aa r X i v : . [ m a t h . A T ] A ug DECOMPOSITION OF TOPOLOGICAL AZUMAYA ALGEBRAS
NINY ARCILA-MAYAA
BSTRACT . Let A be a topological Azumaya algebra of degree mn over aCW complex X . Firstly, we give conditions for the positive integers m and n , and the space X so that A can be decomposed as the tensor productof topological Azumaya algebras of degrees m and n . Then we prove thatif m < n and the dimension of X is higher than 2 m + A has no suchdecomposition. Keywords.
Topological Azumaya algebra, projective unitary group.
1. I
NTRODUCTION
The classical theory of central simple algebras over a field was general-ized by Azumaya [Azu51] and Auslander-Goldman [AG60] by introducingthe concept of Azumaya algebra over a local commutative ring and over anarbitrary commutative ring, respectively. This concept was generalized byGrothendieck [Gro66, 1.1] to the notion of topological Azumaya algebra.
Remark . The notion of Azumaya algebra can be defined over any locallyringed topos, [Gro66].
Definition 1.2. A topological Azumaya algebra of degree n over a topologicalspace X is a bundle of complex algebras over X that is locally isomorphic tothe matrix algebra M n ( C ). Remark . For a deeper discussion on topological Azumaya algebras andthe topological Brauer group, we refer to [AW14].The tensor product of complex algebras can be extended to topological Azu-maya algebras by performing the operation fiberwise. Saltman asked in[Sal99, page 35] whether there is prime decomposition for topological Azu-maya algebras under the tensor product operation, as there is for centralsimple algebras over a field. Antieau–Williams answered this question in[AW14b, Corollary 1.3] by showing the following result:
Theorem 1.4.
For n > an odd integer, there exist a 6-dimensional CW com-plex X and a topological Azumaya algebra A on X of degree n and period such that A has no decomposition A ∼= A ⊗ A n for topological Azumayaalgebras of degrees and n, respectively. The aim of this paper is to provide conditions on a positive integer n and atopological space X such that a topological Azumaya algebra of degree n on X has a tensor product decomposition. The main result of this paper is thefollowing theorem: Theorem 1.5.
Let m and n be positive integers such that m and n are rela-tively prime and m < n. Let X be a CW complex such that dim( X ) ≤ m + . If A is a topological Azumaya algebra of degree mn over X , then there existtopological Azumaya algebras A m and A n of degrees m and n, respectively,such that A ∼= A m ⊗ A n . Theorem 1.5 is a corollary of a more general result. Let a and m be positiveintegers. Let µ m ⊂ SU am be the central subgroup consisting of m -th roots ofunity. There is a short exact sequence of Lie groups:1 µ a SU am / µ m PU am ρ The homomorphism ρ induces a map of classifiying spaces B SU am / µ m → B PU am .From this, given a map from a topological space X to the space B SU am / µ m ,there is a degree- am topological Azumaya algebra over X . We prove inTheorem 3.3 that a map X → B SU abmn / µ mn can be lifted to B SU am / µ m × B SU bn / µ n when the dimension of X is less than 2 am + a , b , m and n are such that am is relatively prime to bn and am < bn .The proof of Theorem 3.3 relies significantly in the description of the homo-morphisms induced on homotopy groups by the r -fold direct sum of matrices L r : U n −→ U rn in the range {
0, 1, . . ., 2 n + } . We call this set “the stablerange” for U n .This paper is organized as follows. The second section presents some pre-liminaries on operations on unitary groups and the description of the homo-morphisms these operations induce on homotopy groups. The third sectionis devoted to the proof of Theorem 3.3. We explain why the decomposition inTheorem 1.5 is not unique up to isomorphism. Acknowledgements.
The author would like to express her deep gratitudeto Ben Williams, her thesis advisor, for having proposed this research topic,pointing out relevant references, and having devoted a great deal of time todiscuss details of the research with the author.
Notation.
