ272)2Θ27c,7k:

7k2(Θ272)2=7k2(Θ27c+7l)2=(Θ27c)7k2(Θ27c)+27k1l+7kl2.

Since c=7l+2, (3l)(Θ272)2+10(Θ272)+7=(3l)(Θ27c+7l)2+10(Θ27c+7l)+7. The right hand side is equal to (Θ27c)((3l)(Θ27c+14l)+10)7(7l321l210l1) which is in the ideal Θ27c,7k since we observed 7l321l210l10(mod 7k1).

Now we prove Θ27c,7k(3l)(Θ272)2+10(Θ272)+7,7k2(Θ272)2. We consider two equalities

7k=7k1((3l)(Θ272)2+10(Θ272)+7)((3l)(Θ272)+10)7k1(Θ272) 7k1=7k2((3l)(Θ272)2+10(Θ272)+7)((3l)(Θ272)+10)7k2(Θ272).

From (a) and (b), 7k1(Θ272) and 7k are in (3l)(Θ272)2+10(Θ272)+7,7k2(Θ272)2.

It remains to prove Θ27c(3l)(Θ272)2+10(Θ272)+7,7k2(Θ272)2. Since 7k1 and 7(10l1)(3l)+100 are coprime, it suffices to prove

7k1(Θ27c)(3l)(Θ272)2+10(Θ272)+7,7k2(Θ272)2. (7(10l1)(3l)+100)(Θ27c)(3l)(Θ272)2+10(Θ272)+7,7k2(Θ272)2.

By (b), (c) directly follows. Consider

(7(10l1)(3l)+100)(Θ27c)=(70l(3l)+1007(3l))(Θ27c)= 10((3l)(Θ272)+7l(3l)+10)(Θ27c)(3l)(10(Θ272)+7)(Θ27c)=10((3l)(Θ27c)+14l(3l)+10)(Θ27c)(3l)(10(Θ272)+7)(Θ27c).

Therefore, to prove (d), it suffices to prove that the following two terms in (e) are in the ideal (3l)(Θ272)2+10(Θ272)+7,7k2(Θ272)2:

((3l)(Θ27c)+14l(3l)+10)(Θ27c) and (10(Θ272)+7)(Θ27c).

The first term ((3l)(Θ27c)+14l(3l)+10)(Θ27c) of (e) is equal to

(3l)(Θ272)2+10(Θ272)+7((3l)49l2+70l+7).

Recall that we observed 7l321l210l10(mod 7k1) in the beginning of the proof. It follows that (3l)49l2+70l+7=7(7l321l210l1)0(mod 7k). Since we proved 7k(3l)(Θ272)2+10(Θ272)+7,7k2(Θ272)2, the term ((3l)(Θ27c)+14l(3l)+10)(Θ27c) is also in (3l)(Θ272)2+10(Θ272)+7,7k2(Θ272)2.

To show the second term (10(Θ272)+7)(Θ27c) of (e) is in (3l)(Θ272)2+10(Θ272)+7,7k2(Θ272)2, consider

(10(Θ272)+7)(Θ27c)=(10(Θ272)+7)(Θ2727l)=10(Θ272)2+7(Θ272)70l(Θ272)49l=10(Θ272)2+7(Θ272)+(3l)70(Θ272)+49(3l)210(Θ272)147=10(Θ272)2+7(Θ272)+(3l)(21(Θ272)2+70(Θ272)+49)  21(3l)(Θ272)2210(Θ272)147=10(Θ272)2+7(Θ272)+(3l)(Θ272)3  21((3l)(Θ272)2+10(Θ272)+7)=(Θ2723)((3l)(Θ272)2+10(Θ272)+7).

Note that we used the equality (Θ272)3=21(Θ272)2+70(Θ272)+49. This completes the proof.

Theorem 4.17. The ideal class monoid C([Θ27]) consists of the following 7 elements,

I0=[Θ272,7],I1=[Θ277,17],I2=[Θ274,5],I3=[Θ2711,13],I4=[Θ2710,11],I5=[Θ2714,19],I6=[Θ27,1,1].

The multiplication table of C([Θ27]) is given in Table 1.

Table 1 . The multiplication table of C([Θ27])..

