A Topology on Milnor’s Group of a Topological Field and Continuous Joint Determinants

Abstract

For the tuple set of commuting invertible matrices with coefficients in a given field, the joint determinants are defined as generalizations of the determinant map for the square matrices. We introduce a natural topology on Milnor’s -groups of a topological field as the quotient topology induced by the joint determinant map and investigate the existence of a nontrivial continuous joint determinant by utilizing this topology, generalizing the author’s previous results on the continuous joint determinants for the commuting invertible matrices over and .

1. Introduction

In [1], a joint determinant is introduced as a generalization of the determinant map for invertible matrices. More precisely, for a field , a joint determinant () is defined as a map from the set of -tuples of commuting matrices in () into some abelian group which satisfies the following properties.

(i) Multilinearity: for commuting matrices and in for some , we have .

(ii) Block diagonal matrices: for commuting and commuting for some , we have

(iii) Similar matrices: for commuting matrices and any , we have .

(iv) Polynomial homotopy: for commuting , we have .

Using the standard inclusion we define as the direct limit of these groups . Using the above inclusions, we may identify the direct limit of the set of -tuples of commuting matrices in over as a subset of . Then, a joint determinant may be thought of as a map from into an abelian group .

The main result in [1] about the joint determinants is that there exists a one-to-one correspondence between the set of joint determinants from into an abelian group and the set of group homomorphisms from Milnor’s -group into . Milnor’s -group is introduced in [2] as the quotient group of the tensor product by the subgroup generated by elements of the form , where for some (). It is a major object of study in algebraic -theory and appears in numerous literatures. For example, Voevodsky’s proof of Bloch-Kato conjecture [3] relates Milnor’s -group of a field with its étale cohomology. The element of represented by is typically denoted by a symbol .

To describe the “universal” joint determinant , we need the Goodwillie group which is defined to be the abelian group generated by -tuples of commuting matrices ( for various ), subject to the following 4 kinds of relations.

(i) Identity matrices: when for some is equal to the identity matrix .

(ii) Similar matrices: for commuting and any .

(iii) Direct sum: for commuting and commuting .

(iv) Polynomial homotopy: for commuting matrices in , where is the polynomial ring over with the indeterminate .

The universal joint determinant map is then the composite of the natural map , which sends an -tuple of commuting matrices to a generator of and the isomorphism , which is described in the proof of Theorem of [1]. From the fact that is an isomorphism follows easily the one-to-one correspondence between the set of joint determinants from into an abelian group and the set of group homomorphisms from Milnor’s -group into .

When , and the universal joint determinant is nothing but the traditional determinant map (Proposition of [1]).

The definition of joint determinant maps is given in purely algebraic terms and so there are possibilities of very complicated joint determinants; for example, when is the field of complex numbers or of real numbers, Milnor’s -groups for are known to be uniquely divisible or a direct sum of a cyclic group of order and a uniquely divisible group, respectively [2].

Thus, if we disregard the topological continuity of a joint determinant map, the joint determinants are far from trivial, but if we require a joint determinant to be continuous, then the situation becomes drastically different. It is proven that, for , there exists only one nontrivial joint determinant from into , which is continuous when restricted to the set of commuting matrices in , for each , with the standard topology (Corollary of [1]).

In the present article, we generalize this result to determine all possible continuous joint determinants from or to a topological abelian group . For this purpose, we introduce a natural topology on Milnor’s -groups for a topological field as the quotient topology induced by the joint determinant map and show that, in case of or , the natural topology on is disjoint union of two indiscrete components or indiscrete topology, respectively. This indicates that, for or , the “universal” continuous joint determinant turns out to be or , respectively.

2. A Natural Topology on

For a topological field , is a topological group with the direct limit topology, that is, a subset of is open if and only if is open for each (e.g., of [4]). The topology on is given by the subspace topology regarding it as a subspace of the product space . Then it coincides with the direct limit topology if we think of as the direct limit of the subspace of -tuples of commuting matrices in the space over .

