aa r X i v : . [ m a t h . A T ] J un THE TOPOLOGICAL PERIOD-INDEX PROBLEM OVER -COMPLEXES, II XING GU
Abstract.
We complete the study of the topological period-index problemover 8 dimensional finite CW complexes started in a preceding paper. Moreprecisely, we determine the sharp upper bound of the index of a topologicalBrauer class α ∈ H ( X ; Z ), where X is of the homotopy type of an 8 dimen-sional finite CW complex and the period of α is divisible by 4. introduction This paper is a sequel to Gu [9], where we investigated the topological period-index problem (TPIP) over finite CW complexes of dimension 8, and determinedthe upper bound of indices of topological Brauer classes with period n not divisibleby 4. In this paper we give a complete answer to the TPIP over finite 8-complexesby studying the case 4 | n .For a path-connected topological space X , let Br( X ) be the topological Brauergroup defined in [2], whose underlying set is the Azumaya algebras (i.e. bundles ofcomplex matrix algebras over X ) modulo the Brauer equivalence: A and A arecalled Brauer equivalent if there are vector bundles E and E such that A ⊗ End( E ) ∼ = A ⊗ End( E ) . The multiplication is given by tensor product.Azumaya algebras over X of degree r are classified up to isomorphism by thecollection of isomorphism classes of P U r -principal bundles over X , i.e., the coho-mology set H ( X ; P U r ), where P U r is the projective unitary group of degree r .Consider the short exact sequence of Lie groups(1.1) 1 → S → U r → P U r → S → U r is the inclusion of scalars. Then the Bockstein homo-morphism(1.2) H ( X ; P U r ) → H ( X ; S ) ∼ = H ( X ; Z )associates an Azumaya algebra A to a class α ∈ H ( X ; Z ). The exactness of thesequence (1.1) implies that(1) α ∈ H ( X ; Z ) tor , the subgroup of torsion elements of H ( X ; Z ), and(2) the class α only depends on the Brauer equivalence class of A .An Azumaya algebra associated to a Brauer class may alternatively be describedas follows. Let A be an Azumaya algebra of degree r over a finite CW complex X . As discussed above, one can associate a P U r -bundle, i.e., a homotopy class of Mathematics Subject Classification.
Primary 55S45; Secondary 55R20.
Key words and phrases.
Brauer groups, twisted K -theory, period-index problems. maps X → B P U r , where B P U r is the classifying space of P U r . An elementarycomputation yields H ( B P U r ; Z ) ∼ = Z /r . The Azumaya algebra A is associated toa class α ∈ H ( X ; Z ) if and only if the following homotopy commutative diagram(1.3) B P U r X K ( Z , α exists such that the dashed arrow represents the (isomorphism class) P U r -bundleclassifying A , the vertical arrow corresponds to a generator of H ( B P U r ) ∼ = Z /r ,and the bottom arrow corresponds to the cohomology class α .Therefore, we established a homomorphism Br( X ) → H ( X ; Z ) tor , and we call H ( X ; Z ) tor the cohomological Brauer group of X , and sometimes denote it byBr ′ ( X ). Serre [8] showed that when X is a finite CW complex, this homomorphismis an isomorphism. Hence, for any α ∈ H ( X ; Z ) tor , there is some r such that a P U r -torsor over X is associated to α via the homomorphism (1.2).Let per( α ) denote the order of α as an element of the group H ( X ; Z ). Serre [8]also showed per( α ) | r for all r such that there is a P U r -torsor over X associated to α in the way described above. Let ind( α ) denote the greatest common divisor ofall such r , then in particular we have(1.4) per( α ) | ind( α ) . The preceding definitions are motivated by their algebraic analogs. We refer tothe introduction of [1] for the algebraic version of the definitions as well as morealgebraic backgrounds. The period-index conjecture, stated as follows, plays a keyrole in the study of Brauer groups:
