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Linear systems over Z[[Q.sub.16]] and roots of maps of some 3-complexes into [mathematical expression not reproducible].

1 Introduction

Given a map f : W [right arrow] M between topological spaces, and an arbitrary point a [member of] M, recall that MR[f,a] = min{#([g.sup.-1](a))|g [member of] [f]}, where [-] means a homotopy class. We say that a map f : W [right arrow] M is strongly surjective, if any map homotopic to it is surjective or, equivalently, if MR[f,a] [not equal to] 0 for some a [member of] M. The problem of the existence of a map f : W [right arrow] M which is strongly surjective has been studied in the paper [1] when W is a CW-complex and M is a closed manifold, both of dimension 3. The main results are:

i) There is no map f : W [right arrow] M which is a strongly surjective map if W is a CW-complex with [H.sup.3](W;Z) = 0 and M is either [S.sup.1] x [S.sup.2], [S.sup.1] x [S.sup.1] x [S.sup.1], or a lens space;

ii) There is no map f : W [right arrow] M which is a strongly surjective map if W is a CW-complex with [H.sup.3](W; [bar.Z]) = 0 and M is either [S.sup.1] x [P.sub.2], where [P.sub.2] denotes the 2-dimensional real projective space, or MA = [S.sup.2] X [0, 1]/(x, 0) ~ (-x,1). Here, [H.sup.3](W; [bar.Z]) is the cohomology with an arbitrary local coefficient system [bar.Z].

iii) There exists a strongly surjective map f : W [right arrow] [M.sub.A], where W is a certain 3-complex with [H.sup.3](W;Z) = 0.

The above results were obtained making use of the obstruction theory. The existence or non-existence of a strongly surjective map f : W [right arrow] M may be determined by an obstruction class [[omega].sup.3](f) [member of] [H.sup.3](W;Z[[pi]]), where [H.sup.3](W;Z[[pi]]) is the cohomology group of W with a local coefficient system and [pi] = [[pi].sub.1](M). In [1], the vanishing of the obstruction to deform a map f : W [right arrow] M to a root free map is described in terms of solutions of a liner system PX = K over the group ring Z [[pi]]. Let [Q.sub.8] = (x,y|[x.sup.2] = [y.sup.2], xyx = y) be the quaternion group of order 8 and [mathematical expression not reproducible] the orbit space of the 3-sphere [S.sup.3] with respect to the action determined by the inclusion [Q.sub.8] [??] [S.sup.3]. In [2] we studied the problem of the existence of a map f : W [right arrow] [mathematical expression not reproducible] which is strongly surjective, we obtained:

iv) If W is a three dimensional CW-complex with [H.sup.3](W;Z) = 0 then there is no strongly surjective map f : W [right arrow] [mathematical expression not reproducible].

This case differs from the previous one because [[pi].sub.1] ([mathematical expression not reproducible]) = [Q.sub.8] is nonabelian. To prove the statement iv) is equivalent to demonstrate that: If PX = K(x - 1) and PX = K(-xy + 1) have solutions over Z[[Q.sub.8]] where P = [[p.sub.ij]] is an m x n matrix with m [less than or equal to] n, and all m x m minors of [epsilon](P) = [[epsilon]([p.sub.ij])] are relatively prime, then the system PX = K has a solution over Z[[Q.sub.8]]. The condition about [epsilon](P) is obtained from the hypothesis that [H.sup.3](W;Z) = 0.

In this work we consider linear systems over Z[[Q.sub.16]] and maps f : W [right arrow] [mathematical expression not reproducible]. The main results are:

Theorem 1.1. Let PX = K be a linear system over Z[[Q.sub.16]], where P = [[p.sub.ij]] is an m x n with m < n. If PX = K(x - 1) and PX = K(-xy + 1) have solutions over Z[[Q.sub.16]] and all m x m minors of[epsilon](P) = [[epsilon]([p.sub.ij])] are relatively prime, then the system PX = K has a solution over Z[[Q.sub.16]].

Theorem 1.2. If W is a three dimensional CW-complex with [H.sup.3](W; Z) = 0, then there is no strongly surjective map f : W [right arrow] [mathematical expression not reproducible].

The techniques used in [2] unfortunately do not apply to this case. Therefore we introduce new techniques to study the case [Q.sub.16]. Let [M.sub.2](Z) be the ring of 2 x 2 matrices with entries in Z and H(Q[[square root of (2)]]) the quaternion field. Throughout this paper, the problem of solving the linear system PX = K will be converted to solving linear systems over Z, [M.sub.2](Z) and H(Q[[square root of (2)]]). The problem of solving linear equations over the quaternion field H(R), has been studied in [3] and [8] making use of quaternionic determinant and inverse square matrix, subjects that will be used in this work.

The paper is organized as follows. Sections 2, 3, 4, 5, 6, 7 are dedicated to prove theorem 1.1. Section 8 contains the proof of theorem 1.2 and discusses the case Q32.

2 Linear systems over Z[[Q.sub.16]]

Consider the quaternion group [Q.sub.16] = (x, y|x = [y.sup.2], xyx = y) of order 16. Any element w [member of] [Q.sub.16] has a unique canonical form w = [x.sup.[mu]][y.sup.[delta]], with 0 [less than or equal to] [mu] < 8 and [delta] = 0, 1. The function [epsilon] : Z[[Q.sub.16]] [right arrow] Z given by [mathematical expression not reproducible] is a ring homomorphism. Let Q be the field of rational numbers, [M.sub.2](Q) the ring of 2 x 2 matrices with entries in Q, Q[[square root of (2)]] = {a + b [[square root of (2)]] | a, b [member of] Q}, H(Q[[[square root of (2)]]]) the quaternion field and [Q.sup.4] [direct sum] [M.sub.2] (Q) [direct sum] H(Q[[square root of (2)]]) the ring with component-wise addition and multiplication. The quaternion field H(Q[[square root of (2)]]) has dimension 8 over Q with basis [mathematical expression not reproducible] where [i.sup.2] = [j.sup.2] = [k.sup.2] = -1, ij = k = -ji, jk = i = -kj and ki = j = -ik. Consider the isomorphism of rings T : Q([Q.sub.16]) [right arrow] [Q.sup.4] [direct sum] [M.sub.2](Q) [direct sum] H(Q[[square root of (2)]]) given by:

[mathematical expression not reproducible]

Given an element q = [q.sub.0] + [q.sub.1]x + *** + [q.sub.7][x.sup.7] + [q.sub.8]y + [q.sub.9]xy + *** + [q.sub.15][x.sup.7] y [member of] Z[[Q.sub.16]], denote by T(q) = ([epsilon](q),[q.sup.1],[q.sup.2],[q.sup.3],[q.sup.4],[q.sup.5]),where

[mathematical expression not reproducible]

and

[mathematical expression not reproducible]

Let [phi] : Z [right arrow] [Z.sub.2] be the quotient map, [phi](n), n [member of] Z will be abbreviated n and should not be confused with [bar.p], p [member of] H(R) the conjugate of the quaternions. Notice that [phi] will be used also as the obvious extension map from the matrix sets [M.sub.n](Z) to [M.sub.n]([Z.sub.2]).

