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

Authors: Sung Myung
Journal: International Journal of Mathematics and Mathematical Sciences
 (2017)

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## 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’sK-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 R and C.


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## Body

## 1. Introduction

In [1], a joint determinant is introduced as a generalization of the determinant map for invertible matrices. More precisely, for a field k, a joint determinant D(=Dl) (l≥1) is defined as a map from the set of l-tuples of commuting matrices in GLn(k) (n≥1) into some abelian group (G,+) which satisfies the following properties.(i) Multilinearity: forl+1 commuting matrices A1,…,Al and B in GLn(k) for some n≥1, we have D(A1,…,AiB,…,Al)=D(A1,…,Ai,…,Al)+D(A1,…,B,…,Al).(ii) Block diagonal matrices: for commutingA1,…,Al∈GLm(k) and commuting B1,…,Bl∈GLn(k) for some m,n≥1, we have (1)DA100B1,…,Al00Bl=DA1,…,Al+DB1,…,Bl.(iii) Similar matrices: for commuting matricesA1,…,Al∈GLn(k) and any S∈GLn(k), we have D(SA1S-1,…,SAlS-1)=D(A1,…,Al).(iv) Polynomial homotopy: for commutingA1(t),…,Al(t)∈GLn(k[t]), we have D(A1(0),…,Al(0))=D(A1(1),…,Al(1)).Using the standard inclusionGLn(k)↪GLn+1(k)(2)a11⋯a1n⋮⋱⋮an1⋯ann⟼a11⋯a1n0⋮⋱⋮0an1⋯ann00⋯01,we define GL(k) as the direct limit of these groups GL(k)=∪n→∞GLn(k). Using the above inclusions, we may identify the direct limit Comml(k) of the set of l-tuples of commuting matrices in GLn(k)l=GLn(k)×⋯×GLn(k) over n as a subset of GL(k)l=GL(k)×⋯×GL(k). Then, a joint determinant may be thought of as a map from Comml(k) into an abelian group G.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 Comml(k) into an abelian group G and the set of group homomorphisms from Milnor’s K-group KlM(k) into G. Milnor’s K-group is introduced in [2] as the quotient group of the tensor product k∗⊗⋯⊗k∗ by the subgroup generated by elements of the form a1⊗⋯⊗al, where ai+aj=1 for some i,j (1≤i<j≤l). It is a major object of study in algebraic K-theory and appears in numerous literatures. For example, Voevodsky’s proof of Bloch-Kato conjecture [3] relates Milnor’s K-group of a field with its étale cohomology. The element of KlM(k) represented by a1⊗⋯⊗al is typically denoted by a symbol {a1,…,al}.To describe the “universal” joint determinantComml(k)→KlM(k), we need the Goodwillie group GWl(k) which is defined to be the abelian group generated by l-tuples of commuting matrices (A1,…,Al) (A1,…,Al∈GLn(k) for various n≥1), subject to the following 4 kinds of relations.(i) Identity matrices:(A1,…,Al)=0 when Ai for some i is equal to the identity matrix In∈GLn(k).(ii) Similar matrices:(A1,…,Al)=(SA1S-1,…,SAlS-1) for commuting A1,…,Al∈GLn(k) and any S∈GLn(k).(iii) Direct sum:(A1,…,Al)+(B1,…,Bl)=A100B1,…,Al00Bl for commuting A1,…,Al∈GLn(k) and commuting B1,…,Bl∈GLm(k).(iv) Polynomial homotopy:(A1(0),…,Al(0))=(A1(1),…,Al(1)) for commuting matrices A1(t),…,Al(t) in GLn(k[t]), where k[t] is the polynomial ring over k with the indeterminate t.The universal joint determinant mapΦl:Comml(k)→KlM(k) is then the composite of the natural map Comml(k)→GWl(k), which sends an l-tuple of commuting matrices to a generator of GWl(k) and the isomorphism ϕl:GWl(k)→~KlM(k), which is described in the proof of Theorem 6.7 of [1]. From the fact that ϕ is an isomorphism follows easily the one-to-one correspondence between the set of joint determinants from Comml(k) into an abelian group G and the set of group homomorphisms from Milnor’s K-group KlM(k) into G.Whenl=1, GWl(k)≃k∗ and the universal joint determinant is nothing but the traditional determinant map (Proposition 4.4 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, whenk is the field C of complex numbers or R of real numbers, Milnor’s K-groups KlM(k) for l≥2 are known to be uniquely divisible or a direct sum of a cyclic group of order 2 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, forl≥2, there exists only one nontrivial joint determinant from Comml(R) into R×, which is continuous when restricted to the set of commuting matrices in GLn(R), for each n, with the standard topology (Corollary 7.3 of [1]).In the present article, we generalize this result to determine all possible continuous joint determinants fromComml(R) or Comml(C) to a topological abelian group G. For this purpose, we introduce a natural topology on Milnor’s K-groups KlM(k) for a topological field k as the quotient topology induced by the joint determinant map and show that, in case of k=R or C, the natural topology on KlM(k) is disjoint union of two indiscrete components or indiscrete topology, respectively. This indicates that, for k=R or C, the “universal” continuous joint determinant turns out to be Comml(R)→Z2 or Comml(C)→{1}, respectively.