Conjecture 1.1 ([6], Colliot-Th´el`ene) . Let k be either a C d -field or the functionfield of a d -dimensional variety over an algebraically closed field. Let α ∈ Br( k ),and suppose that per( α ) is prime to the characteristic of k . Thenind( α ) | per( α ) d − . This conjecture has the following topological analog first considered by Antieauand Williams in [1]:
For a given Bruaer class α of a finite CW complexes X , find the sharp lowerbound of e such that ind( α ) | per( α ) e holds for all finite CW complex X in C and all elements α ∈ Br ( X ) . Notice that such an e exists if and only if per α and ind( α ) have the same primedivisors, which follows from Corollary 3.2, [2].For further explanations of the preceding definitions and backgrounds on thetopological period-index problem, see [1], [2], [7] and [9]. All definitions and nota-tions in this paper are consistent with those in [9]. More precisely, the expression ǫ p ( n ) denotes the greatest common divisor of a prime p and a positive integer n ,and we let β n be the canonical generator of H ( K ( Z /n, Z ), i.e., the image of theidentity class in H ( K ( Z /n, Z /n ) under the Bockstein homomorphism.In [9] we have shown the following HE TOPOLOGICAL PERIOD-INDEX PROBLEM OVER 8-COMPLEXES, II 3
Theorem 1.2 (Theorem 1.6, [9]) . Let X be a topological space of homotopy typeof an -dimensional connected finite CW-complex, and let α ∈ H ( X ; Z ) tor be atopological Brauer class of period n . Then (1.5) ind( α ) | ǫ ( n ) ǫ ( n ) n . In addition, if X is the -th skeleton of K ( Z /n, , and α is the restriction of thefundamental class β n ∈ H ( K ( Z /n, , Z ) , then ( ind( α ) = ǫ ( n ) ǫ ( n ) n , ∤ n , ǫ ( n ) n | ind( α ) , | n .In particular, the sharp lower bound of e such that ind( α ) | n e for all X and α is . The goal of this paper is to improve Theorem 1.2 and show the following
Theorem 1.3.
Let X be a topological space of homotopy type of an -dimensionalconnected finite CW-complex, and let α ∈ H ( X ; Z ) tor be a topological Brauer classof period n . Then (1.6) ( ind( α ) | ǫ ( n ) n , if n ≡ , ind( α ) | ǫ ( n ) n , otherwise . In addition, if X is the -th skeleton of K ( Z /n, and α is the restriction of thefundamental class β n ∈ H ( K ( Z /n, , Z ) , then ( ind( α ) = 2 ǫ ( n ) n , if n ≡ , ind( α ) = ǫ ( n ) n , otherwise . In particular, the sharp lower bound of e such that ind( α ) | n e for all X and α is . One readily sees that the only thing remains to show is the following
Proposition 1.4.
Let X be the 8-th skeleton of K ( Z /n,
2) and α ∈ H ( X ; Z ) therestriction of β n . When 4 | n , we haveind( α ) = ǫ ( n ) n . Theorem 1.3 may be interpreted in terms of the twisted complex K -theory andthe twisted Atiyah-Hirzebruch spectral sequences (AHSS). For a CW complex X and a class α ∈ H ( X ; Z ), the twisted complex K -theory associates the pair ( X, α )to a graded ring K ∗ ( X ) α in a contravariant manner. For α = 0, it is the usual com-plex K -theory K ∗ ( X ). The readers may refer to [4] and [5] for more backgrounds.As for the usual complex K -theory, there is a twisted AHSS of the pair ( X, α )converging to the twisted K-theory K ∗ ( X ) α . We denote it by ( e E ∗ , ∗∗ , e d ∗ , ∗∗ ). Thespectral sequence satisfies e E s,t ∼ = H s ( X ; K t (pt))where K ∗ denotes the complex topological K -theory in the usual sense. By theBott Periodicity Theorem, we have(1.7) e E s,t ∼ = ( H s ( X ; Z ) , t even , , t odd . The twisted AHSS is considered in [1], [2], [3] and [9]. The following result can bederived immediately from [2].
XING GU − − − Z Z /n Z /ǫ ( n ) n Z /ǫ ( n ) n Figure 1.