Lemma 2.1. The set of all the elements

[mathematical expression not reproducible]

where a, c, e, g, 2b, 2d, 2f, 2h [member of] Z with [bar.2b] = [bar.2d] and [bar.2f] = [bar.2h], denoted by [bar.H](Q[[square root of (2)]]), is a subring of H(Q[[square root of (2)]]).

Proof Let [mathematical expression not reproducible] and [mathematical expression not reproducible] be elements of [mathematical expression not reproducible]. The element [q.sub.1] - [q.sub.2] lies in [mathematical expression not reproducible]. Consider the product

[mathematical expression not reproducible],

where

i) a = [a.sub.1][a.sub.2] - [c.sub.1][c.sub.2] - [e.sub.1][e.sub.2] - [g.sub.1][g.sub.2] + 2([b.sub.1][b.sub.2] - [d.sub.1][d.sub.2] - [f.sub.1][f.sub.2] - [h.sub.1][h.sub.2])

ii) b = [a.sub.1][b.sub.2] + [b.sub.1][a.sub.2] - [c.sub.1][d.sub.2] - [c.sub.2][d.sub.1] - [e.sub.1][f.sub.2] - [f.sub.1][e.sub.2] - [g.sub.1][h.sub.2] - [g.sub.2][h.sub.1]

iii) c = [a.sub.1][c.sub.2] + [c.sub.1][a.sub.2] + [e.sub.1][g.sub.2] - [g.sub.1][e.sub.2] + 2([b.sub.1][d.sub.2] + [d.sub.1][b.sub.2] + [f.sub.2][h.sub.1] - [f.sub.1][h.sub.2])

iv) d = [a.sub.1][d.sub.2] + [b.sub.1][c.sub.2] - [c.sub.1][b.sub.2] - [a.sub.2][d.sub.1] - [e.sub.1][h.sub.2] + [f.sub.1][g.sub.2] - [g.sub.1][f.sub.2] - [e.sub.2][h.sub.1]

v) e = [a.sub.1][e.sub.2] - [c.sub.1][g.sub.2] + [e.sub.1][a.sub.2] + [g.sub.1][c.sub.2] + 2([b.sub.1][f.sub.2] - [d.sub.1][h.sub.2] + [f.sub.1][b.sub.2] + [h.sub.1][d.sub.2])

vi) f = [a.sub.1][f.sub.2] + [b.sub.1][e.sub.2] - [c.sub.1][h.sub.2] - [g.sub.2][d.sub.1] + [e.sub.1][b.sub.2] + [f.sub.1][a.sub.2] + [g.sub.1][d.sub.2] + [c.sub.2][h.sub.1]

vii) g = [a.sub.1][g.sub.2] + [c.sub.1][e.sub.2] - [e.sub.1][c.sub.2] + [g.sub.1][a.sub.2] + 2([b.sub.1][h.sub.2] + [d.sub.1][f.sub.2] + [f.sub.1][d.sub.2] + [h.sub.1][b.sub.2])

viii) h = [a.sub.1][h.sub.2] + [b.sub.1][g.sub.2] + [c.sub.1][f.sub.2] + [e.sub.2][d.sub.1] + [e.sub.1][d.sub.2]+ [f.sub.1][c.sub.2] +[g.sub.1][b.sub.2] + [a.sub.2][h.sub.1].

Note that 2[b.sub.1]2[b.sub.2] - 2[d.sub.1]2[d.sub.2] - 2[f.sub.1]2[f.sub.2] - 2[h.sub.1]2[h.sub.2] = 2k with k[member of]Z, since [bar.2[b.sub.1]] = [bar.2[d.sub.1]], [bar.2[f.sub.1]] = [bar.2[h.sub.1]], [bar.2[b.sub.2]] = [bar.2[d.sub.2]] and [bar.2[f.sub.2]] = 2[h.sub.2]. Hence, 2([b.sub.1][b.sub.2] - [d.sub.1][d.sub.2] - [f.sub.1][f.sub.2] - [h.sub.1][h.sub.2]) =k[member of]Z and a [member of] Z. By a similar argument, c, e, g [member of] Z.

Now,

2b = [a.sub.1]2[b.sub.2] + 2[b.sub.1][a.sub.2] - [c.sub.1]2[d.sub.2] - [c.sub.2]2[d.sub.1] - [e.sub.1]2[f.sub.2] - 2[f.sub.1][e.sub.2] - [g.sub.1]2 [h.sub.2] ~ [g.sub.2]2[h.sub.1]

and

2d = [a.sub.1]2[d.sub.2] + 2[b.sub.1][c.sub.2] - [c.sub.1]2[b.sub.2] - [a.sub.2]2[d.sub.1] - [e.sub.1]2[h.sub.2] + 2[f.sub.1][g.sub.2] - [g.sub.1]2[f.sub.2] ~ [e.sub.2]2[h.sub.1].

Because [mathematical expression not reproducible], [bar.[e.sub.1]2[f.sub.2]] = [bar.[e.sub.1]2[h.sub.2]], [bar.[e.sub.2]2[f.sub.1]] = [bar.[e.sub.2]2[h.sub.1]], [bar.[g.sub.1]2[h.sub.2]] = [bar.[g.sub.1]2[f.sub.2]] and [bar.2[h.sub.1][g.sub.2]] = [bar.2[f.sub.1g2], then [bar.2b] = [bar.2d]. Analogously, [bar.2f] = [bar.2h].

The restriction T : Z[[Q.sub.16]] [right arrow] [Z.sup.4] [direct sum] [M.sub.2](Z) [direct sum] [bar.H](Q[[square root of (2)]]) gives an embedding of Z[[Q.sub.16]] in [Z.sup.4] [direct sum] [M.sub.2](Z) [direct sum] [bar.H](Q[[square root of (2)]]). To study the solutions of a linear system PX = K over Z[[Q.sub.16]], where P = [[p.sub.ij]] is an m x n matrix with entries in Z[[Q.sub.16]], and X = [[x.sub.1] ***[[x.sub.n]].sup.t] and K = [[k.sub.1]***[[k.sub.m]].sup.t] are column vectors with coordinates in Z[[Q.sub.16]], we will consider first the problem in [Z.sup.4] [direct sum] [M.sub.2](Z) [direct sum] [bar.H](Q[[square root of (2)]]).

Theorem 2.2. If the system [bar.P]X = [bar.K] has a solution over [Z.sup.4] [direct sum] [M.sub.2](Z) [direct sum] [bar.H](Q[[square root of (2)]]), where [bar.P] = [T([p.sub.ij])] and [bar.K] = [T([k.sub.1]) *** T ([[k.sub.m])].sup.t], then the system PX = 16K has a solution over Z[[Q.sub.16]].