Throughout this paper, all topological spaces will be CW com-plexes. We fix basepoints for connected topological spaces, and for topolog-ical groups we take the identities as basepoints. We write π i ( X ) in place of π i ( X , x ). If f : X → Y is a continous map, the induced homomorphism onhomotopy groups is denoted by π i ( f ) : π i ( X ) → π i ( Y ), for all i ∈ N .2. S TABILIZATION OF OPERATIONS ON U n Let m , n ∈ N , we consider the following matrix operations:(1) The direct sum of matrices , L : U m × U n −→ U m + n defined by A ⊕ B = µ A B ¶ .(2) The r-fold direct sum , L r : U n −→ U rn given by A ⊕ r = A ⊕ · · · ⊕ A | {z } r -times .(3) The tensor product of matrices , N : U m × U n −→ U mn defined by A ⊗ B = a B · · · a m B ... . . . ... a m B · · · a mm B , ECOMPOSITION OF TOPOLOGICAL AZUMAYA ALGEBRAS 3 for A = ( a i j ) ∈ U m .(4) The r-fold tensor product , N r : U n −→ U n r given by A ⊗ r = A ⊗ · · · ⊗ A | {z } r -times .The homomorphism of homotopy groups induced by the operations abovewill be denoted by L ∗ , L r ∗ , N ∗ and N r ∗ , respectively.We recall the homotopy groups of U n and SU n in low degrees, and computethe homotopy groups of U am / µ m and SU am / µ m in low degrees.The homotopy groups of the unitary group can be calculated by using Bottperiodicity. [Bot58] proves that π i (U n ) ∼= i < n is even, Z if i < n is odd, Z / n ! if i = n .Since there is a fibration SU n , → U n det −−→ S , we can use the long exact se-quence associated to it to see that π i (SU n ) ∼= ( i = π i (U n ) otherwise.Note that SU am is a simply connected m -cover of SU am / µ m . We deducethat π i (SU am / µ m ) ∼= ( Z / m if i = π i (SU am ) otherwise.Observe diagram (1) below(1) µ m µ m { } SU am U am S SU am / µ m U am / µ m S . detdet All column as well as the two top rows of diagram (1) are exact. The nine-lemma implies that the bottom row is also exact. Therefore, π i (U am / µ m ) ∼= π i (SU am / µ m ) for all i > am / µ m . By exactness ofthe bottom row of diagram (1), the induced sequence on fundamental groupsis exact,(2) 0 → π ¡ SU am / µ m ¢ → π ¡ U am / µ m ¢ → π ¡ S ¢ → π ¡ S ¢ ∼= Z , then sequence (2) splits. Hence, π ¡ U am / µ m ¢ ∼= Z ⊕ Z / m .2.1. Stabilization.
For n ∈ N , the standard inclusion of unitary groups U n , → U n + is 2 n -connected. Lemma 2.1.
For m , n ∈ N and m ≤ n. Let st m , n : U m −→ U m + n be the mapdefined by st m , n ( A ) = µ A I n ¶ . ARCILA-MAYA
The map st m , n : U m −→ U m + n is m-connected.Proof. The map st m , n is equal to the composite of a series of standard inclu-sions U m , → U m + , → U m + , → · · · , → U m + n . (cid:3) Lemma 2.2.
Let n , r ∈ N . For all j =
1, . . ., r define s j : U n −→ U rn by s j ( A ) = diag( I n , . . ., I n , A , I n , . . ., I n ), where A is in the j-th position. The map s j is n-connected for all j =
1, . . ., r.Proof.
Note that we can perform elementary row and column operations ons j ( A ) to transform it into s j + ( A ), this is s j + ( A ) = R s j ( A ) C where R and C are permutation matrices. This can be used to define a homotopy throughgroup homomorphisms between s j and s j + .Then all homomorphisms induced by s , . . ., s r on homotopy groups areequal. It suffices to study the homomorphism induced by s .Note that s = st n ,( r − n . By applying Lemma 2.1 the proof is completed. (cid:3) Remark . The matrix multiplication m : U n × U n −→ U n induces a homo-morphism of homotopy groups π i (m) : π i (U n × U n ) −→ π i (U n ) by pre-composition,for all i ∈ N . By using the isomorphism of homotopy groups π i ( X ) × π i ( Y ) π i ( X × Y ) ¡ α , β ¢ α × β , ∼= we can define a group operation m ∗ : π i (U n ) × π i (U n ) −→ π i (U n ). Therefore, π i (U n ) have two operations, the usual sum and the one induced by matrixmultiplication. The Eckmann–Hilton argument implies these operations areequal, this is m ∗ ( α , β ) = α + β .2.2. Operations.Proposition 2.4.