The twisted Atiyah-Hirzebruch spectral sequence asso-ciated to K ( Z /n,
2) and β n . Theorem 1.5 (Theorem 3.1, [9]) . Let X be a connected finite CW-complex and let α ∈ Br ( X ) . Consider e E ∗ , ∗∗ , the twisted Atiyah-Hirzebruch spectral sequence of thepair ( X, α ) with differentials e d s,tr with bi-degree ( r, − r + 1) . In particular, e E , ∼ = Z ,and any e E , r with r > is a subgroup of Z and therefore generated by a positiveinteger. The subgroup e E , (resp. e E , ∞ ) is generated by per( α ) (resp. ind( α )) . Following Theorem 1.5, it can be easily deduced from Theorem B of [1] andTheorem 1.2 that when X = K ( Z /n,
2) and α = β n , the differentials e d , r aresurjective onto entries on the E -page for r <
7, and when 4 ∤ n , also for r = 7.Theorem 1.3, however, provides the first known example of a e d , r that is NOTsurjective onto some e E s,t . Proposition 1.6.
Suppose 4 | n . In the twisted Atiyah-Hirzebruch spectral se-quence ( e E ∗ , ∗∗ , e d ∗ , ∗∗ ) of the pair ( K ( Z /n, , β n ), the image of e d , is 2 e E , − . Proof assuming Theorem 1.3.
This is easily deduced from the preceeding paragraphand Theorem 1.3, once we recall the cohomology of K ( Z /n, (cid:3) The technical core of this paper is to solve a homotopy lifting problem of thesame nature as the one displayed in (1.3), with some modifications for technical con-venience. We recall some more notations from [9]. Let m, n be integers. Then Z /n is a closed normal subgroup of SU mn in the sense of the following monomorphismof Lie groups: Z /n ֒ → SU mn : t e π √− t/n I mn , where I r is the identity matrix of degree r . We denote the quotient group by P ( n, mn ), and consider its classifying space B P ( n, mn ). It follows immediately HE TOPOLOGICAL PERIOD-INDEX PROBLEM OVER 8-COMPLEXES, II 5 from Bott’s periodicity theorem that we have(1.8) π i ( B P ( n, mn )) ∼ = Z /n, if i = 2 , Z , if 2 < i < mn + 1, and i is even,0 , if 0 < i < mn , and i is odd.In the context of this paper, the space B P ( n, mn ) may be considered as a refinementof B P U mn . For a finite CW-complex X and a topological Brauer class α of period n , there is an element α ′ in H ( X ; Z /n ), which is sent to α by the Bocksteinhomomorphism. Consider the following lifting diagram(1.9) B P ( n, mn ) X K ( Z /n, α ′ where the vertical arrow is the projection from B P ( n, mn ) to the bottom stage ofits Postnikov tower. We have the following Proposition 1.7. [Proposition 4.3, [9]] Let X , α be as above, and suppose that H ( X ; Z ) = 0. Then α is classified by an Azumaya algebra of degree mn if andonly if the lifting in diagram (1.9) exists.In Section 2 we recall some facts on Eilenberg-Mac Lane spaces. Section 3 is acollection of lemmas on the cohomology of B P ( n, mn ). In Section 4 we consider ak-invariant in the Postnikov decomposition of B P ( n, mn ), for 4 | n and suitable m ,leading to the proof of Proposition 1.4 and Theorem 1.3.2. recollection of facts on eilenberg - mac lane spaces All the facts recalled here are either well-known or easily deduced from [11].The integral cohomology of K ( Z /n,
2) in degree ≤ Z [ β n , Q n , R n , ρ n ] / ( nβ n , ǫ ( n ) β n , ǫ ( n ) nQ n , ǫ ( n ) nR n , ǫ ( n ) ρ n ) , where deg( β n ) = 3 , deg( Q n ) = 5 , deg( R n ) = 7, and deg( ρ n ) = 8. In other words,there is exactly one generator in each of the degrees 3 , , ,
7, which are, respectively, β n , Q n , β n , R n , of order n, ǫ ( n ) n, ǫ ( n ) , ǫ ( n ) n, and 2 generators in degree 8, β n Q n and ρ n , of order ǫ ( n ) and ǫ ( n ), respectively.(See (2.5) of [9].)For n ≥
3, the ring H ∗ ( K ( Z , n ); Z ) in degree ≤ n + 3 is isomorphic to thefollowing graded rings:(2.2) Z [ ι n , Γ n ] / (2Γ n ) , n > , even , Z [ ι n , Γ n ] / (2 ι n , n ) , n > , odd , Z [ ι , Γ ] / (2Γ , Γ − ι ) , n = 3 , where ι n , of degree n , is the so-called fundamental class, and Γ n , of degree n + 3,is a class of order 2.(See (2.1) of [9].) XING GU
Proposition 2.1.