Proof: Observe that the isomorphism T has the following property:

[mathematical expression not reproducible]

Thus, every element of [Z.sup.4] [direct sum] [M.sub.2](Z) [direct sum] [bar.H](Q[[square root of (2)]]) multiplied by

[mathematical expression not reproducible]

is an element of T(Z[[Q.sub.16]]). By hypothesis [bar.P]X = [bar.K] has a solution over [Z.sup.4] [direct sum] [M.sub.2](Z) [direct sum] [bar.H](Q[[square root of (2)]]), that is, there exists a column vector [X.sub.0] with coordinates in [Z.sup.4] [direct sum] [M.sub.2](Z) [direct sum] [bar.H](Q[[square root of (2)]]) such that [bar.P][X.sub.0] = [bar.K]. As T(16) belongs to the center of [Z.sup.4] [direct sum] [M.sub.2](Z) [direct sum] [bar.H](Q[[square root of (2)]]), then [bar.P](T(16)[X.sub.0]) = [bar.T](16)K. Let [X.sub.1] be a column vector with coordinates in Z[[Q.sub.16]] such that T([X.sub.1]) = T(16)[X.sub.0], then P[X.sub.1] = 16K. m

Let M(q) be the 16 x 16 matrix given by multiplication in Z[[Q.sub.16]], that is,

[mathematical expression not reproducible],

where

[mathematical expression not reproducible]

and

[mathematical expression not reproducible]

The linear system PX = K over Z[[Q.sub.16]] is equivalent to a linear system M(P)X = V(K) over Z, where M(P) = [M([p.sub.ij])] is a block matrix and V(K) = [v([k.sub.1]) *** v([[k.sub.m])].sup.t] with v(q) = [[q.sub.0] *** [[q.sub.15]].sup.t] is a column vector with coordinates in Z.

Lemma 2.3. Let AX = K be a linear system over Z, where Ais an m x n matrix with m [less than or equal to] n. If the system AX = [2.sup.r]K has a solution over Z, where r [greater than or equal to] 1 is an integer and the matrix A has at least one odd m x m-minor, then AX = K has solution over Z.

Proof: By hypothesis, there exists an m x m-submatrix B = [[b.sub.ij]] of A with an odd det B. Therefore, the system AX = (det B)K has a solution over Z. Suppose that A[X.sub.0] = [2.sup.r]K, A[X.sub.1] = (detB)K and s * [2.sup.r] + p * (det B) = 1 with s, p [member of] Z. Then, A(s[X.sub.0] + p[X.sub.1]) = s * [2.sup.r]K + p * (detB)K = (s * [2.sup.r] + p * detB)K = K.

3 Parity between det M(q) and [epsilon](q)

Let [T] be the matrix of T, then

[mathematical expression not reproducible]

where

[mathematical expression not reproducible]

Theorem 3.1. If A = [[A.sub.ij]] is a square block matrix, where the A[i.sub.j] are mutually commutative m x m matrices, and B is them x m matrix obtained by taking the determinant of A with the Aij as elements, then det A = det B.

Proof: See theorem 1 of [7].

Let q = a+b [square root of (2)] 2 + ci +d [square root of (2i)] + ej + f[square root of (2j)]+ gk+h [square root of (2k)] be an element of [bar.H](Q[[square root of (2)]]) and s (q) = [a.sup.2] + [c.sup.2] + [e.sup.2] + [g.sup.2] + 2[b.sup.2] + 2[d.sup.2] + 2[f.sup.2] + 2[h.sup.2].

Lemma 3.2. [[square root of (detQ(q))]] = -8[(ba + dc + fe + hg).sup.2] + s[(q).sup.2]

Proof: Consider the mutually commutative matrices,

[mathematical expression not reproducible]

Therefore

[mathematical expression not reproducible]

and applying the theorem 3.1, we have

det Q(q) = det[([A.sup.2] + [B.sup.2] + [C.sup.2] + [D.sup.2]).sup.2] = (det[([A.sup.2] + [B.sup.2] + [C.sup.2] + [D.sup.2])).sup.2].

If S = [A.sup.2] + [B.sup.2] + [C.sup.2] + [D.sup.2], then

[mathematical expression not reproducible]

Thus, [mathematical expression not reproducible].

Theorem 3.3. The numbers det Q(q) and [epsilon](q) have the same parity.

Proof: The term -8[(ba + dc + fe + hg).sup.2] is always even, then the parity det Q(q) is equal to parity s(q) = [a.sup.2] + [c.sup.2] + [e.sup.2] + [g.sup.2] + 2[b.sup.2] + 2[d.sup.2] + 2[f.sup.2] + 2[h.sup.2]. The numbers [epsilon](q) and a + c + e + g + 2b + 2f have the same parity. We have the following possibilities:

i) [epsilon](q) odd, 2b and 2f even.

In this case, a + c + e + g is odd, 2b = 2[n.sub.1], 2f = 2[n.sub.2], 2d = 2[n.sub.3] and 2h = 2[n.sub.4] with [n.sub.1], [n.sub.2], [n.sub.3], [n.sub.4] [member of] Z. Therefore, 4[b.sup.2] = 4[k.sub.1], 4[f.sup.2] = 4[k.sub.2], 4[d.sup.2] = 4[k.sub.3], 4[h.sup.2] = 4[k.sub.4] with [k.sub.1], [k.sub.2], [k.sub.3], [k.sub.4] [member of] Z and 2[b.sup.2] + 2[d.sup.2] + 2[f.sup.2] + 2[h.sup.2] = 2([k.sub.1] + [k.sub.2] + [k.sub.3] + [k.sub.4]) is even. Because [a.sup.2] + [c.sup.2] + [e.sup.2] + [g.sup.2] is odd, then s(q) is odd.

ii) [epsilon](q) odd, 2b and 2f odd.

In this case, a + c + e + g is odd, 2b = 2[n.sub.1] + 1, 2f = 2[n.sub.2] + 1, 2d = 2[n.sub.3] + 1 and 2h = 2[n.sub.4] + 1 with [n.sub.1], [n.sub.2], [n.sub.3], [n.sub.4] [member of] Z. Thus, 4[b.sup.2] = 4[k.sub.1] + 1, 4[f.sup.2] = 4[k.sub.2] + 1, 4[d.sup.2] = 4[k.sub.3] + 1,4[h.sup.2] = 4[k.sub.4] + 1 with [k.sub.1], [k.sub.2], [k.sub.3], [k.sub.4] [member of] Z and

2[b.sup.2] + 2[d.sup.2] + 2[f.sup.2] + 2[h.sup.2] = 2 ([k.sub.1] + [k.sub.2] + [k.sub.3] + [k.sub.4]) + 2

As [a.sup.2] + [c.sup.2] + [e.sup.2] + [g.sup.2] is odd, then s (q) is odd.

iii) [epsilon](q) odd, 2b odd and 2f even.