I0I1I2I3I4I5I6
I0I0I0I0I0I0I0I0
I1I0I2I3I4I5I6I1
I2I0I3I4I5I6I1I2
I3I0I4I5I6I1I2I3
I4I0I5I6I1I2I3I4
I5I0I6I1I2I3I4I5
I6I0I1I2I3I4I5I6


Proof. In Section 5.3, we observe that Pic([Θ27]) consists of I1,I2,,I6, and the multiplication is given by IiIj=Ii+j for all 1i,j6 where subscripts are understood modulo 6. It suffices to analyze equivalence classes of non-invertible ideals of [Θ27]. By Proposition 4.12, every non-zero, non-invertible ideal of [Θ27] is equivalent to Θ272,7J for some invertible ideal J. Let I0 be the equivalence class of the non-invertible ideal Θ272,7.

Since Pic([Θ27]) consists of Ii for i=1,,6, the equivalence class of J is Ii for some 1i6. In Section 5.3, we observe the following.

  • (1) For i=1,,6, each Ii is represented by the ideal Θ2727k,49 for some k=0,1,3,4,5,6.

  • (2) Θ272,49 is a principal ideal.

  • (3) For any k=1,3,4,5,6, Θ272,7Θ272,49=Θ272,7Θ2727k,49.

Since Θ272,49 is a principal ideal and Θ272,7 represents I0, these observations imply that

I0Ii=IiI0=I0

for any i=1,,6. To obtain Table 1, it remains to show that I0I0=I0. For this, we show that

Θ272,7Θ272,7Θ272,7.

Recall that f27(x)=x327x2+26x1=(x2)37(3(x2)2+10(x2)+7). It follows that (Θ272)3=7(3(Θ272)2+10(Θ272)+7). Then we have

Θ272,7(Θ272)3,7(Θ272)2    =7(3(Θ272)2+10(Θ272)+7),7(Θ272)2.

On the other hand, we have

Θ272,7Θ272,7=(Θ272)2,7(Θ272),49          (Θ272)3,7(Θ272)2,49(Θ272)          =7(3(Θ272)2+10(Θ272)+7),7(Θ272)2,49(Θ272).

Note that

49(Θ272)=7(3(Θ272)2+10(Θ272)+7)(Θ272)7(3(Θ272)+10)(Θ272)2    7(3(Θ272)2+10(Θ272)+7),7(Θ272)2.

It follows that

Θ272,7Θ272,77(3(Θ272)2+10(Θ272)+7),7(Θ272)2Θ272,7,

and this completes the proof.

In this section, we use MAGMA to find representatives of elements of Pic([Θn]).

Definition 5.1. Let x be an element of C([Θn]). We say a tuple (c,d,n)CS is a representative of x if the integers c and d satisfy 1cd and x=[Θnc,d]. (Recall that (c,d,n)CS if and only if fn(c)0 (mod d).) We say a representative (c,d,n) of x is minimal if (c',d',n) is another representative of x, then either d>d or d'=d and c>c.

Remark 5.2. Since Θn1,1 is principal, the minimal representative of the trivial element in C([Θn]) is (1,1,n). Every element x of C([Θn]) has a representative by Proposition 2.14. The minimal representative of x is minimal with respect to the colexicographic order on the set of representatives of x. Each tuple (c,d,n) corresponds to the standard Cappell-Shaneson matrix Xc,d,n.

5.1. The maximal order case

Here we assume that [Θn] is a Dedekind domain, that is, Pic([Θn])=C([Θn]) is a group. We give two pseudocodes each of which computes the following:

  • (1) The list of minimal representatives (c,d,n)CS such that d≤ N for any given integers N>0 and n such that [Θn] is a Dedekind domain.

  • (2) The representatives (c,d,n) of x such that d≤ N for a given representative (c0,d0,n) of an element x of C([Θn]) and a given integer N>0. (Equivalently, for a given standard Cappell-Shaneson matrix Xc0,d0,n and an integer N>0, the pseudocode computes the set of standard Cappell-Shaneson matrices Xc,d,n such that dN.)

We give the corresponding MAGMA codes in Section 5.2, and these MAGMA codes will be used in the proof of Theorem B given in Section 6.

5.2. MAGMA codes

In this subsection, we give MAGMA codes. One can execute the codes by pasting them to the online MAGMA calculator http://magma.maths.usyd.edu.au/calc/.

We first give a MAGMA code for Algorithm 1.

Now we give a MAGMA code for Algorithm 2.

5.3. Representatives of elements of the Picard group when n = 27

From the MAGMA code given below, Pic([Θ27])6 with a generator is represented by the ideal

I=1+18601Θ272,Θ27+3672Θ272,26737Θ272.