The classes Γ n ∈ H n +3 ( K ( Z , n ); Z ) as above for all n ≥ Z ∈ H ( K ( Z ); Z ) of order 2,where K ( R ) denotes the Eilenberg - Mac Lane spectrum associated to a unit ring R . Moreover, the mod 2 reduction of Sq Z is the well-understood Steenrod squareSq . In other words, the following diagram in the homotopy category of spectracommutes:(2.3) K ( Z ) Σ K ( Z ) K ( Z /
2) Σ K ( Z / Sq Z Sq where the vertical arrows are the obvious ones.For a proof see Lemma 2.1 of [9].3. the group H ( B P ( n, mn ); Z )As in the introduction we denote by P ( n, mn ) the quotient group of the followinginclusion of Lie groups: Z /n ֒ → SU mn : t e π √− t/n I mn , where I r is the identity matrix of degree r , as well as its classifying space B P ( n, mn ).In this section we study the cohomology group H ( B P ( n, mn ); Z ).Let us remind ourselves of the following notation: for a simply connected space X , let X [ k ] denote the k th level of the Postnikov tower of X . Consider the Postnikovtower of B P ( n, mn ) for ǫ ( n ) n | m , n > B P ( n, mn )[6] K ( Z , B P ( n, mn )[5] ≃ K ( Z /n, × K ( Z , K ( Z , K ( Z /n, K ( Z , = κ κ =0 where κ and κ are the k-invariant of the space B P ( n, m ). The equation κ =0 follows from Proposition 4.11 of [9]. Consequently, B P ( n, mn )[5] is a trivialfibration over K ( Z /n,
2) with fiber K ( Z , B P ( n, mn ) → B P ( n, mn )[6] ≃ K ( Z /n, × K ( Z , , where ǫ ( n ) n | m . For future reference we take notes of the induced homomorphismbetween integral cohomology rings as follows:(3.2) β n × x ′ , R n × R n ( x ′ ) , × ι e ′ , × Γ Sq Z ( e ′ ) , where x ′ and e ′ are the additive generators of H ( B P ( n, mn ); Z ) ∼ = Z /n and H ( B P ( n, mn ); Z ) ∼ = Z , respectively. Here R n is the generator of H ( K ( Z /n, Z )as in (2.1), regarded as a cohomology operation in the obvious way.By the construction of Postnikov towers we have HE TOPOLOGICAL PERIOD-INDEX PROBLEM OVER 8-COMPLEXES, II 7
Lemma 3.1.
Suppose ǫ ( n ) n | m . Then we have H ( B P ( n, mn ); Z ) ∼ = H ( K ( Z /n, × K ( Z , / ( κ ) . In particular, the group H ( B P ( n, mn ); Z ) is generated by R n ( x ′ ) , x ′ e ′ , and Sq Z ( e ′ ) . Consider the short exact sequence of Lie groups1 → Z /n → SU mn → P ( n, mn ) → , from which arises a fiber sequence B SU mn → B P ( n, mn ) → K ( Z /n, S E ∗ , ∗∗ , S d ∗ , ∗∗ ).For k >
2, it is well-known that H ∗ ( B SU k ; Z ) = Z [ c , · · · , c k ] , where c i is the i th universal Chern class of degree 2 i . Lemma 3.2.
Suppose ǫ ( n ) n | m . The differential S d , = 0 . In particular, c ∈ S E , is a permanent cocycle.Proof. Diagram (3.1) implies H ( B P ( n, mn ); Z ) ∼ = H ( K ( Z /n, Z ) ∼ = Z /ǫ ( n ) n. Hence we have S d , = 0. There is no other non-trivial differential out of S E , ∗ forobvious degree reasons, so c is a permanent cocycle. (cid:3) Lemma 3.3 (Lemma 6.1, [9]) . Suppose ǫ ( n ) n | m . Recall that H ( K ( Z /n, Z )) ∼ = Z /n is generated by an element β n , and that H ( K ( Z /n, Z )) ∼ = Z /ǫ ( n ) n isgenerated by R n . In the spectral sequence S E ∗ , ∗∗ , we have S d , ( c ) = 2 c β n withkernel generated by nǫ ( n ) c , and S d , ( nǫ ( n ) c ) = ǫ ( n ) ǫ ( m/n ) ǫ ( mn ) nR n . All the other differentials out of S E , ∗ are trivial. See Figure 2 for the differentials mentioned in the lemmas above.