In this case, a + c + e + g is even, 2b = 2[n.sub.1] + 1, 2f = 2[n.sub.2], 2d = 2[n.sub.3] + 1 and 2h = 2[n.sub.4] with [n.sub.1], [n.sub.2], [n.sub.3], [n.sub.4] [member of] Z. So, 4[b.sup.2] = 4[k.sub.1] + 1,4[f.sup.2] = 4[k.sub.2],4[d.sup.2] = 4[k.sub.3] + 1 and 4[h.sup.2] = 4[k.sub.4], com [k.sub.1], [k.sub.2], [k.sub.3], [k.sub.4] [member of] Z and

2[b.sup.2] + 2[d.sup.2] + 2[f.sup.2] + 2[h.sup.2] = 2 ([k.sub.1] + [k.sub.2] + [k.sub.3] + [k.sub.4]) + 1

As [a.sup.2] + [c.sup.2] + [e.sup.2] + [g.sup.2] is even, then s (q) is odd.

If [epsilon](q) is even, the argument is analogous.

In particular, it was demonstrated that the numbers [epsilon](q) and s(q) have the same parity.

Theorem 3.4. The numbers [epsilon](q) and detN(q) have the same parity, where

[mathematical expression not reproducible]

Proof: Because the matrix N(q) is formed by commutative blocks, by theorem 3.1

det N(q) = [([n.sub.1][n.sub.4] - [n.sub.3][n.sub.2]).sup.2].

Notice that [bar.[n.sub.1]] = [bar.[n.sub.4]] and [bar.[n.sub.3]] = [bar.[n.sub.2]]. We have two cases to consider:

i) [epsilon](q) odd.

In this case, we have [bar.[n.sub.1]] [not equal to] [bar.[n.sub.3]], then detN(q) is odd.

ii) [epsilon](q) even.

In this case, we have [bar.[n.sub.1]] = [bar.[n.sub.3]], then det N(q) is even.

Theorem 3.5. The numbers det M(q) and [epsilon](q) have the same parity.

Proof: The determinant of the matrix M(q) is equal to the determinant of the matrix [T]M(q)[[T].sup.-1]. Therefore,

det M(q) = [epsilon](q) * [q.sup.1] * [q.sup.2] * [q.sup.3] * detN(q) * det Q(q).

Now apply theorem 3.3 and theorem 3.4.

4 Linear systems over [M.sub.2](Z)

Let Z be the ring of integer numbers and [M.sub.2](Z) the ring of 2 x 2 matrices with entries in Z. Consider the matrices

[mathematical expression not reproducible]

in [M.sub.2](Z). The system AX = K over [M.sub.2](Z) is equivalent to the system N(A)X = v[(K).sup.t] over Z, where

[mathematical expression not reproducible] and v (K = [[X.sub.0] [y.sub.0] [z.sub.0] [t.sub.0]].

Let [Z.sub.2] = {[bar.0], [bar.1]} be the ring of integers modulo 2 and [E.sub.4]([Z.sub.2]) be the set of all 4 x 4 matrices over [Z.sub.2] of the form:

[mathematical expression not reproducible]

The set [E.sub.4]([Z.sub.2]) is a commutative subring with identity of the matrix ring [M.sub.4]([Z.sub.2]). Since det(E) = a + b for any E [member of] [E.sub.4]([Z.sub.2]), we have

det([E.sub.1] + [E.sub.2]) = det([E.sub.1]) + det([E.sub.2])

for all [E.sub.1],[E.sub.2] [member of] [E.sub.4]([Z.sub.2]).

Consider [N.sub.2](Z) the set of matrices A = [[a.sub.ij]] [member of] [M.sub.2](Z) such that [bar.[a.sub.11]] = [bar.[a.sub.22]] and [bar.[a.sub.21]] = [bar.[a.sub.12]]. Observe that, if A [member of] [N.sub.2](Z), then [phi](N(A)) [member of] [E.sub.4]([Z.sub.2]).

Lemma 4.1. Let A = [[A.sub.ij]] be ann x n matrix over [E.sub.4]([Z.sub.2]), i.e, a block matrix. If A is regarded as a matrix over [Z.sub.2], then its determinant will be

det(A) = [summation over ([sigma][member of][S.sub.n])]det([A.sub.1[sigma](1)]) *** det([A.sub.n[sigma](n)]),

where [S.sub.n] is the symmetric group on n symbols.

Proof: As blocks [A.sub.ij] of A are commutative pairwise, we can apply theorem 3.1. Thus,

[mathematical expression not reproducible].

Since the determinant on [E.sub.4]([Z.sub.2]) is additive, the result follows.

Lemma 4.2. Let B = [[B.sub.ij]] be an m x m matrix with entries in [N.sub.2](Z). If D(B) = [det([B.sub.ij])] and N(B) = [N([B.sub.ij])], then det D(B) = det N(B).

Proof: As [phi](N([B.sub.ij])) [member of] [E.sub.4]([Z.sub.2]), from lemma 4.1

[mathematical expression not reproducible]

Since det N([B.sub.ij]) = [(det[B.sub.ij]).sup.2], then [bar.det(N([B.sub.ij]))] = [bar.det[B.sub.ij]] and

[mathematical expression not reproducible]

Hence, [[bar.det N(B)] = vdet D(B)].

Theorem 4.3. Consider the linear system AX = K over [M.sub.2](Z), where A = [[A.sub.ij]] is an m x n matrix with m [less than or equal to] n and [A.sub.ij] [member of] [N.sub.2](Z). If the system AX = [mathematical expression not reproducible] has solution over [M.sub.2] Z and the matrix D(A) = [det] ([A.sub.ij]) has at least one odd m x m-minor, then the system AX = K has a solution over [M.sub.2](Z).

Proof: Certainly, the system AX = K over [M.sub.2](Z) is equivalent to the system N(A)X = V(K) over Z, where N(A) = [N([A.sub.ij])] is a block matrix and V(K) = [v([k.sub.1]) *** v[([k.sub.m])].sup.t] is a column vector. Multiplying the equation AX = [mathematical expression not reproducible]. by [mathematical expression not reproducible] from the right, we have

[mathematical expression not reproducible].

Hence, a solution of [mathematical expression not reproducible] over [M.sub.2] (Z) yields a solution of the system N(A)X = 2V(K) over Z. If we find a 2m x 2m odd minor of N(A), we can apply lemma 2.3 and conclude that N(A)X = V(K) has solution over Z. By hypothesis, there exists an m x m-submatrix B = [[B.sub.ij]] of A with an odd det D(B). From lemma 4.2, detN(B) is odd.

5 Quaternionic determinants

In the case studied in [2] the 4x4 matrix equivalent to Q(q) had entries in Z and the techniques of section 4 were sufficient to obtain the expected results. Here Q(q) has rational entries, being necessary the introduction of new concepts such as quaternionic determinants. Let R be the field of real numbers, we can write A [member of] [M.sub.n](H(R)) uniquely as A = [A.sub.0] + [A.sub.1]i + [A.sub.2]j + [A.sub.3]k where [A.sub.0], [A.sub.1], [A.sub.2] and [A.sub.3] are real n x n matrices. Consider the homomorphism [mu] : Mn(H(R)) [right arrow] [M.sub.4n](R) given by

[mathematical expression not reproducible]

According to [5], the Study's determinant is Sdet(A) = [square root of (detR [mu](A))]. For any a = [a.sub.0] + [a.sub.1]i + [a.sub.2]j + [a.sub.3]k [member of] H (R) define

[mathematical expression not reproducible]

Given an matrix A = [[a.sub.ij]] [member of] [M.sub.n](H(R)), the matrix [psi] (A) = [[psi]([a.sub.ij])] [member of] [M.sub.4n](R) can be transformed by means of exchanging lines, columns and signs in the matrix [mu] (A). Therefore, Sdet(A) = [square root of (detR [psi](A))].