For all i ∈ N , the homomorphism L ∗ : π i (U m ) × π i (U n ) −→ π i (U m + n ) is equal to the composite of π i (st m , n ) × π i (st n , m ) and m ∗ .Proof. It is enough to observe that the direct sum factors asU m × U n U m + n × U m + n U m + n ( A , B ) µµ A I n ¶ , µ I m B ¶¶ µ A B ¶ . (st m , n ,st n , m ) m (cid:3) Remark . Observe that Lemma 2.1 lets us identify π i (U m ) with π i (U m + n )for all i < m . Corollary 2.6.
If m < n, then L ∗ ( x , y ) = x + y for all i < m.Proof. By Lemma 2.1, the homomorphisms π i (st m , n ) and π i (st n , m ) are isomor-phisms for i < m and i < n , respectively.Then from Proposition 2.4, L ∗ ( x , y ) = m ∗ ( x , y ) = x + y for x ∈ π i (U m ), y ∈ π i (U n ) and i < m . (cid:3) ECOMPOSITION OF TOPOLOGICAL AZUMAYA ALGEBRAS 5
Proposition 2.7.
For all i ∈ N , the homomorphism L r ∗ : π i (U n ) −→ π i (U rn ) isequal to the composite of the product of the stabilization maps π i (s j ) : π i (U n ) −→ π i (U rn ) , j =
1, . . ., r and multiplication by r.Proof.
The r -block summation factors asU n (U n ) × r (U rn ) × r U rn A ( A , . . ., A ) ¡ s ( A ), . . ., s r ( A ) ¢ s ( A ) · · · s r ( A ) = A ⊕ r . ∆ Q s j m By the Eckmann–Hilton argument m ∗ : π i (U rn ) r −→ π i (U rn ) is given bym ∗ ( a , . . ., a r ) = a + · · · + a r , for a j ∈ π i (U rn ) and j =
1, . . ., r .As a result, L r ∗ takes the form π i (U n ) π i (U n ) × r π i (U rn ) × r π i (U rn ) a ( a , . . ., a ) ¡ s ∗ ( a ), . . ., s ∗ ( a ) ¢ s ∗ ( a ) + · · · + s ∗ ( a ) = r s ∗ ( a ), ∆ Q π i (s j ) m where s ∗ denotes π i (s j ) with j =
1, . . ., r . This notation makes sense becauseLemma 2.2 yields the equality π i (s ) = · · · = π i (s r ). This proves the statement. (cid:3) Corollary 2.8.
If i < n, then L r ∗ ( x ) = rx.Proof. By Lemma 2.2, π i (s j ) : π i (U n ) −→ π i (U rn ) is an isomorphism for all j =
1, . . ., r and i < n , then Q π i (s j ) : π i (U n ) × r −→ π i (U rn ) × r is an isomorphismfor all i < n . By Proposition 2.7 we conclude L r ∗ ( x ) = rx for x ∈ π i (U n ) and i < n . (cid:3) Lemma 2.9.
The homomorphisms − ⊗ I n : U m −→ U mn and I n ⊗ − : U m −→ U mn are homotopic.Proof. Let A ∈ U m . Let E i j denote the matrix obtained by swapping the i throw and the j th row of I mn ∈ M mn ( C ).Observe that after perfoming a finite number of row and column operationson the matrix I n ⊗ A ∈ U mn , namely row interchange and column interchange,we obtain the the matrix A ⊗ I n . In other words, A ⊗ I n = R ( I n ⊗ A ) C where R and C are products of elementary matrices E i j .Since U mn is path-connected, there exist paths α R and α C in U mn from I mn to R and C , respectively.We define a homotopy H : U m × [0, 1] −→ U mn between − ⊗ I n and I n ⊗ − by H ( A , t ) = α R ( t )( I n ⊗ A ) α C ( t ). (cid:3) Proposition 2.10.