Lemma 3.4.
Suppose ǫ ( n ) n | m . (1) The element Sq Z ( e ′ ) is a linear combination of x ′ e ′ and R n ( x ′ ) . (2) The element R n ( x ′ ) is of order ǫ ( n ) ǫ ( m/n ) nǫ ( mn ) . (3) The order of the group H ( B P ( n, mn ); Z ) is ǫ ( n ) ǫ ( n ) ǫ ( m/n ) nǫ ( mn ) . XING GU
ZZ Z /n Z /ǫ ( n ) n Z /ǫ ( n ) n × Figure 2.
Low dimensional differentials of the spectral sequence S E ∗ , ∗∗ , when ǫ ( n ) n | m , n >
1. The dashed arrows represent trivialdifferentials.
Proof.
As indicated in Figure 2, we have the exact sequence0 → S E , ∞ → H ( B P ( n, mn ); Z ) → S E , ∞ → , where S E , ∞ and S E , ∞ are generated by R n ( x ′ ) and x ′ e ′ , respectively. Hence (1)follows. The statement (2) is an immediate a consequence of Lemma 3.3. To verifythe statement (3), it suffices to check the order of S E , ∞ and S E , ∞ , which alsofollows from Lemma 3.3. (cid:3) the k-invariant κ Consider the space B P ( n, mn )[6], the 6th level of the Postnikov tower of B P ( n, mn ).We assume throughout this section that ǫ ( n ) n | m , n >
1, for when this holds, wehave the homotopy equivalence (Proposition 4.11, [9])(4.1) B P ( n, mn )[6] = B P ( n, mn )[5] ≃ K ( Z /n, × K ( Z , . The goal of this section is to prove Proposition 1.4, therefore Theorem 1.3, bydetermining the k-invariant κ ∈ H ( B P ( n, mn )[5]; Z ) ≃ H ( K ( Z /n, × K ( Z , Z )for ǫ ( n ) n | m and 4 | n .The equations (2.1) and (2.2) together with the K¨unneth formula give us H ( K ( Z /n, × K ( Z , Z )=( R n × ⊕ ( β n × ι ) ⊕ (1 × Γ ) ∼ = Z /ǫ ( n ) n ⊕ Z /n ⊕ Z / , (4.2)where R n × β n × ι and 1 × Γ generate the three summands, respectively. HE TOPOLOGICAL PERIOD-INDEX PROBLEM OVER 8-COMPLEXES, II 9
When n is even, it follows from Theorem 6.8 of [9] that up to an invertiblecoefficient, we have(4.3) κ ≡ ǫ ( n ) mǫ ( m ) ǫ ( m ) n λR n × λ β n × ι + 1 × Γ mod 2 − torsions , an element in H ( K ( Z /n, × K ( Z , Z ), where λ is invertible in Z /ǫ ( n ) n .To narrow down the choices of κ , we have the following Lemma 4.1.
Suppose ǫ ( n ) n | m , n > . In H ( K ( Z /n, × K ( Z , Z ) we have κ ≡ × Γ mod ( R n × , β n × ι ) . Proof.