We check that I5Θ277,17 and Θ2723,49Θ277,17 are principal ideals. Since I6 is a principal ideal, we can conclude that IΘ277,17 and I5Θ2723,49. Similarly, we have

IΘ277,17Θ2744,49,I2Θ274,5Θ279,49,I3Θ2711,13Θ2730,49,I4Θ2710,11Θ2737,49,I5Θ2714,19Θ2723,49,I6Θ271,1Θ272,49.

Consequently, the elements of Pic([Θ27])6 have representatives

(1,1,27),(4,5,27),(10,11,27),(11,13,27),(7,17,27),(14,19,27).

Note that these representatives are actually in CS since

f27(1)=132712+2611=10(mod 1),f27(4)=432742+2641=2650(mod 5),f27(10)=10327102+26101=14410(mod 11),f27(11)=11327112+26111=16510(mod 13),f27(7)=732772+2671=7990(mod 17),f27(14)=14327142+26141=21850(mod 19).

To prove Theorem 4.17, for k=1,3,4,5,6, we also observe that

Θ272,7Θ272,49=Θ272,7Θ2727k,49

Here is the MAGMA code used in above.

5.4. Representatives the ideal class monoids

Table 2 . Representatives of elements of C([Θn]) for 3n34..

n#C([Θn])Representatives of elements of C([Θn])
31(1,1,3)
41(1,1,4)
51(1,1,5)
61(1,1,6)
71(1,1,7)
81(1,1,8)
91(1,1,9)
102(1,1,10), (2,3,10)
111(1,1,11)
122(1,1,12), (4,5,12)
133(1,1,13), (2,3,13), (3,5,13)
142(1,1,14), (5,7,14)
152(1,1,15), (2,5,15)
163(1,1,16), (2,3,16), (4,7,16)
173(1,1,17), (4,5,17), (6,7,17)
182(1,1,18), (3,5,18)
196(1,1,19), (2,3,19), (3,7,19), (5,9,19), (8,11,19), (9,11,19)
203(1,1,20), (2,5,20), (2,7,20)
213(1,1,21), (5,7,21), (9,13,21)
226(1,1,22), (2,3,22), (4,5,22), (8,9,22), (6,11,22), (14,17,22)
235(1,1,23), (3,5,23), (4,7,23), (5,11,23), (6,13,23)
244(1,1,24), (6,7,24), (3,11,24), (17,23,24)
259 (1,1,25), (2,3,25), (2,5,25), (2,9,25), (7,11,25), (7,13,25), (8,13,25), (10,13,25), (13,17,25)
264(1,1,26), (3,7,26), (4,11,26), (12,17,26)
277(1,1,27), (4,5,27), (2,7,27), (10,11,27), (11,13,27), (7,17,27), (14,19,27)
2810(1,1,28), (2,3,28), (3,5,28), (5,7,28), (5,9,28), (8,15,28), (13,19,28), (16,19,28), (19,23,28), (23,27,28)
294(1,1,29), (4,17,29), (8,19,29), (27,37,29)
308(1,1,30), (2,5,30), (4,7,30), (2,11,30), (8,11,30), (9,11,30), (5,13,30), (15,17,30)
317(1,1,31), (2,3,31), (6,7,31), (8,9,31), (9,17,31), (11,17,31), (15,23,31)
326(1,1,32), (4,5,32), (3,13,32), (4,13,32), (12,13,32), (18,23,32)
337(1,1,33), (3,5,33), (3,7,33), (6,11,33), (12,19,33), (16,23,33), (35,43,33)
3412(1,1,34), (2,3,34), (2,7,34), (2,9,34), (5,11,34), (9,13,34),(5,17,34), (9,19,34), (10,19,34), (15,19,34), (20,27,34), (26,41,34)

Table 3 . Representatives of elements of C([Θn]) for 35n50..