The conjugate of a quaternion q = [q.sub.0] + [q.sub.1]i + [q.sub.2]j + [q.sub.3]k is defined to be [bar.q] = [q.sub.0] - [q.sub.1]i - [q.sub.2]j - [q.sub.3] k so that [bar.p + q] = [bar.p] + [bar.q], [bar.p * q] = [bar.q] *[bar.p] for all p, q [member of] H(R). The norm of q is defined by [eta] (q) = q * [bar.q] = [q.sup.2.sub.0] + [q.sup.2.sub.1] + [q.sup.2.sub.2] + [q.sup.2.sub.3] such that [eta](p * q) = [eta](p) * [eta](q) and the trace of q is t(q) = q + [bar.q].

Given a matrix A [member of] Mn(H(R)) define A* = [[bar.A].sup.t], if A = A*, we say that A is Hermitian. For a Hermitian matrix, the Moore's determinant is defined by specifying a certain ordering of the factors in the n! terms in the sum. Let [sigma] be a permutation of n. Write it as a product of disjoint cycles. Permute each cycle cyclically until the smallest number in the cycle is in front. Then sort the cycles in decreasing order according to the first number of each cycle. In other words, write

[mathematical expression not reproducible]

where for each i, we have [n.sub.i1] < [n.sub.ij] for all j > 1, and [n.sub.11] > [n.sub.21] > *** > [n.sub.r1]. Then we define

[mathematical expression not reproducible]

Observe that, for any matrix A [member of] Mn(H(R)), the matrix AA* is Hermitian.

Theorem 5.1. For any quaternionic matrix A, we have

SdetA = Mdet(AA*).

Proof: See theorem 10 of [5].

If A is Hermitian, then Mdet(A) is a real number. In particular, if A is Hermitian and A [member of] [M.sub.n] ([bar.H](Q[[square root of (2)]])), then Mdet (A) [member of] [bar.H](Q[[square root of (2)]]) and Mdet(A) is a real number. Therefore, Mdet(A) = a +b [square root of (2)], with a, b [member of] Z, because b [member of] Q with 2b even. Notice that, if q [member of] [bar.H](Q[[square root of (2)]]), then

[mathematical expression not reproducible]

Therefore, [eta](q) = s(q) with s(q) + r [square root of 2], r [member of] Z.

Lemma 5.2. For any [q.sub.], [q.sub.2] [member of] [bar.H](Q[[square root of (2)]]) we have [bar.s([q.sub.1] + [q.sub.2])] = [bar.s([q.sub.1]) + s([q.sub.2])] and [bar.s([q.sub.1] * [q.sub.2])] = [bar.s([q.sub.1]) * s([q.sub.2])].

Proof: Notice that,

[mathematical expression not reproducible]

Hence, s([q.sub.1] + [q.sub.2]) = s([q.sub.1]) + s([q.sub.2]) + 2m, that is, [bar.s([q.sub.1] + [q.sub.2])] = [bar.s([q.sub.1]) + s([q.sub.2])].

Moreover,

[mathematical expression not reproducible]

Thus, s([q.sub.1] * [q.sub.2]) = s([q.sub.1]) * s([q.sub.2]) + 2[r.sub.1][r.sub.2], that is, [bar.s([q.sub.1] * [q.sub.2])] = [bar.s([q.sub.1]) * s([q.sub.2])].

Lemma 5.3. If [mathematical expression not reproducible] and [mathematical expression not reproducible], then Mdet(AA*) = S + R [square root of (2)] with R, S [member of] Z and [bar.S] = [bar.det s (A)].

Proof: Consider the matrix

[mathematical expression not reproducible]

The Moore's determinant is

[mathematical expression not reproducible]

Because [q.sub.1][bar.[q.sub.1]][q.sub.3][bar.[q.sub.3]] = [q.sub.3][bar.[q.sub.1]][q.sub.1][bar.[q.sub.3]], [q.sub.2][bar.[q.sub.2]][q.sub.4][bar.[q.sub.4]] = [q.sub.4][bar.[q.sub.2]][q.sub.2][bar.[q.sub.4]], [q.sub.3][bar.[q.sub.1]][q.sub.2][bar.[q.sub.4]] = [bar.[q.sub.4] [bar.[q.sub.2]][q.sub.1][bar.[q.sub.3]]], then

[mathematical expression not reproducible]

If S = s([q.sub.1]) * s([q.sub.4]) + s([q.sub.3]) * s([q.sub.2]) + 2[k.sub.2], then Mdet(AA*) = S + R [square root of (2)] with [bar.det s (A) = S].

The technique used to prove the lemma 5.3 requires the analysis of an excessively large number of combinations to n greater than two, so we will use Ivan Kyrchei's ideas in [8] to extend it. In [8], for an A [member of] [M.sub.n](H(R)) is defined the ith row determinant, denoted by rdetiA and jth column determinant, denoted by cdetiA. For the purpose of this work it is only necessary to define the ith row determinant. In [8], definition 2.4, the ith row determinant of A [member of] [M.sub.n](H(R)) is defined as the alternating sum of n! products of entries of A, during which the index permutation of every product is written by the direct product of disjoint cycles. That is

[mathematical expression not reproducible]

The cycle notation of the permutation [sigma] is written as follows

[mathematical expression not reproducible]

Here the index i opens the first cycle from the left and the other cycles satisfy the following conditions

[mathematical expression not reproducible]

Theorem 5.4. If A [member of] [M.sub.n](H(R)) is a Hermitian matrix, then

rde[t.sub.1]A = *** = rde[t.sub.n]A = cde[t.sub.1]A = *** = cde[t.sub.n]A [member of] R

Proof: See theorem 3.1 of [8].

Since all column and row determinants of a Hermitian matrix over H(R) are equal, in [8] remark 3.1, put

det(A) = rde[t.sub.i]A = cde[t.sub.i]A for i = 1,..., n.

Given a matrix A = [a.sub.ij] [member of][M.sub.n]([bar.H](Q[[square root of (2)]])), we define [A.sub.2] = [[a.sup.2.sub.ij]]. The next result was inspired by theorem 3.1 of [8].

Theorem 5.5. For any matrix A=[a.sub.ij] [member of] [M.sub.n], ([bar.H](Q[[square root of (2)]]))if

[mathematical expression not reproducible],

with [S.sub.1], [R.sub.1], [S.sub.2], [R.sub.2] [member of] Z, then [bar.[S.sub.1]] = [bar.[S.sub.2]].