Assume that the lemma is false. Since 1 × Γ is of order 2, we have κ ∈ ( R n × , β n × ι ) . Therefore, we have H ( B P ( n, mn ); Z ) ∼ = H ( K ( Z /n, × K ( Z , Z ) / ( κ ) ∼ =( R n × ⊕ ( β n × ι ) ⊕ (1 × Γ ) / ( κ ) ∼ =[( R n × ⊕ ( β n × ι )] / ( κ ) ⊕ (1 × Γ ) , but this violates (1) of Lemma 3.4, which asserts that H ( B P ( n, mn ); Z ) is gener-ated by R n ( x ′ ) and Sq Z ( e ′ ). (cid:3) To prove Proposition 1.4, we only consider the case 4 | n . More precisely, it sufficesto consider the case that X is the 8th skeleton of K ( Z /n,
2) for n divisible by 4,and determine ind( α ) for α the restriction of β n ∈ H ( K ( Z /n, Z ).As a consequence of Theorem 1.2, the above ind( α ) is either 2 ǫ ( n ) n or ǫ ( n ) n .We write α = P p α p where p runs over all prime divisors of per( α ) and the periodof α p is a power of p . Then Theorem 3 of [3] asserts ind( α ) = Q p ind( α p ), witheach ind( α p ) a power of p . Hence it suffices to determine ind( α ).In other words, we have reduced the problem to the case n = 2 r , m = 2 r or2 r +1 , with r >
1. We assume this for the rest of this section:
Convention 4.2. n = 2 r , m = 2 r or 2 r +1 , r > . Lemma 4.3. In H ( K ( Z /n, × K ( Z , Z ) , We have κ ≡ λ β n × ι mod ( R n × , × Γ ) , where the coefficient λ is invertible in Z /n .Proof. We argue by contradiction. Suppose that λ is not invertible in Z /n . Noticethat, for our choice of m and n , equation (4.3) implies κ ≡ r − λR n × λ β n × ι + 1 × Γ mod 2 − torsions . Since λ is not invertible, κ has order less than 2 r . On the other hand, it followsfrom (4.2) that the group H ( K ( Z /n, × K ( Z , Z ) has order 2 r +1 . Thereforethe group H ( B P ( n, mn ); Z ) ∼ = H ( K ( Z /n, × K ( Z , Z ) / ( κ )has order greater than 2 r +1 , contradicting Lemma 3.4. (cid:3) Notice that κ is determined by the Postnikov tower merely up to multiplicationby an invertible coefficient. By the choice we have made of n, m , this means thatwe are free to multiply κ by any odd integer. Hence, we are enabled by Lemma4.3 to normalize (4.3) by fixing λ = 1: κ = 2 r − λR n × β n × ι + 1 × Γ mod a 2-torsion in ( R n × , × Γ ) , where λ is odd. However, since R n is of order 2 r , and 2 r − ≡ r − λ mod 2 r for allodd integer λ , the preceding equation becomes(4.4) κ = 2 r − R n × β n × ι + 1 × Γ mod a 2-torsion in ( R n × , × Γ ) . Combining (4.4) with Lemma 4.1, we have(4.5) κ ≡ r − R n × β n × ι + 1 × Γ mod a 2-torsion in ( R n × . Moreover, we have the following
Lemma 4.4.
The abelian group H ( B P ( n, mn ); Z ) is additively generated by R n ( x ′ ) , x ′ e ′ and Sq Z ( e ′ ) , modulo the relation (4.6) µR n ( x ′ ) + x ′ e ′ + Sq Z ( e ′ ) = 0 , where µ is either or r − . Moreover, only one of the two possible relations holds.Proof. Only the uniqueness requires a proof. Indeed, if both relations hold, thenwe have 2 r − R n ( x ′ ) = 0 and x ′ e ′ + Sq Z ( e ′ ) = 0 . Hence the abelian group H ( B P ( n, mn ); Z ) is generated by R n ( x ′ ) and Sq Z ( e ′ ),whose orders divide 2 r − and 2, respectively. Therefore the order of the group H ( B P ( n, mn ); Z ) divides 2 r , violating Lemma 3.4. (cid:3) We turn to the hard work of determining