n#C([Θn])Representatives of elements of C([Θn])
3510(1,1,35), (2,5,35), (5,7,35), (3,11,35), (2,13,35), (3,17,35), (4,19,35), (17,25,35), (13,29,35), (17,37,35)
365(1,1,36), (7,11,36), (6,13,36), (8,17,36), (11,19,36)
3715(1,1,37), (2,3,37), (4,5,37), (4,7,37), (5,9,37), (4,11,37), (14,15,37), (6,17,37), (11,21,37), (5,23,37), (7,23,37), (5,27,37), (26,33,37), (14,45,37), (23,51,37)
3812(1,1,38), (3,5,38), (6,7,38), (10,11,38), (7,13,38), (8,13,38), (10,13,38), (10,17,38), (6,19,38), (8,25,38), (22,29,38), (43,55,38)
396(1,1,39), (14,17,39), (17,29,39), (25,29,39), (26,29,39), (21,37,39)
4016(1,1,40), (2,3,40), (2,5,40), (3,7,40), (8,9,40), (11,13,40), (2,15,40), (17,19,40), (17,21,40), (12,23,40), (20,29,40), (12,31,40), (25,31,40), (14,37,40), (30,41,40), (17,57,40)
419(1,1,41), (2,7,41), (2,11,41), (8,11,41), (9,11,41), (7,19,41), (21,23,41), (9,29,41), (28,43,41)
4210(1,1,42), (4,5,42), (5,7,42), (13,17,42), (16,17,42), (20,23,42), (19,25,42), (21,31,42), (19,43,42), (31,71,42)
4316(1,1,43), (2,3,43), (3,5,43), (2,9,43), (5,13,43), (8,15,43), (12,17,43), (5,19,43), (6,23,43), (13,25,43), (10,43,43), (38,45,43), (29,51,43), (15,53,43), (54,61,43), (10,67,43)
449(1,1,44), (4,7,44), (6,11,44), (7,17,44), (23,29,44), (9,31,44), (13,31,44), (22,31,44), (8,37,44)
4514(1,1,45), (2,5,45), (6,7,45), (5,11,45), (3,13,45), (4,13,45), (12,13,45), (2,17,45), (3,19,45), (12,25,45), (27,31,45), (27,35,45), (42,53,45), (24,61,45)
4612(1,1,46), (2,3,46), (5,9,46), (3,11,46), (4,17,46), (14,19,46), (3,23,46), (10,23,46), (14,27,46), (4,29,46), (11,29,46), (14,33,46)
4716(1,1,47), (4,5,47), (3,7,47), (7,11,47), (9,13,47), (15,17,47), (13,19,47), (16,19,47), (18,19,47), (17,23,47), (14,25,47), (28,31,47), (24,35,47), (18,41,47), (32,43,47), (39,83,47)
4818(1,1,48), (3,5,48), (2,7,48), (4,11,48), (2,13,48), (9,17,48), (11,17,48), (8,19,48), (8,23,48), (9,23,48), (18,25,48), (5,29,48), (20,31,48), (23,35,48), (25,43,48), (15,47,48), (54,67,48), (39,71,48)
4920(1,1,49), (2,3,49), (5,7,49), (8,9,49), (10,11,49), (6,13,49), (5,21,49), (4,23,49), (11,23,49), (8,27,49), (15,29,49), (25,37,49), (29,37,49), (32,37,49), (32,39,49), (13,43,49), (38,43,49), (50,69,49), (33,73,49), (41,89,49)
5012(1,1,50), (2,5,50), (2,19,50), (14,23,50), (22,25,50), (19,31,50), (11,37,50), (15,37,50), (24,37,50), (13,41,50), (9,43,50), (48,61,50)

Table 4 . Representatives of elements of C([Θn]) for 51n61..