Proof: Initially suppose that the matrix A = [[a.sub.ij]] is Hermitian, then [a.sub.ii] [member of] R and [a.sub.ij] = [bar.[a.sub.ij]]. We divide the monomials of some rde[t.sub.i]A into two subsets. If the indices of the coefficients of a monomial form a permutation as product of disjoint cycles of lengths 1 and 2 then we include this monomial in the first subset, the other monomials belong to the second subset. If the indices of the coefficients form a disjoint cycle of length 1, then these coefficients are entries of the principal diagonal of the Hermitian matrix A, hence they are real number of the form [s.sub.1] + [r.sub.1] [square root of (2)] with [s.sub.1], [r.sub.1] [member of] Z. If the indices of the coefficients form a disjoint cycle of length 2, then these elements are conjugated, [mathematical expression not reproducible], and their product takes on a real value number as well,

[mathematical expression not reproducible]

Let d be a monomial from the second subset and assume that the indices of its coefficients form a permutation as a product of r disjoint cycles, by the proof of the theorem 3.1 of [8], there exist another [2.sup.p] - 1 monomials, where p = r - [rho] and [rho] is the number of disjoint cycles of length 1 and 2, such that the sum of these [2.sup.p] - 1 monomials and d is given by

[mathematical expression not reproducible]

Here [alpha] is the product of the coefficients whose indices form disjoint cycles of lengths 1 and 2, [v.sub.k] [member of] {1,..., r} and k = 1, ... p. As t(q) = 2m+n[square root of (2)] with m, n [member of] Z, if det A = [s.sub.2] + [r.sub.2], [square root of (2)] then the parity of [s.sub.1] is determined by the monomials of the first subset.

Now we consider the Hermitian matrix C = AA* and the Hermitian matrix D = [A.sub.2][A*.sub.2]. Note that,

[mathematical expression not reproducible].

From lemma 5.2, [bar.s([c.sub.ij])] = [bar.s ([d.sub.ij])]. Let us look at the elements of the principal diagonal,

[mathematical expression not reproducible]

and

[mathematical expression not reproducible],

since [bar.s([c.sub.ii])] = [bar.s ([d.sub.ii])], if [c.sub.11] *** [c.sub.nn] = [s.sub.3] + [r.sub.3] [square root of (2)] and [d.sub.11] *** [d.sub.nn] = [s.sub.4] + [r.sub.4] [square root of (2)], then [bar.[s.sub.3]] = [bar.[s.sub.4]]. Now, let's consider the monomials whose indices of coefficients form a disjoint cycle of length 2, in this case these elements are conjugated, [mathematical expression not reproducible] and [mathematical expression not reproducible], then

[mathematical expression not reproducible]

and

[mathematical expression not reproducible],

with [mathematical expression not reproducible].

Therefore, if det(AA*) = [S.sub.1] + [R.sub.1] [square root of (2)] and det([A.sub.2][A*.sub.2]) = [S.sub.2] + [R.sub.2] [square root of (2)], then [bar.[S.sub.1]] = [bar.[S.sub.2]].

In [8], definition 7.2, for any A [member of] [M.sub.n](H(R)), the determinant of its corresponding Hermitian matrix is called its double determinant, that is,

ddetA = det(A*A) = det(AA*).

Corollary 5.6. For any matrix A = [[a.sub.ij]] [member of] [M.sub.n]([bar.H](Q[[square root of (2)]])), if

[mathematical expression not reproducible],

with [S.sub.1], [R.sub.1], [S.sub.2], [R.sub.2] [member of] Z, then [S.sub.1] = [S.sub.2].

Proof: By remark 7.2 of [8], we have ddetA = Mdet(A* A).

6 Linear systems over [bar.H](Q[[square root of (2)]])

Lemma 6.1. If q [member of] [bar.H](Q[[square root of (2)]]), then [q.sup.2] [member of] [bar.H](Q[[square root of (2)]]).

Proof Let q = (a + b [square root of (2)]) + (c + d [square root of (2)])i + (e + f [square root of (2)])j + (g + h [square root of (2)])k be an element of q [member of] [bar.H](Q[[square root of (2)]]), we have

[mathematical expression not reproducible],

where

i) [mathematical expression not reproducible], with A [member of] Z and B [member of] Z

ii) [mathematical expression not reproducible], with C [member of] Z and D [member of] Z

iii) [mathematical expression not reproducible], with E [member of] Z and F [member of] Z

iv) [mathematical expression not reproducible], with G [member of] Z and H [member of] Z.

Consider the ring homomorphism [phi] : Q[[square root of (2)]] [right arrow] [M.sub.2](Q) given by

[mathematical expression not reproducible].

Lemma 6.2. If A = [[a.sub.ij]] is an element of [M.sub.n] ([bar.H](Q[[square root of (2)]]), then

detQ(A) = (det[([phi](Mdet(AA*)))).sup.2],

where Q(A) = [Q([a.sub.ij])].

Proof The matrix [psi](A) is an element of [mathematical expression not reproducible]. Notice that [mathematical expression not reproducible], then detQ(A) = det([phi](de[t.sub.R] [psi](A))). Because detR [psi](A) = [Sdet.sup.2] (A) and Mdet(AA*) = Sdet(A), we have

detQ(A) = (det([phi][(Mdet(AA*)))).sup.2].

Lemma 6.3. If [q.sub.1], [q.sub.2] [member of] H(Z[[square root of (2)]]), then the matrices [phi](Q([q.sub.1])) and [phi](Q([q.sub.2])) are mutually commutative and det([phi](Q([q.sub.1])) + [phi](Q([q.sub.2]))) = det[phi](Q([q.sub.1])) +det[phi](Q([q.sub.2])).

Proof: The matrices

[mathematical expression not reproducible]

where [A.sub.i], [B.sub.i], [C.sub.i], [D.sub.i] [member of] [M.sub.2] (Z) and are mutually commutative for i = 1,2. Hence

[mathematical expression not reproducible],

and [phi](Q([q.sub.1])) * [phi](Q([q.sub.2])) = [phi](Q([q.sub.2])) * [phi](Q([q.sub.1])). Moreover

det([phi](Q([q.sub.1])) + [phi](Q([q.sub.2]))) = det([phi](Q([q.sub.1]) + Q([q.sub.2]))) = [phi](det(Q([q.sub.1]) + Q([q.sub.2]))).

Now, Q([q.sub.1]) + Q([q.sub.2]) = Q([q.sub.1] + [q.sub.2]) and from lemma 6.2,

detQ([q.sub.1] + [q.sub.2]) = (det([phi]([eta][([q.sub.1] + [q.sub.2])))).sup.2].

From lemma 5.2, [eta]([q.sub.1] + [q.sub.2]) = s([q.sub.1]) + s([q.sub.2]) + 2m + ([r.sub.1] + [r.sub.2] + n) [square root of (2)]. Hence,

(det([phi]([eta]([q.sub.1] + q2)))) = ((s([q.sub.1]) + s[(q2) +2m).sup.2] - 2[([r.sub.1] +r2 + n)).sup.2].

Thus,

Proof.

[phi](detQ([q.sub.1] + [q.sub.2])) = [phi](s([q.sub.1])) + [phi](s([q.sub.2])) = [phi](detQ([q.sub.1])) + [phi](detQ([q.sub.2])).