Let i ∈ N , then the homomorphism N ∗ : π i (U m ) × π i (U n ) −→ π i (U mn ) is given by N ∗ ( x , y ) = L n ∗ ( x ) + L m ∗ ( y ) for x ∈ π i (U m ) and y ∈ π i (U n ) .Proof. Consider the compositesU m U m × { I n } U mn and U n { I m } × U n U mn A ( A , I n ) A ⊗ I n B ( I m , B ) I m ⊗ B . N N
ARCILA-MAYA
Since I m ⊗ B = B ⊕ m , then the second composite is equal to L m . By Lemma 2.9the first composite is equivalent to the map A I n ⊗ A , then it is equivalentto L n . From this we get the commutative diagram belowU m × { I n } U m × U n { I m } × U n U mn . N L m L n Thus the induced diagram on homotopy groups takes the form π i (U m ) × { } π i (U m ) × π i (U n ) { } × π i (U n ) π i (U mn ). N ∗ L m ∗ L n ∗ (cid:3) Corollary 2.11.
If m < n, then N ∗ ( x , y ) = nx + m y for i < m.Proof. The statement follows from Corollary 2.8 and Proposition 2.10. (cid:3)
Proposition 2.12.
Let i ∈ N , then the homomorphism N r ∗ : π i (U n ) −→ π i (U n r ) is given by N r ∗ ( x ) = r L n r − ∗ ( x ) for x ∈ π i (U n ) .Proof. It can be proven by induction and Proposition 2.10. (cid:3)
Corollary 2.13.
If i < n, the map N r ∗ ( x ) = rn r − x.Proof. Corollary 2.8 and Proposition 2.12 yield the result. (cid:3)
Tensor product on the quotient.
Let a , b , m and n be positive integers.The tensor product operation N : U am × U bn −→ U abmn sends the group µ m × µ n to µ mn . In consequence, the operation descends to the quotient(3) ⊗ : U am / µ m × U bn / µ n −→ U abmn / µ mn . Proposition 2.14.
The homomorphism ⊗ ∗ : π i (U am / µ m ) × π i (U bn / µ n ) −→ π i (U abmn / µ mn ) is given by N ∗ ( x , y ) = L bn ∗ ( x ) + L am ∗ ( y ) for i > and by N ∗ ( x + α , y + β ) = bnx + am y + αβ for i = .Proof. There exits a map of fibrations(4) µ m × µ n U am × U bn U am / µ m × U bn / µ n µ mn U abmn U abmn / µ mn , m N N where m is multiplication. Then there exits a homomorphism between thelong exact sequences associated to the fibrations in diagram (4). We obtain acommutative square π i (U am ) × π i (U bn ) π i (U am / µ m ) × π i (U bn / µ n ) π i (U abmn ) π i (U abmn / µ mn ). ∼= N ∗ N ∗ ∼= ECOMPOSITION OF TOPOLOGICAL AZUMAYA ALGEBRAS 7
From this diagram and Proposition 2.10 we have that for all i > N ∗ ( x , y ) = L bn ∗ ( x ) + L am ∗ ( y ) where x ∈ π i (U am / µ m ) and y ∈ π i (U bn / µ n ).There exists a similar map of fibrations as the one in diagram (4), but withthe spaces SU am and SU bn instead of U am and U bn , respectively. In this casewe obtain the commutative square, π (SU am / µ m ) × π (SU bn / µ n ) µ m × µ n