µ , for which we will need an auxiliaryspace Y to be defined and studied as follows. Recall the homotopy equivalence(4.1) B P ( n, mn )[6] = B P ( n, mn )[5] ≃ K ( Z /n, × K ( Z , . Consider the following map B P ( n, mn ) → B P ( n, mn )[6] → K ( Z , e ′ of H ( B P ( n, mn ); Z ). We denote by Y its homotopy fiber. In other words, we have a fiber sequence(4.7) Y → B P ( n, mn ) e ′ −→ K ( Z , e ′ of H ( B P ( n, mn ); Z ). Thespace Y plays a key role in determining the coefficient µ . By construction thesecond arrow in (4.7) induces an isomorphism of homotopy groups π ( B P ( n, mn )) ∼ = π ( K ( Z , . This isomorphism lies in the long exact sequence of homotopy groups of the fibersequence (4.7), from which, together with (1.8), we deduce(4.8) π i ( Y ) ∼ = Z /n, if n = 2 , Z , if 6 ≤ i < mn + 1, and i is even , , if 0 < i < mn , and i is odd, or i = 4 . HE TOPOLOGICAL PERIOD-INDEX PROBLEM OVER 8-COMPLEXES, II 11
By de-looping the last term of (4.7) we obtain another fiber sequence(4.9) K ( Z , h −→ Y → B P ( n, mn ) . Consider the projection from Y to the bottom level of its Postnikov tower g : Y → K ( Z /n, . Lemma 4.5. (1)
The induced homomorphisms g ∗ : H k ( K ( Z /n, Z ) → H k ( Y ; Z ) are isomorphisms for k ≤ . (2) The homomorphism H ( g ; Z ) is injective. Furthermore, we have H ( Y ; Z ) = g ∗ ( H ( K ( Z /n, Z )) ⊕ ( ω ) ∼ = H ( K ( Z /n, Z ) ⊕ Z , where ω generates the summand Z . (3) The induced homomorphism g ∗ : H ( K ( Z /n, Z ) → H ( Y ; Z ) is surjective. (4) H ( Y ; Z /
2) = g ∗ ( H ( K ( Z /n, Z / ω ) , where an integral cohomology class with an overhead bar denotes its reduc-tion in cohomology with coefficients in Z / .Proof. It follows from (4.8) that we have the following partial picture of its Post-nikov tower Y [7] K ( Z , K ( Z , Y [6] Y [2] = K ( Z /n, K ( Z , = The statements (1), (2) and (3) follows from a simple observation on the fibersequence K ( Z , → Y [6] → K ( Z /n, (cid:3) Consider the fiber sequence (4.9). Let ( E ∗ , ∗∗ ( Z ) , d ∗ , ∗∗ ) and ( E ∗ , ∗∗ ( Z / , ¯ d ∗ , ∗∗ ) de-note the associated cohomological Serre spectral sequences with coefficients in Z and Z /
2, respectively.We first consider the case with coefficients in Z /
2. This is easier since µ ≡ h ∗ : H ( Y ; Z / → H ( K ( Z , Z / E ∗ , ∗∗ ( Z / h ∗ : H ( Y ; Z ) → H ( K ( Z , Z ) to determine a particular differential of the spectral sequence E ∗ , ∗∗ ( Z ), which inturn determines the coefficient µ .As in Lemma 4.5, we use overhead bars to denote the mod 2 reductions of integralcohomology classes. Lemma 4.6.
The homomorphism h ∗ : H ( Y ; Z / → H ( K ( Z , Z / is surjective. More precisely, we have (4.10) h ∗ (¯ ω ) = ¯ ι . Proof.
Consider the spectral sequence ( E ∗ , ∗∗ ( Z ) , d ∗ , ∗∗ ). Refer to Figure 3 for therelevant differentials.It follows from (1) of Lemma 4.5 that d , is the first nontrivial differential outof the bidegree (0 ,