Theorem 6.4. If A = [[a.sub.ij]] is an element of [M.sub.n]([bar.H](Q[[square root of (2)]]), then dets(A) and detQ(A) have the same parity.

Proof. From lemma 6.1, the matrix Q([A.sub.2]) = [Q([a.sup.2.sub.j])] has integer entries. Using the techniques of section 4 and the lemma 6.3, we conclude that detQ([A.sub.2]) and dets([A.sub.2]) have the same parity. As [bar.s(q)] = [bar.s([q.sup.2])],then [bar.detQ([A.sub.2])] = [bar.dets(A)]. From lemma 6.2

detQ([A.sub.2]) = (det([phi](Mdet[([A.sub.2][A*.sub.2])))).sup.2] = [([S.sup.2.sub.1] - 2[R.sup.2.sub.1]).sup.2]

and

detQ(A) = [(det([phi](Mdet(AA*)))).sup.2] = [([S.sub.2.sup.2] - 2[R.sup.2.sub.2]).sup.2]

From theorem 5.4, [bar.[S.sub.1]] = [bar.[S.sub.2]]. Therefore

[bar.dets(A)] = [bar.detQ([A.sub.2])] = [bar.[S.sub.1]] = [bar.[S.sub.2]] = [bar.det Q(A)].

Corollary 6.5. If A = [[a.sub.ij]] is an element of [M.sub.n]([bar.H](Q[[square root of (2)]])), then

Mdet(AA*) = S + R [square root of (2)],

with [bar.S] = [bar.det s (A)].

Lemma 6.6. If [mathematical expression not reproducible] and K is column vector with entries in [bar.H](Q[[square root of (2)]]), then the system AX = Mdet(AA*) * K has a solution in [bar.H](Q[[square root of (2)]]).

Proof: If Mdet(AA*) = 0, then X = 0. On the other hand, if Mdet(AA*) [not equal to] 0, by theorem 8.1 and remark 8.1 of [8], there exists [mathematical expression not reproducible] such that A * Adj[[A]] = Mdet(AA*) * [I.sub.n], where [I.sub.n] is the identity matrix. Hence, A * (Adj[[A]]K) = Mdet(AA*) * K. As [bar.H](Q[[square root of (2)]]) is a subring, then [X.sub.0] = Adj[[A]]K is a column vector with coordinates in [bar.H](Q[[square root of (2)]]) and A[X.sub.0] = Mdet(AA*) * K.

Theorem 6.7. Consider the system AX = K over [bar.H](Q[[square root of (2)]]), where A = [[a.sub.ij]] is a over [bar.H](Q[[square root of (2)]]) and the matrix s (A) = [s([a.sub.ij])] has at least one odd m x m-minor, then the system AX = K has a solution over [bar.H](Q[[square root of (2)]]).

Proof: Multiplying the equation [mathematical expression not reproducible] by [mathematical expression not reproducible] from the right, we have [mathematical expression not reproducible]. Therefore, the equation AX = 2K has a solution over [bar.H](Q[[square root of (2)]]). By hypothesis, there exists an m x m-submatrix B = [[b.sub.ij]] of A with an odd dets(B). From corollary 6.5, Mdet(BB*) = S + R[[square root of (2)]] with S odd. From lemma 6.6, the system AX = (S + R[square root of (2)]) has solution over [bar.H](Q[[square root of (2)]]). Multiplying the equation AX = (S + R[square root of (2)]) by R-S[square root of (2)], we have AX (R-S[square root of (2)]) = ([R.sup.2] - 2[S.sup.2]). Thus the equation AX = ([R.sup.2] -2[S.sup.2])K has solution over [bar.H](Q[[square root of (2)]]). As [R.sup.2] - 2[S.sup.2] is odd, then AX = K has solution over [bar.H](Q[[square root of (2)]]).

7 Main Result

Consider matrices P = [[q.sub.ij]] [member of] [M.sub.n](Z[[Q.sub.16]]), M(P) = [M([q.sub.ij])], N(P) = [N([q.sub.ij])], [epsilon](P) = [[epsilon]([q.sub.ij])] and[member of](P) = [[member of]([q.sub.ij])], where

[mathematical expression not reproducible]

Let T be a diagonal block 16n x 16n-matrix, whose blocks of the diagonal is the matrix [T], then [T.sup.-1] is the diagonal block matrix, whose blocks of the diagonal are [[T].sup.-1].Consider the matrix

[mathematical expression not reproducible]

Exchanging rows and columns of TM(P)[T.sup.-1], we have

[mathematical expression not reproducible]

Theorem 7.1. The numbers det M(P) and det[epsilon](P) have the same parity.

Proof: The number detM(P) is equal to the number det M(P). Now,

detM(P) = det[member of](P) * detN(P) * det Q(P)

i) Using the same techniques of section 4, we have [bar.[bar.det[member of](P)]] = [bar.det[epsilon](P)].

ii) From lemma 4.2 and theorem 3.4, it follows [bar.det N(P)] = [bar.det[epsilon](P)].

iii) From theorem 6.4 and theorem 3.3, it follows [bar.det Q(P)] = [bar.det[epsilon](P)].

Therefore, [bar.det M(P)] = [bar.det [epsilon](P)].

Theorem 7.2. Let PX = K be a linear system over Z[[Q.sub.16]], where P = [[p.sub.ij]] is am x n matrix with m [less than or equal to] n. If PX = K(x - 1) and PX = K(-xy + 1) has a solution over Z[[Q.sub.16]] and all m x m minors of [epsilon](P) = [[epsilon]([p.sub.ij])] are relatively prime, then the system PX = K has a solution over Z[[Q.sub.16]].

Proof First, we look at the system [bar.P]X = [bar.K] over [Z.sup.4] [direct sum] [M.sup.2](Z) [direct sum] [bar.H](Q[[square root of (2)]]). Observe that this system is equivalent to the systems, [epsilon](P) = [epsilon](K), [P.sub.1] X = [K.sup.1], [P.sub.2]X = [K.sup.2], [P.sub.3]X = [K.sup.3], [P.sub.4] = [K.sup.4] and [P.sub.5]X = [K.sup.5], where [P.sub.1] = [[p.sup.1.sub.ij]], [P.sub.2] = [[p.sup.2.sub.ij]], [P.sub.3] = [[p.sup.3.sub.ij]], [P.sub.4] = [[p.sup.4.sub.ij]], [P.sub.5] = [[p.sup.5.sub.ij]] and [K.sup.i] = [[k.sup.i.sub.1] *** [k.sup.i.sub.m]]t for i = 1,..., 5. The first four systems must be solved over Z, the fifth system must be solved over [M.sub.2] (Z), and the last one over [bar.H](Q[[square root of (2)]]). Since the m x m minors of [epsilon](P) are relatively prime, the system [epsilon](P)X = [epsilon](K) has a solution over Z. Notice that

[mathematical expression not reproducible]

and

[mathematical expression not reproducible].