π (SU abmn / µ mn ) µ mn . ∼= N ∗ m ∼= Thus the homomorphism ⊗ ∗ : π (SU am / µ m ) × π (SU bn / µ n ) → π (SU abmn / µ mn )is equal to the multiplication.Consider the map of fibrationsSU am / µ m × SU bn / µ n U am / µ m × U bn / µ n S × S SU abmn / µ mn U abmn / µ mn S , N det × det N φ det where φ ( x , y ) = x bn y am . From this diagram we have a map of short exactsequences, π (SU am / µ m ) × π (SU bn / µ n ) π (U am / µ m ) × π (U bn / µ n ) π ( S ) × π ( S ) π (SU abmn / µ mn ) π (U abmn / µ mn ) π ( S ). i m N ∗ φ ∗ j Let s : π ( S ) × π ( S ) → π (U am / µ m ) × π (U bn / µ n ) be a section of the mapinduced on homotopy by det × det, and let ψ : π ( S ) → π ( S ) × π ( S ) be asection of φ ∗ . Define t : π ( S ) → π (SU abmn / µ mn ) by t : = ⊗ ∗ s ψ . The homo-morphism t is a section of π (SU abmn / µ mn ) → π ( S ) and makes the squarebelow commute π ( S ) × π ( S ) π (U am / µ m ) × π (U bn / µ n ) π ( S ) π (SU abmn / µ mn ). s φ ∗ N ∗ t Therefore, the short exact sequences above split and π (U am / µ m ) × π (U bn / µ n ) Ker i ⊕ Im s π (SU abmn / µ mn ) Ker j ⊕ Im t . N ∗ ∼= m + φ ∗ ∼= This is N ∗ ( x + α , y + β ) = φ ∗ ( x , y ) + m( α , β ) for x + α ∈ Z ⊕ Z / m , y + β ∈ Z ⊕ Z / n .By Remark 2.3 and Corollary 2.11, N ∗ ( x + α , y + β ) = bnx + am y + αβ . (cid:3) Remark . Observe that we can also define operations on SU m × SU n thatsatisfy analogous properties to those proved for the operations we defined onU m × U n . ARCILA-MAYA
3. P
ROOF OF T HEOREM n over X and principal PU n -bundles.Therefore, a topological Azumaya algebra on X of degree n will be consideredas a homotopy class in [ X , B PU n ].Consider the space B SU am / µ m and the map B SU am / µ m → K( Z / m , 2) whichis the projection of B SU am / µ m on the the first non-trivial stage of its Post-nikov tower.Given a map A : X −→ B SU am / µ m , we define the Brauer class of A asfollows. Let χ m denote the composite of A and B SU am / µ m → K( Z / m , 2). The Brauer class of A is cl( A ) = ˜ β m ( χ m ), where ˜ β m : K( Z / m , 2) → K( Z , 3) is thereduced Bockstein map. As illustrated in diagram (5).(5) B SU am / µ m X K( Z / m , 2) K( Z , 3). A χ m ˜ β m A left homotopy inverse.
Let a , b , m and n be positive integers. Byapplying the classifying-space functor to the homomorphism (3) we get a map F ⊗ : B U am / µ m × B U bn / µ n → B U abmn / µ mn . Similarly, we obtain the map F ⊗ : B SU am / µ m × B SU bn / µ n → B SU abmn / µ mn .(6)If we take the quotient by µ am and µ bn in (6), we write f ⊗ instead of F ⊗ . Proposition 3.1.