Definition 1. For a topological field , the topology on Milnor’s -group is the quotient topology with respect to the map , which is the composite of a natural map followed by the group isomorphism which is described in the proof of Theorem of [1].

The obvious map is actually surjective by Corollary of [1] and so is a surjection.

Theorem 2. is a topological group with respect to the topology given in Definition 1.

Proof. By the definition of the Goodwillie group , the group law on is given via by the direct sum rule: for commuting and commuting (). This addition rule is not expressed by a continuous map , but the following continuous map actually induces the group operation on : To prove that the two elements of map to the same element under , it is enough to verify that an -tuple of commuting matrices in represents the same element in which is represented by the -tuple of matrices which is obtained by simultaneously changing th and th rows and also th and th columns of all matrices . For notational convenience, we will prove this for 1st and 2nd rows and columns of matrices and the proof is easily generalized to matrices. Let us write the th entry of the matrix as (). In , we have Using the polynomial homotopy which results in interchanging the 1st and 4th rows with negative sign to the new 4th row and then interchanging 1st and 4th columns with negative sign to the new 4th column, we see that, in , Again, by applying the polynomial homotopy which results in interchanging the 2nd and 3rd rows with negative sign to the new 3rd row and then interchanging 2nd and 3rd columns with negative sign to the new 3rd column, we have, in , which is equal to in .

3. The Topological Structures of and

Theorem 3. For , the topological space is a disjoint union of two indiscrete open sets.

Proof. Note that we have , where the first direct factor is generated by and is a uniquely divisible group [2]. For where are negative for all , we have which is equal to the sum of and various symbols of the form where at least one of is positive.
Every element of can be written as a sum of symbols of the form , where at least one of is positive. By writing a positive real number as a square of its square root, we may assume that is positive for every (e.g., in case ).
Let be any open set of containing the identity element and consider its inverse image in . Let be any element. Then contains an -tuple of diagonal matrices of the formwhere is positive for every and . By taking th root of for sufficiently large , we may assume that is arbitrarily close to . Then contains an element which is arbitrarily close to with equal to the identity matrix. So, contains an element which is contained in the open set . Hence must contain .
Similarly, the coset is also an indiscrete subspace. In fact, is the image under of the set of -tuples of commuting matrices in () such that the determinants of are positive for some . On the other hand, is the image under of the set of -tuples of commuting matrices where the determinants of are negative for all . Therefore, the proper open sets of are and .

Corollary 4. For , the topological space is indiscrete (trivial).

Proof. Let be an open set of containing the identity element and let . For any element , contains an -tuple of diagonal matrices. Write each diagonal element of as a product of a positive real number and a complex number with absolute value . Any complex number with absolute value is arbitrarily close to a root of unity and any symbol containing a root of unity is trivial since, for example, if is an th root of unity. Combining this fact with the arguments given in the proof of Theorem 3, we see that contains an element which is contained in . This shows that the natural topology on is indiscrete.

4. Applications to Joint Determinants

When is a topological field, a joint determinant from into a topological abelian group is called continuous if is given the subspace topology of with the direct limit topology as described in Section 2. Since the natural topology on in Definition 1 is the quotient topology, any continuous joint determinant induces a continuous map from into and vice versa.

Corollary 5. For and any topological abelian group which is , any continuous joint determinant from into factors through the discrete group .

Proof. This follows directly from Theorem 3. Note that a topological group is Hausdorff if it is (cf. Lemma of [5]).

The following is a direct consequence of Corollary 4.

Corollary 6. For and any topological abelian group which is , any continuous joint determinant from into is trivial.

We summarize our results on the continuous joint determinants for and in the following theorem, which is virtually equivalent to Theorem 3 and Corollary 4.

Theorem 7. For or and a topological abelian group , let be a continuous surjective joint determinant. When , is a composite of the usual determinant map followed by a canonical epimorphism with equipped with a coarser topology than the quotient topology induced by the epimorphism. When and , either is an indiscrete space or has an indiscrete subgroup of index . If , then has the indiscrete topology.

Conflicts of Interest

The author declares that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MEST) (no. NRF-2010-0006083).