By hypothesis PX = (x - 1)K and PX = (-xy + 1)K has a solution over Z[[Q.sub.16]], then the systems [P.sub.1] = 2[K.sup.1], [P.sub.2]X = 2[K.sup.2], [P.sub.3]X = -2[K.sup.3] have solutions over Z,

[mathematical expression not reproducible]

has a solution over [M.sub.2](Z) and

[mathematical expression not reproducible]

has a solution over [bar.H](Q[[square root of (2)]]). Since the integers [epsilon](pij), [p.sup.1.sub.ij], [p.sup.2.sub.ij], [p.sup.3.sub.ij], det([p.sup.4.sub.ij]) and s([q.sub.ij]) has the same parity, the matrices [P.sub.1], [P.sub.2], [P.sub.3], D = [det([p.sup.4.sub.ij])] and s([P.sub.5]) have at least one odd m x m-minor. From lemma 2.3, the first three systems have a solution over Z. From theorem 4.3, the system [P.sub.4]X = [K.sup.4] has a solution over [M.sub.2](Z). From theorem 6.7, the last system has a solution over [bar.H](Q[[square root of (2)]]). Hence, the system [bar.P]X = [bar.K] has a solution over [Z.sup.4] [direct sum] [M.sub.2] (Z) [direct sum][bar.H](Q[[square root of (2)]]). By theorem 2.2, the system PX = 16K has a solution over Z[[Q.sub.16]]. Now, by theorem 7.1, there exists a 16m x 16m submatrix B of M(P) with an odd det B, from lemma 2.3 the system PX = K has a solution over Z[[Q.sub.16]].

8 Strongly surjective map

The orbit space of the 3-sphere [S.sup.3] with respect to the action of the quaternion group Q16 determined by the inclusion [Q.sub.16] [??] [S.sup.3] is a compact orientable manifold of dimension 3, denoted by [mathematical expression not reproducible]. Let [mathematical expression not reproducible] be its universal covering. The CW-complex structure of [mathematical expression not reproducible] has one 3-cell and two 2-cells, and the boundary operator

[mathematical expression not reproducible]

is given by [mathematical expression not reproducible](see [6] pg. 253 and [10]). Let W be a finite connected CW-complex of dimension 3, with m cells of dimension 3, n cells of dimension 2, and W its universal covering. Suppose that the boundary operator

[mathematical expression not reproducible]

is given by the matrix

[mathematical expression not reproducible]

with columns defined by [mathematical expression not reproducible] for i = 1,..., m. We have

[mathematical expression not reproducible].

Here, [intersection] = [[pi].sub.1] (W) and [epsilon] : Z[[intersection]] [right arrow] Z is given by [mathematical expression not reproducible] and [epsilon][(A).sup.t] is the transpose of [epsilon](A) = [[epsilon](3[[~.a].sub.ij])]. Given a map f : W [right arrow] M, consider the matrix

[mathematical expression not reproducible]

where the ring homomorphism f# : Z[[intersection]] [right arrow] Z [[pi]] is defined by

[mathematical expression not reproducible]

is the extension of the induced homomorphism f# : [intersection] [right arrow] [pi] of fundamental groups.

Theorem 8.1. If W is a three dimensional CW-complex with [H.sup.3](W; Z) = 0, then there is no strongly surjective map f : W [right arrow] [mathematical expression not reproducible].

Proof: From corollary 5.5 of [1], we have to show that the system PX = K has a solution over Z[[Q.sub.16]], where P is the matrix described above. From theorem 5.6 of [1], the systems PX = K(x - 1) and PX = K(-xy + 1) have solutions over Z[[Q.sub.16]]. The hypothesis [H.sup.3](W;Z) = 0 implies that m [less than or equal to] n and all m x m minors {[[epsilon].sub.1],..., [[epsilon].sub.r]} of [epsilon](P) are relatively prime (see chapter 3, proposition 15 of [9]). From theorem 7.2, with these hypotheses the system PX = K has a solution over Z[[Q.sub.16]].

Finally, let [mathematical expression not reproducible] be the orbit space of 3-sphere [S.sup.3] with respect to the action of the quaternion group [Q.sub.32] determined by the inclusion [Q.sub.32] [??] [S.sup.3]. In the same way, one can consider the problem of the existence of a strongly surjective map f : W [right arrow] M[Q.sub.32]. This problem is equivalent to solving the linear system PX = K over Z[[Q.sub.32]] satisfying the following assumptions: PX = K(x - 1) and PX = K(-xy + 1) have solutions over Z[[Q.sub.32]] and all m x m minors of [epsilon](P) are relatively prime. The techniques used in this work rely heavily on the decomposition [mathematical expression not reproducible]. According to [4],

[mathematical expression not reproducible],

Here, [mathematical expression not reproducible] is the quaternion algebra over the field [mathematical expression not reproducible], generated by i, j, where [i.sup.2] = 2 - [square root of (2)], [j.sup.2] = -1 and ij = -ji. Therefore, the case [Q.sub.32], is not a simple generalization of the case [Q.sub.16].

Acknowledgments

I would like to thank Daciberg Lima Goncalves and Oziride Manzoli Neto for reading and suggestions which helped to improve the manuscript.

References

[1] C. Aniz, Strong surjectivity of mapping of some 3-complexes into 3-manifolds, Fundamenta Mathematicae 192 (2006), 195-214.

[2] C. Aniz, Strong surjectivity of mapping of some 3-complexes into [mathematical expression not reproducible], Central European Journal of Mathematics (2008), 1-7.

[3] C. Longxuan, Definition of determinant and cramer solutions over the quaternion field, Acta Mathematica Sinica 7 (1991) 171- 180.

[4] C. R. G. Vergara and F. E. B. Martinez, Wedderburn decomposition of some special rational group algebras, Lecturas Matematicas 23 (2002) 99-106.

[5] H. Aslaksen, Quaternionic determinants, The Mathematical Intelligencer (1996), 57-65.

[6] H. Cartan and S. Eilenberg, Homological Algebra, Princeton University Press, 1956.

[7] I. Kovacs, D. S. Silver and S. G. Williams, Determinants of commuting-block matrices, Amer. Math. Monthly 106 (1999), 950-952.

[8] I. Kyrchei, Cramer rule over quaternion skew field, arXiv:math/0702447v1 [math.RA] 2007.

[9] R. Brooks, Coincidences, roots and fixed points, Doctoral Dissertation, University of California, Los Angeles, 1967.

[10] R. G. Swan, Periodic resolutions for finite groups, Ann. of Math.72 (1960), 267-291.

Claudemir Aniz

Received by the editors in August 2016 - In revised for in December 2016.

Communicated by K. Dekimpe, D.L. Goncalves and P. Wong.

2010 Mathematics Subject Classification : 55M20, 55S35, 55N25.

Key words and phrases : quaternion group; quaternionic determinant; strongly surjective map.

Universidade Federal de Mato Grosso do Sul Instituto de Matematica Caixa Postal 549 79070-900/Campo Grande, MS, Brasil

E-mail: claudemir.aniz@ufms.br
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Author:Aniz, Claudemir
Publication:Bulletin of the Belgian Mathematical Society - Simon Stevin
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Date:Oct 1, 2017
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