## 2. A Natural Topology onKlM(k)

For a topological fieldk, GL(k)=∪n→∞GLn(k) is a topological group with the direct limit topology, that is, a subset U of GL(k) is open if and only if U∩GLn(k) is open for each n≥1 (e.g., 3.1 of [4]). The topology on Comml(k) is given by the subspace topology regarding it as a subspace of the product space GL(k)l=GL(k)×⋯×GL(k). Then it coincides with the direct limit topology if we think of Comml(k) as the direct limit of the subspace of l-tuples of commuting matrices in the space GLn(k)l=GLn(k)×⋯×GLn(k) over n.Definition 1.
For a topological fieldk, the topology on Milnor’s K-group KlM(k) is the quotient topology with respect to the map Φl:Comml(k)→KlM(k), which is the composite of a natural map Comml(k)→GWl(k) followed by the group isomorphism ϕl:GWl(k)→~KlM(k) which is described in the proof of Theorem 6.7 of [1].The obvious mapComml(k)→GWl(k) is actually surjective by Corollary 4.3 of [1] and so Φl is a surjection.Theorem 2.
K
l
M
(
k
) is a topological group with respect to the topology given in Definition 1.Proof.
By the definition of the Goodwillie groupGWl(k), the group law on KlM(k) is given via ϕl:GWl(k)→~KlM(k) by the direct sum rule: (A1,…,Al)+(B1,…,Bl)=A100B1,…,Al00Bl for commuting A1,…,Al∈GLp(k) and commuting B1,…,Bl∈GLq(k) (p,q≥1). This addition rule is not expressed by a continuous map Comml(k)×Comml(k)→Comml(k), but the following continuous map Comml(k)×Comml(k)→Comml(k) actually induces the group operation on KlM(k): (3)a11a12a13⋯a21a22a23⋯a31a32a33⋯⋮⋮⋮⋱,…,b11b12b13⋯b21b22b23⋯b31b32b33⋯⋮⋮⋮⋱,…⟼a110a120a130⋯0b110b120b13⋯a210a220a230⋯0b210b220b23⋯a310a320a330⋯0b310b320b33⋯⋮⋮⋮⋮⋮⋮⋱,….To prove that the two elements (4)a110a120⋯0b110b12⋯a210a220⋯0b210b22⋯⋮⋮⋮⋮⋱,…,a11a12⋯00⋯a21a22⋯00⋯⋮⋮⋱⋮⋮⋮00⋯b11b12⋯00⋯b21b22⋯⋮⋮⋯⋮⋮⋱,…of Comml(k) map to the same element under Φl:Comml(k)→KlM(k), it is enough to verify that an l-tuple (A1,…,Al) of commuting matrices in GLn(k)l represents the same element in GWl(k) which is represented by the l-tuple of matrices which is obtained by simultaneously changing ith and jth rows and also ith and jth columns of all ln×n matrices A1,…,Al. For notational convenience, we will prove this for 1st and 2nd rows and columns of 2×2 matrices and the proof is easily generalized to n×n matrices. Let us write the (i,j)th entry of the matrix Ak as aijk (k=1,2,…,l). In GWl(k), we have (5)A1,…,Al=a111a121a211a221,…,a11la12la21la22l=a111a12100a211a2210000100001,…,a11la12l00a21la22l0000100001.Using the polynomial homotopy (6)1-t200t01000010t3-2t001-t2a111a12100a211a22100001000011-t200-t010000102t-t3001-t2,…,1-t200t01000010t3-2t001-t2a11la12l00a21la22l00001000011-t200-t010000102t-t3001-t2,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 GWl(k), (7)a111a12100a211a2210000100001,…,a11la12l00a21la22l0000100001=10000a2210-a21100100-a1210a111,…,10000a22l0-a21l00100-a12l0a11l.Again, by applying the polynomial homotopy (8)100001-t2t00t3-2t1-t20000110000a2210-a21100100-a1210a111100001-t2-t002t-t31-t200001,…,100001-t2t00t3-2t1-t20000110000a22l0-a21l00100-a12l0a11l100001-t2-t002t-t31-t200001,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 GWl(k), (9)10000a2210-a21100100a1210a111,…,10000a22l0-a21l00100a12l0a11l=1000010000a221a21100a121a111,…,1000010000a22la21l00a12la11l,which is equal to a221a211a121a111,…,a22la21la12la11l in GWl(k).