n#C([Θn])Representatives of elements of C([Θn])
5113(1,1,51), (4,7,51), (7,13,51), (8,13,51), (10,13,51), (5,17,51), (13,23,51), (19,23,51), (12,29,51), (24,31,51), (31,41,51), (38,47,51), (43,47,51)
5228(1,1,52), (2,3,52), (4,5,52), (6,7,52), (2,9,52), (2,11,52), (8,11,52), (9,11,52), (14,15,52), (3,17,52), (12,19,52), (20,21,52), (9,25,52), (11,27,52), (27,29,52), (16,31,52), (8,33,52), (20,33,52), (36,41,52), (29,45,52), (20,51,52), (24,55,52), (20,63,52), (41,71,52), (59,73,52), (59,75,52), (56,87,52), (32,103,52)
53 15(1,1,53), (3,5,53), (11,13,53), (8,17,53), (9,19,53), (10,19,53), (15,19,53), (23,25,53), (8,29,53), (21,29,53), (24,29,53), (7,31,53), (24,41,53), (37,53,53), (45,83,53)
5412(1,1,54), (3,7,54), (6,17,54), (4,19,54), (15,23,54), (19,29,54), (4,31,54), (5,31,54), (14,31,54), (31,37,54), (28,41,54), (25,53,54)
5527(1,1,55), (2,3,55), (2,5,55), (2,7,55), (5,9,55), (6,11,55), (2,15,55), (10,17,55), (11,19,55), (2,21,55), (18,23,55), (7,25,55), (23,27,55), (17,33,55), (39,43,55), (32,45,55), (39,47,55), (44,51,55), (44,53,55), (17,55,55), (11,57,55), (17,61,55), (29,67,55), (41,69,55), (20,73,55), (32,75,55), (27,85,55)
5615(1,1,56), (5,7,56), (5,11,56), (5,13,56), (14,17,56), (16,23,56), (15,31,56), (16,37,56), (38,41,56), (29,43,56), (11,47,56), (20,47,56), (73,89,56), (75,101,56), (78,107,56)
5716(1,1,57), (4,5,57), (3,11,57), (6,19,57), (22,23,57), (4,25,57), (3,29,57), (7,29,57), (18,29,57), (12,37,57), (10,41,57), (43,53,57), (41,73,57), (15,79,57), (25,89,57), (15,109,57)
5836(1,1,58), (2,3,58), (3,5,58), (4,7,58), (8,9,58), (7,11,58), (3,13,58), (4,13,58), (12,13,58), (8,15,58), (11,21,58), (3,25,58), (17,27,58), (17,31,58), (18,31,58), (23,31,58), (29,33,58), (18,35,58), (23,37,58), (33,37,58), (17,39,58), (29,39,58), (19,41,58), (20,43,58), (8,45,58), (36,47,58), (29,53,58), (8,61,58), (53,63,58), (53,75,58), (56,79,58), (25,91,58), (20,109,58), (107,117,58), (101,123,58), (83,141,58)
5914(1,1,59), (6,7,59), (4,11,59), (13,17,59), (16,17,59), (17,19,59), (10,29,59), (26,31,59), (28,37,59), (41,49,59), (42,59,59), (26,61,59), (38,61,59), (55,67,59)
6016(1,1,60), (2,5,60), (10,11,60), (9,13,60), (12,17,60), (7,19,60), (2,23,60), (5,23,60), (7,23,60), (17,25,60), (6,37,60), (6,43,60), (5,47,60), (41,53,60), (32,55,60), (48,83,60)
6121(1,1,61), (2,3,61), (3,7,61), (2,9,61), (2,13,61), (7,17,61), (17,21,61), (20,27,61), (19,37,61), (30,43,61), (32,47,61), (35,47,61), (41,47,61), (41,51,61), (51,59,61), (53,59,61), (46,73,61), (23,79,61), (80,91,61), (85,103,61), (26,139,61)

Table 5 . Representatives of elements of C([Θn]) for 62n69..

n#C([Θn])
Representatives of elements of C([Θn])
6218(1,1,62), (4,5,62), (2,7,62), (6,13,62), (2,17,62), (5,19,62), (24,25,62), (14,29,62), (9,35,62), (22,37,62), (23,41,62), (24,43,62), (21,53,62), (14,59,62), (28,61,62), (19,65,62), (55,71,62), (63,73,62)
6324(1,1,63), (3,5,63), (5,7,63), (2,11,63), (8,11,63), (9,11,63), (4,17,63), (12,23,63), (8,25,63), (10,31,63), (11,31,63), (33,35,63), (4,41,63), (6,41,63), (12,41,63), (33,49,63), (13,55,63), (60,73,63), (68,77,63), (74,83,63), (28,97,63), (60,97,63), (72,97,63), (79,107,63)
6430(1,1,64), (2,3,64), (5,9,64), (7,13,64), (8,13,64), (10,13,64), (15,17,64), (3,19,64), (21,23,64), (5,27,64), (6,29,64), (13,29,64), (16,29,64), (8,39,64), (20,39,64), (23,39,64), (5,43,64), (17,43,64), (18,47,64), (28,47,64), (32,51,64), (40,53,64), (41,57,64), (37,59,64), (32,67,64), (51,71,64), (74,87,64), (62,97,64), (75,109,64), (146,159,64)
6521(1,1,65), (2,5,65), (4,7,65), (9,17,65), (11,17,65), (14,19,65), (20,23,65), (2,25,65), (32,35,65), (15,41,65), (16,41,65), (34,41,65), (31,47,65), (11,49,65), (53,61,65), (39,67,65), (52,67,65), (61,79,65), (58,83,65), (62,89,65), (90,113,65)
6620(1,1,66), (6,7,66), (6,11,66), (11,13,66), (13,19,66), (16,19,66), (18,19,66), (6,23,66), (8,31,66), (29,31,66), (9,37,66), (27,37,66), (30,37,66), (29,41,66), (40,43,66), (33,47,66), (20,49,66), (23,53,66), (68,79,66), (64,109,66)
6728(1,1,67), (2,3,67), (4,5,67), (8,9,67), (5,11,67), (14,15,67), (8,19,67), (19,25,67), (26,27,67), (22,29,67), (5,33,67), (5,37,67), (7,37,67), (18,37,67), (21,41,67), (31,43,67), (34,43,67), (40,47,67), (39,53,67), (49,55,67), (44,75,67), (26,81,67), (78,97,67), (71,99,67), (44,111,67), (92,111,67), (41,173,67), (50,179,67)
6824(1,1,68), (3,5,68), (3,7,68), (3,11,68), (5,17,68), (13,25,68), (17,29,68), (25,29,68), (26,29,68), (3,35,68), (33,43,68), (45,49,68), (12,53,68), (20,53,68), (36,53,68), (23,61,68), (34,61,68), (15,67,68), (36,67,68), (23,73,68), (25,79,68), (37,89,68), (80,97,68), (126,197,68)
6918(1,1,69), (2,7,69), (7,11,69), (5,13,69), (3,17,69), (2,19,69), (3,23,69), (10,23,69), (20,29,69), (20,37,69), (22,53,69), (36,61,69), (49,67,69), (32,71,69), (57,73,69), (24,107,69), (60,127,69), (80,181,69)