Let a, b, m and n be positive integers such that ma and nbare relatively prime and am < bn. There exist N > and a homomorphism T : U am × U bn −→ U N such that (1) The homomorphism T descends to e T : U am / µ m × U bn / µ n −→ U N . (2) The map ¡ F ⊗ , B e T ¢ : B U am / µ m × B U bn / µ n −→ B U abmn / µ mn × B U N is (2 am + -connected.Proof. Existence of T.Since am and bn are relatively prime, so are m ( am ) m and n ( bn ) n . Hencethere exist positive integers u and v such that n ( bn ) n v − m ( am ) m u = ± N denote u ( am ) m + v ( bn ) n . We define T using the operations describedin Section 2, as the compositeT : U am × U bn U ( am ) m × U ( bn ) n U u ( am ) m × U v ( bn ) n U N . ( ⊗ m , ⊗ n ) ( ⊕ u , ⊕ v ) ⊕ The homomorphism T descends to e T : U am / µ m × U bn / µ n −→ U N . We must show that µ m × µ n is contained in Ker(T). Let α and β be m -thand n -th roots of unity, respectively. Note that the element at the leftmostside of diagram (7) is sent to the identity matrix in U N .(7) ¡ α I am , β I bn ¢ ¡ α m I ( am ) m , β n I ( bn ) n ¢ ¡ I u ( am ) m , I v ( bn ) n ¢ I N . The map ¡ F ⊗ , B e T ¢ is (2 am + -connected. ECOMPOSITION OF TOPOLOGICAL AZUMAYA ALGEBRAS 9
We want to prove that the induced homomorphism on homotopy groups(8) ¡ F ⊗ , B e T ¢ i : π i (B U am / µ m ) × π i (B U bn / µ n ) −→ π i (B U abmn / µ mn ) × π i (B U N )is an isomorphism for ll i < am + i = am + am +
2, it suffices to prove that ¡ F ⊗ , B e T ¢ i is an isomorphismfor all i even and i < am + Case 1:
Let i < am + i
2. The homomorphism (8) takes the form ¡ F ⊗ , B e T ¢ i : Z × Z −→ Z × Z .Proposition 2.14 and Corollary 2.11 show F ⊗ ( x , y ) = N ∗ ( x , y ) = bnx + am y .Proposition 2.14 and Corollaries 2.6, 2.8 and 2.13 imply B e T i : π i (B U am / µ m ) × π i (B U bn / µ n ) −→ π i (B U N ) is given by B e T i ( x , y ) = m ( am ) m − ux + n ( bn ) n − v y .Thereby, the homomorphism (8) is represented by the matrix µ bn amm ( am ) m − u n ( bn ) n − v ¶ ,which is invertible. This proves ¡ F ⊗ , B e T ¢ i is bijective. Case 2:
When i =
2, the homomorphism (8) takes the form ¡ F ⊗ , B e T ¢ : ( Z ⊕ Z / m ) × ( Z ⊕ Z / n ) −→ ( Z ⊕ Z / mn ) × Z .The homomorphism B e T : π (B U am / µ m ) × π (B U bn / µ n ) −→ π (B U N ) is suchthat B e T ( x + α , y + β ) = B e T i ( x , y ) for all x + α ∈ Z ⊕ Z / m and y + β ∈ Z ⊕ Z / n . To seethis we can use that µ m × µ n ⊂ Ker(T) and an argument similar to the one usedin Proposition 2.14. Thus ¡ F ⊗ , B e T ¢ ( x + α , y + β ) = ¡ F ⊗ ( x , y ) + αβ , B e T i ( x , y ) ¢ .Consequently, ( F ⊗ , B e T) is bijective. (cid:3) Proposition 3.2.
Let a, b, m and n be positive integers such that ma and nbare relatively prime and am < bn. There exist N > and a homomorphism T : SU am × SU bn −→ SU N such that (1) The homomorphism T descends to e T : SU am / µ m × SU bn / µ n −→ SU N . (2) The map ( F ⊗ , B e T) : B SU am / µ m × B SU bn / µ n −→ B SU abmn / µ mn × B SU N is (2 am + -connected.Proof. Using Remark 2.15 and proceding as we did in Proposition 3.1 theresult follows. (cid:3)
Factorization through F ⊗ : B SU am / µ m × B SU bn / µ b → B SU abmn / µ mn .Theorem 3.3. Let a, b, m and n be positive integers such that ma and nbare relatively prime and am < bn. Let X be a CW complex such that dim( X ) ≤ am + .Every map A : X → B SU abmn / µ mn can be lifted to B SU am / µ m × B SU bn / µ n along the map F ⊗ .Proof. Diagramatically speaking, we want to find a map A m × A n : X → B SU am / µ m × B SU bn / µ n such that diagram (9) commutes up to homotopy(9) B SU am / µ m × B SU bn / µ n X B SU abmn / µ mn . F ⊗ AA m × A n Proposition 3.2 yields a map J : B SU am / µ m × B SU bn / µ n −→ B SU N where N is some positive integer. Observe that F ⊗ factors through B SU abmn / µ mn × B SU N , so we can write F ⊗ as the composite of ( F ⊗ , J ) and the projection proj shown in diagram (10).(10) B SU am / µ m × B SU bn / µ n B SU abmn / µ mn × B SU N B SU abmn / µ mn ( F ⊗ , J ) F ⊗ proj Since ( F ⊗ , J ) is (2 am + X ) < am +
1, then by White-head’s theorem( F ⊗ , J ) : [ X , BSU am / µ m × B SU bn / µ n ] → [ X , BSU abmn / µ mn × B SU N ]is a surjection, [Spa81, Corollary 7.6.23].Let s denote a section of proj . The bijectivity of ( F ⊗ , J ) implies s ◦ A has apreimage A m × A n : X → B U am / µ m × B U bn / µ n such that ( F ⊗ , J ) ◦ ( A m × A n ) ≃ s ◦ A .Commutativity of diagram (9) follows from commutativity of diagram (10).Thus, the result follows. (cid:3) Factorization through f ⊗ : B PU am × B PU bn → B PU abmn .Theorem 3.4.