## 3. The Topological Structures ofKlM(R) and KlM(C)

Theorem 3.
Forl≥2, the topological space KlM(R) is a disjoint union of two indiscrete open sets.Proof.
Note that we haveKlM(R)≃(Z/2)⊕H, where the first direct factor Z/2 is generated by {-1,…,-1} and H is a uniquely divisible group [2]. For {a1,…,al} where ai are negative for all i=1,…,l, we have {a1,…,al}={-1,a2,…,al}+{-a1,a2,…,al}={-1,-1,a3,…,al}+{-1,-a2,…,al}+{-a1,a2,…,al}=⋯ which is equal to the sum of {-1,-1,-1,…,-1} and various symbols of the form {b1,…,bl} where at least one of bi is positive.
Every element ofH can be written as a sum of symbols of the form b1,…,bl, where at least one of bi is positive. By writing a positive real number as a square of its square root, we may assume that bi is positive for every i=1,…,l (e.g., {b1,b2}={b12,b2} in case b2>0).
LetU be any open set of KlM(R) containing the identity element and consider its inverse image V=Φl-1(U) in Comml(R). Let h∈H be any element. Then Φl-1({h}) contains an l-tuple of diagonal matrices of the form(10)a11a21⋱an1,…,a1la2l⋱anl,where aki is positive for every k=1,…,n and i=1,…,l. By taking mth root of ak1 for sufficiently large m, we may assume that ak1 is arbitrarily close to 1. Then Φl-1({h}) contains an element which is arbitrarily close to (A1,…,Al) with A1 equal to the identity matrix. So, Φl-1({h}) contains an element which is contained in the open set V=Φl-1(U). Hence U must contain H.
Similarly, the cosetH+{-1,-1,…,-1} is also an indiscrete subspace. In fact, H is the image under Φl of the set of l-tuples (A1,…,Al) of commuting matrices in GLn(R)l (n≥1) such that the determinants of Ai are positive for some i∈{1,…,l}. On the other hand, H+{-1,-1,…,-1} is the image under Φl of the set of l-tuples (A1,…,Al) of commuting matrices where the determinants of Ai are negative for all i=1,…,l. Therefore, the proper open sets of KlM(R) are H and H+{-1,-1,…,-1}.Corollary 4.
Forl≥2, the topological space KlM(C) is indiscrete (trivial).Proof.
LetU be an open set of KlM(C) containing the identity element and let V=Φ-1(U)⊂Comml(C). For any element h∈KlM(C), Φ-1({h}) contains an l-tuple (A1,…,Al) of diagonal matrices. Write each diagonal element of Ai as a product of a positive real number and a complex number with absolute value 1. Any complex number with absolute value 1 is arbitrarily close to a root of unity and any symbol containing a root of unity is trivial since, for example, {ζ,b}=1/m{ζm,b}=0 if ζ is an mth root of unity. Combining this fact with the arguments given in the proof of Theorem 3, we see that Φ-1({h}) contains an element which is contained in V=Φ-1(U). This shows that the natural topology on KlM(C) is indiscrete.


## 4. Applications to Joint Determinants

Whenk is a topological field, a joint determinant from Comml(k) into a topological abelian group G is called continuous if Comml(k) is given the subspace topology of GL(k)=∪n→∞GLn(k) with the direct limit topology as described in Section 2. Since the natural topology on KlM(k) in Definition 1 is the quotient topology, any continuous joint determinant induces a continuous map from KlM(k) into G and vice versa.Corollary 5.
Forl≥2 and any topological abelian group G which is T0, any continuous joint determinant from Comml(R) into G factors through the discrete group Z2.Proof.
This follows directly from Theorem3. Note that a topological group is Hausdorff if it is T0 (cf. Lemma 10.1 of [5]).The following is a direct consequence of Corollary4.Corollary 6.
Forl≥2 and any topological abelian group G which is T0, any continuous joint determinant from Comml(C) into G is trivial.We summarize our results on the continuous joint determinants fork=R and k=C in the following theorem, which is virtually equivalent to Theorem 3 and Corollary 4.Theorem 7.
Fork=R or k=C and a topological abelian group G, let D:Comml(k)→G be a continuous surjective joint determinant. When l=1, D is a composite of the usual determinant map followed by a canonical epimorphism k∗↠G with G equipped with a coarser topology than the quotient topology induced by the epimorphism. When k=R and l≥2, G either is an indiscrete space or has an indiscrete subgroup of index 2. If k=C, then G has the indiscrete topology.



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