The goal of this section is to prove Theorem B and Corollary D.

6.1. Proof of Theorem B

The statement of Theorem B is that Conjecture 1.2 is true for trace n if n is an integer such that 64n69. By Theorem A, it suffices to check that Conjecture 1.2 is true for trace n where 3n69. We will use the reformulation of Conjecture 1.2 and the notations given in Section 2.4. To simplify the proof, we first prove a lemma.

Lemma 6.1. Let n>3 be an integer. Suppose that Conjecture 1.2 is true for trace m if 3≤ m≤ n-1. If every element of C([Θn]) has a representative (c,d,n) such that nn0(mod d) for some 6nn0n1, then Conjecture 1.2 is true for trace n.

Proof. By Theorem A and the hypothesis, Conjecture 1.2 is true for trace m if 6nmn1. Let x be an element of C([Θn]), and (c,d,n) be a representative of x satisfying that n= n0+kd for some 6nn0n1 and k. Then we have (c,d,n) G(c,d,n0) because n=n0+kd. Since Conjecture 1.2 is true for trace n0, (c,d,n0) (1,1,2). It follows that

(c,d,n) G(c,d,n0) (1,1,2).

Therefore Conjecture 1.2 is true for trace n.

Proof of Theorem B. In Table 2, we give minimal representatives of non-trivial elements of C([Θn]) for 3n32. In particular, C([Θn]) is trivial if 3n9 or n=11, and hence Conjecture 1.2 is true for trace 3n9, and for trace 11. Since 107(mod 3), by applying Lemma 1.2 for n=10, we can see that Conjecture 1.2 is true for trace 10. similarly, 127(mod 5), by applying Lemma 6.1 for n=12, we can see that Conjecture 1.2 is true for trace 12.

We can continue this argument to conclude that Conjecture 1.2 is true for trace n for 3n51. In fact, by Lemma 6.1, it suffices to observe the following statement using Tables 2-4. For any 13n51, every non-trivial element of C([Θn]) has minimal representative (c,d,n) such that nn0 for some 6nn0n1.

When n=52, the inductive argument works for all minimal representatives except (32,103,52). We give a sequence of Gompf equivalences from (32,103,52) to (87,101,50):

(32,103,52) G(32,103,51) S(87,101,51) G(87,101,50).

Since Conjecture 1.2 is true for trace 50, (87,101,50) is also Gompf equivalent to (1,1,2), and hence Conjecture 1.2 is true for trace 52.

As in the above, one can easily check that for any 53n55, every non-trivial element of C([Θn]) has minimal representative (c,d,n) such that nn0 for some 6nn0n1 using Table 4. Conjecture 1.2 is true for trace 53n55. For 56n69, we can similarly continue the inductive argument except few cases. For brevity of our discussion, we just record Gompf equivalences for these exceptional cases.

  • (15,109,57)~G(15,109,52)~S(18,79,52)~G(18,79,27).

  • (107,117,58)~G(107,117,59)~S(29,109,59)~G(29,109,50).

  • (101,123,58)~G(101,123,65)~S(30,47,65)~G(30,47,18).

  • (83,141,58)~S(128,165,58)~G(128,165,107)~S(38,119,107)~G(38,119,12).

  • (26,139,61)~S(119,291,61)~G(119,291,230)~S(302,391,230)~G(302,391,161)~S(114,149,161)~G(114,149,15).

  • (146,159,64)~G(146,159,95)~S(26,89,95)~G(26,89,6).