Let a, b, m and n be positive integers such that ma and nbare relatively prime and am < bn. Let X be a CW complex such that dim( X ) ≤ am + .If A is a topological Azumaya algebra of degree abmn such that cl( A ) has period mn, then there exist topological Azumaya algebras A m and A n ofdegrees am and bn, respectively, such that per(cl( A m )) = m, per(cl( A n )) = nand A ∼= A m ⊗ A n .Proof. In this case we want to solve the lifting problem shown in diagram (11)up to homotopy, with per(cl( A m )) = m , per(cl( A n )) = n .(11) B PU am × B PU bn X B PU abmn . f ⊗ AA m × A n By [Gu19, Proposition 4.3] there exists a map A ′ : X → B SU abmn / µ mn suchthat per(cl( A ′ )) = per(cl( A )) = mn . Then, by Theorem 3.3 there exists a map A ′ m × A ′ n : X −→ B SU am / µ m × B SU bn / µ n . Claim . per(cl( A ′ m )) = m and per(cl( A ′ n )) = n .By Claim 3.5 and [Gu19, Proposition 4.3] there exists a map A m × A n : X −→ BPU am × B PU bn such that per(cl( A m )) = m and per(cl( A n )) = n . ECOMPOSITION OF TOPOLOGICAL AZUMAYA ALGEBRAS 11
It remains to show that diagram (11) commutes. Consider the diagrambelow(12) B SU am / µ m × B SU bn / µ n B PU am × B PU bn X B SU abmn / µ mn B PU abmn A ′ m × A ′ n B ρ × B ρ f ⊗ A m × A n A F ⊗ A ′ Observe that the square, as well as top, bottom and left triangles, of diagram(12) commute. Hence, the right triangle commutes. (cid:3)
Theorem 1.5 is a corollary of Theorem 3.4.
Proof of Claim . Let red m : Z / mn → Z / m be the reduction homomorphism,which induces a map red ∗ m : K( Z / mn , 2) → K( Z / m , 2). Observe that χ ′ m = red ∗ m χ ′ mn .Moreover, there is a commutative diagramK( Z / mn , 2) K( Z / m , 2)K( Z , 3) K( Z , 3). red ∗ m ˜ β mn ˜ β m × n Thus, cl( A ′ m ) = n cl( A ′ ). Given that cl( A ′ ) has period mn , we have per(cl( A ′ m )) = m . A similar argument shows per(cl( A ′ n )) = n . (cid:3) Theorem 3.6.