  • (41,173,67)~G(41,173,106)~S(210,233,106)~G(210,233,127)~S(158,267,127)~G(158,267,140)~S(153,179,140)~G(153,179,39).

  • (50,179,67)~S(272,291,67)~G(272,291,224)~S(142,397,224)~G(142,397,173)~S(14,149,173)~G(14,149,24).

  • (126,197,68)~S(248,265,68)~G(248,265,197)~S(170,407,197)~G(170,407,210)~S(18,277,210)~G(18,277,67)~S(38,205,67)~G(38,205,138)~S(139,227,138)~G(139,227,89)~S(70,97,89)~G(70,97,8).

  • (80,181,69)~S(167,211,69)~G(167,211,142)~S(218,269,142)~G(218,269,127)~S(36,151,127)~G(36,151,24).

This completes the proof.

6.2. Proof of Corollary D

We show Corollary D which says that ΣMkϵ is diffeomorphic to S4, but Mk is not similar to An for any integers k and n where

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

Proof of Corollary D. Note that Mk=X2,7,49k+27. As we mentioned in the introduction, ΣMkϵ is diffeomorphic to S4 for any integer k and ϵ2 by Corollary C or its weaker version given in [20, Theorem 3.2]. It remains to show that Mk is not similar to An for any integers k and n. By Proposition 2.14, the similarity class of Mk corresponds to the ideal class [Θ49k+272,7]C([Θ49k+27]). We proved in Proposition 4.10 that the ideal Θ49k+272,7 is not invertible, and hence represents a non-trivial element in C([Θ49k+27]). As we discussed in Remark 2.23, the similarity class of An corresponds to the trivial element in C([Θn+2]). It follows that Mk is not similar to An for any k and n.

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

Xc,d,c+2=0ab0cd102,

and showed that some of them are Gompf equivalent to A0.

Theorem 7.1 ([16,Theorem 3.1]) The Cappell-Shaneson matrix Xc,d,c+2 is Gompf equivalent to A0 if 0c94 and a19,37, or if 1d35.

Using our method, we generalize Theorem 7.1 by removing the technical conditions on the entry a, and weakening the condition on the entry d as follows:

Theorem 7.2. The Cappell-Shaneson matrix Xc,d,c+2 is Gompf equivalent to A0 if 0c94 or if 1d134.

Proof. By Theorem B, Xc,d,c+2 is Gompf equivalent to A0 if 1d134. It suffices to prove for the cases that a=19 or 37 and 0c94. By Proposition 2.10, fc+2(c)0(mod d) since Xc,d,c+2 is a Cappell-Shaneson matrix. The following tuples (c,d,c+2) in CS give the list of Cappell-Shaneson matrices Xc,d,c+2 satisfying a=19 and 0c94:

(8,3,10),(12,7,14),(27,37,29),(31,49,33),(46,109,48),(50,129,52),(65,219,67),(69,247,71),(84,367,86),(88,403,90).

The tuples in the first row correspond to Cappell-Shaneson matrices with trace 69, and hence Gompf equivalent to A0 by Theorem B. We give Gompf equivalences from the tuples in the second row as we did in the proof of Theorem B to the tuples that are known to Gompf equivalent to (1,1,2) using the MAGMA code for Algorithm 2 given in Section 5.2 as follows:

  • (69,247,71)~S(83,103,71).

  • (84,367,86)~S(102,127,86).

  • (88,403,90)~S(107,133,90).

Similarly, the following tuples (c,d,c+2) in CS give the list of Cappell-Shaneson matrices Xc,d,c+2 satisfying a=37 and 0c94:

(11,3,13),(27,19,29),(48,61,50),(64,109,66),(85,193,87).

As we did before, we give a Gompf equivalence from (85,193,87) to a tuple that is known to Gompf equivalent to (1,1,2) as follows:

(85,193,87) S(198,283,87) G(198,283,196)      S(155,229,196) G(155,229,33).

This completes the proof.