The map F ⊗ : B SU am / µ m × B SU bn / µ n → B SU abmn / µ mn doesnot have any section.Proof. Suppose there exists a section σ of F ⊗ . Then the composite(13) π am + (B SU abmn / µ mn ) π am + (B SU am / µ m ) × π am + (B SU bn / µ n ) π am + (B SU abmn / µ mn ) σ ∗ ( F ⊗ ) ∗ is the identity.Consider the homomorphism(14) ⊗ ∗ : π am + (SU am / µ m ) × π am + (SU bn / µ n ) −→ π am + (SU abmn / µ mn ).From [Mim95], π am + (SU am ) is a torsion group. Then, by Proposition 2.10the homomorphism (14) is given by ( x , y ) am y . Therefore, the composite(13) is not surjective, which contradicts the fact that it is the identity. (cid:3) In Theorem 1.5 it is proven that there exists a tensor product decomposi-tion for topological Azumaya algebras over low dimensional CW complexes.The result of Theorem 3.6 means than such decomposition does not exist fortopological Azumaya algebras over an arbitrary CW complex. Moreover, forpositive integers m , n where m < n , if A is a topological Azumaya algebras of degree mn over a finite CW complex of dimension higher than 2 m +
1, then A has no decomposition A ∼= A m ⊗ A n . Remark . The topological Azumaya algebras A m and A n in Theorem 1.5are not unique up to isomorphism. In order to see this, we consider the rela-tive Postnikov tower of the map f ⊗ :(15) B PU m × B PU n ... F Y [5] F Y [4] K( π F , 6)B PU mn K( π F , 5) k k where k i − : Y [ i − → K ¡ π i F i , i + ¢ is the k -invariant that classifies the fibersequence F i → Y [ i ] → Y [ i − X be a CW complex of dim( X ) ≤
6. Let m and n be as in the hypothesisof Theorem 1.5, and m >
3. Let A be a topological Azumaya algebra of degree mn .We can use the properties of a relative Postnikov tower of map to simplifythe tower in diagram (15). For instance, there are homotopy equivalences Y [4] ≃ Y [5] and Y [6] ≃ Y [7], and the homotopy groups of the homotopy fibers F and F are both isomorphic to the integers.(16) B PU m × B PU n ... Y [4] ≃ B PU mn × K( Z , 4) K( Z , 7) X B PU mn K( Z , 5) k A ( A , ξ ) A m × A n k Observe that the k -invariant k is nullhomotopic because H (B PU mn ; Z )is trivial. Hence there is no obstruction to lift A to Y [4]. Similarly, we canlift the identity map id BPU mn to Y [4], in this case we obtain the splitting Y [4] ≃ B PU mn × K( Z , 4). Then the lifting of A takes the form ( A , ξ ) : X → B PU mn × K( Z , 4).The cohomology groups of X vanish for all degrees greater than 6, giventhat X is 6-dimensional. Thus ( A , ξ ) can be lifted up the Postnikov tower toB PU m × B PU n . See diagram (16).This proves that A can be decomposed as A m ⊗ A n . The lifting ( A , ξ ) is notnecessarily unique. In fact, every cohomology class ξ ∈ H ( X ; Z ) gives rise toa lifting ( A , ξ ). ECOMPOSITION OF TOPOLOGICAL AZUMAYA ALGEBRAS 13 R EFERENCES [AG60] Maurice Auslander and Oscar Goldman,
The Brauer group of a commutative ring ,Trans. Amer. Math. Soc (1960), no. 3, 367–409. ↑ The period-index problem for twisted topolog-ical K–theory , Geom. Topol. (2014), no. 2, 1115–1148. ↑ Unramified division algebras do not always contain Azumaya maximal or-ders , Invent. Math. (2014), no. 1, 47–56. ↑ On maximally central algebras , Nagoya Math. J. (1951), 119–150. ↑ The space of loops on a Lie group , Michigan Math. J. (1958), no. 1,35–61. ↑ Le groupe de Brauer: I. Algèbres d’Azumaya et interpréta-tions diverses , Séminaire bourbaki: années 1964/65 1965/66, exposés 277-312, 1966,pp. 199–219. talk:290. MR1608798 ↑ The topological period–index problem over 8-complexes , J. Topol. (2019),no. 4, 1368–1395. ↑ Chapter 19 - Homotopy theory of Lie groups , Handbook of Alge-braic Topology, 1995, pp. 951–991. ↑ Lectures on division algebras , CBMS Reg. Conf. Ser. Math., vol. 94,AMS, Providence, RI; on behalf of CBMS, Washington, DC, 1999. ↑ Algebraic Topology , Illustrate, Springer New York, New York,NY, 1981. ↑ EPARTMENT OF M ATHEMATICS , U
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