  1. I. R. Aitchison and J. H. Rubinstein. Fibered knots and involutions on homotopy spheres, In Four-manifold theory (Durham, N.H., 1982). volume 35 of Contemp. Math. Providence, RI: Amer. Math. Soc.; 1984:1-74. DOI:10.1090/conm/035/780575.
    CrossRef
  2. S. Akbulut. A fake 4-manifold, In Four-manifold theory (Durham, N.H., 1982). volume 35 of Contemp. Math. Providence, RI: Amer. Math. Soc.; 1984:75-141. DOI:10.1090/conm/035/780576.
    CrossRef
  3. S. Akbulut. On fake S3טS1#S2×S2, In Combinatorial methods in topology and algebraic geometry (Rochester, N.Y., 1982). volume 44 of Contemp. Math. Providence, RI: Amer. Math. Soc.; 195:281-286. DOI:10.1090/conm/044/813119.
    CrossRef
  4. S. Akbulut, Constructing a fake 4-manifold by Gluck construction to a standard 4-manifold, Topology, 27(2)(1988), DOI:10.1016/0040-9383(88) 90041-9, 239-243 pp.
    CrossRef
  5. S. Akbulut, Scharlemann's manifold is standard, Ann. of Math. (2), 149(2)(1999), DOI:10.2307/120972, 497-510 pp.
    CrossRef
  6. S. Akbulut, Cappell-Shaneson's 4-dimensional s-cobordism, Geom. Topol., 6(2002), DOI:10.2140/gt.2002.6.425, 425-494 pp.
    CrossRef
  7. S. Akbulut, Cappell-Shaneson homotopy spheres are standard, Ann. of Math. (2), 171(3)(2010), DOI:10.4007/annals.2010.171.2171, 2171-2175 pp.
    CrossRef
  8. S. Akbulut, The Dolgachev surface. Disproving the Harer-Kas-Kirby conjecture, Comment. Math. Helv., 87(1)(2012), DOI:10.4171/CMH/252, 187-241 pp.
    CrossRef
  9. S. Akbulut. 4-manifolds. volume 25 of Oxford Graduate Texts in Mathematics. Oxford: Oxford University Press; 2016. ISBN 978-0-19-878486-9. DOI:10.1093/acprof:oso/9780198784869.001.0001.
    CrossRef
  10. S. Akbulut and R. C. Kirby, An exotic involution of S4, Topology, 18(1)(1979), DOI:10.1016/0040-9383(79)90015-6, 75-81 pp.
    CrossRef
  11. S. Akbulut and R. C. Kirby, An exotic involution of S4, Topology, 24(4)(1985), DOI:10.1016/0040-9383(85)90010-2, 375-390 pp.
    CrossRef
  12. M. F. Atiyah and I. G. Macdonald. Introduction to commutative algebra. Reading, Mass.-London-Don Mills, Ont.: Addison-Wesley Publishing Co.; 1969.
  13. S. E. Cappell and J. L. Shaneson, Some new four-manifolds, Ann. of Math. (2), 104(1)(1976), 61-72.
    CrossRef
  14. S. E. Cappell and J. L. Shaneson, There exist inequivalent knots with the same complement, Ann. of Math. (2), 103(2)(1976), 349-353.
    CrossRef
  15. H. Cohen, et al. PARI/GP version 2.7.5. The PARI Group, Bordeaux; 2015 Available from http://pari.math.u-bordeaux.fr.
  16. G. Earle.
  17. M. H. Freedman, R. E. Gompf, S. Morrison and K. Walker, Man and machine thinking about the smooth 4-dimensional Poincar'e conjecture, Quantum Topol., 1(2)(2010), 171-208.
    CrossRef
  18. R. E. Gompf, Killing the Akbulut-Kirby 4-sphere, with relevance to the Andrews-Curtis and Schoenflies problems, Topology, 30(1)(1991), DOI:10.1016/0040-9383(91)90036-4, 97-115 pp.
    CrossRef
  19. R. E. Gompf, On Cappell-Shaneson 4-spheres, Topology Appl., 38(2)(1991), DOI:10.1016/0166-8641(91)90079-2, 123-136 pp.
    CrossRef
  20. R. E. Gompf, More Cappell-Shaneson spheres are standard, Algebr. Geom. Topol., 10(3)(2010), DOI:10.2140/agt.2010.10.1665, 1665-1681 pp.
    CrossRef
  21. C. G. Latimer and C. C. MacDuffee, A correspondence between classes of ideals and classes of matrices, Ann. of Math. (2), 34(2)(1933), DOI:10.2307/1968204, 313-316 pp.
    CrossRef
  22. M. Newman. Integral matrices. New York-London: Academic Press; 1972. Pure and Applied Mathematics, Vol. 45.
  23. P. Stevenhagen. The arithmetic of number rings, In Algorithmic number theory: lattices, number fields, curves and cryptography. volume 44 of Math. Sci. Res. Inst. Publ. Cambridge: Cambridge Univ. Press; 2008:209-266.
  24. O. Taussky, On a theorem of Latimer and MacDuffee, Canadian J. Math., 1(1949), 300-302.
    CrossRef