3) and it is an isomorphism:(4.11) d , : E , ( Z ) ∼ = Z ∼ = −→ Z ∼ = E , ( Z )sending the generator ι of H ( K ( Z , Z ) to ± e ′ , the generator of H ( B P ( n, mn ); Z ).Passing to ( E ∗ , ∗∗ ( Z / , ¯ d ∗ , ∗∗ ) as shown in Figure 3, it follows from (4.11) that¯ ι ∈ H ( K ( Z , Z / ∼ = E , ( Z / d , (¯ ι ) = ¯ e ′ ∈ H ( B P ( n, mn ); Z / ∼ = E , ( Z / ∼ = E , ( Z / . Therefore, ¯ ι = Sq (¯ ι ) is also transgressive and we have(4.13) ¯ d , (¯ ι ) = ¯ d , (Sq (¯ ι )) = Sq ( ¯ d , (¯ ι )) = Sq (¯ e ′ ) . It follows from (4.12) that we have(4.14) ¯ d , (¯ x ′ ¯ ι ) = ¯ x ′ ¯ e ′ ∈ H ( B P ( n, mn ); Z / ∼ = E , ( Z / ∼ = E , ( Z / . Taking the mod 2 reduction of κ as in (4.5), we have µ ≡ r > x ′ ¯ e ′ + Sq (¯ e ′ ) = 0 ∈ H ( B P ( n, mn ); Z / . This relation, together with (4.14), showsSq (¯ e ′ ) ≡ ∈ E , ( Z / ∼ = H ( B P ( n, mn ); Z / / (¯ x ′ ¯ e ′ ) . Then it follows from (4.13) that ¯ d , (¯ ι ) = 0. Hence, ¯ ι is a permanent cocycle,which proves that ¯ ι is in the image of h ∗ , and in particular, that h ∗ is surjectivein dimension 6 and with coefficients in Z / g factors as Y → B P ( n, mn ) → K ( Z /n, Y → B P ( n, mn ) induces an isomorphism π ( Y ) ∼ = π ( B P ( n, mn )).In particular, it follows that g ◦ h : K ( Z , → Y → K ( Z /n, H ( Y ; Z /
2) = g ∗ ( H ( K ( Z /n, Z / ω ) , so h ∗ takes the first direct summand to 0. It then follows from the surjectivity of h ∗ that we have h ∗ (¯ ω ) = ¯ ι , and we conclude. (cid:3) HE TOPOLOGICAL PERIOD-INDEX PROBLEM OVER 8-COMPLEXES, II 13 , e ′ ¯ ι ¯ ι ¯ x ′ ¯ e ′ ¯ x ′ , Sq ( ¯ e ′ )¯ ι Figure 3.
Differentials of the spectral sequence E ∗ , ∗∗ ( Z / Lemma 4.7.
The homomorphism h ∗ : H ( Y ; Z ) → H ( K ( Z , Z ) is surjective. Furthermore, h ∗ ( ω ) = ι .Proof. Since H ( K ( Z , Z ) ∼ = Z / ι , we have either h ∗ ( ω ) = ι or h ∗ ( ω ) = 0 . Lemma 4.6 shows that the latter is impossible. (cid:3)
We proceed to determine κ . Proposition 4.8.
Under Convention 4.2, we have κ = β n × ι + 1 × Γ . Proof.
Consider the spectral sequence ( E ∗ , ∗∗ ( Z ) , d ∗ , ∗∗ ). (The picture of the differen-tials looks the same as Figure 3, but with all the overhead bars removed, and Sq replaced by Sq Z .) For obvious degree reasons the only possibly nontrivial differ-entials landing in bidegree (7 ,
0) are d , and d , . It follows from (4.11) and theLeibniz rule that we have d , ( x ′ ι ) = x ′ e ′ , and(4.16) E , ( Z ) = E , ( Z ) ∼ = H ( B P ( n, mn ); Z ) / ( x ′ e ′ )Recall (4.11): d , : E , ( Z ) ∼ = Z ∼ = −→ Z ∼ = E , ( Z ) , ι
7→ ± e ′ , which, in particular, asserts that ι is transgressive. Therefore, so is Sq Z ( ι ) = ι . (See Proposition 2.1.) Furthermore, since transgressions commute with stablecohomology operations, we have(4.17) d , ( ι ) = d , (Sq Z ( ι )) = Sq Z ( d , ( ι )) = Sq Z ( e ′ ) , where the last step follows from (4.11). On the other hand, it follows from Lemma4.7 that ι is a permanent cocycle. Therefore, (4.16) and (4.17) implies that Sq Z ( e ′ )is in the subgroup of H ( B P ( n, mn ); Z ) generated by x ′ e ′ , i.e.,(4.18) νx ′ e ′ + Sq Z ( e ′ ) = 0 , for some integer ν .It follows from Lemma 4.4 that 2 x ′ e ′ = 0. Therefore we only need to chose from ν = 0 and ν = 1. If ν = 0, then Sq Z ( e ′ ) = 0. Applying Lemma 4.4 again, we have x ′ e ′ = µR n ( x ′ ) , which implies that H ( B P ( n, mn ); Z ) is generated by R n ( x ′ ), contradicting (2)and (3) of Lemma 3.4. Therefore, we have ν = 1 and x ′ e ′ + Sq Z ( e ′ ) = 0 ∈ H ( B P ( n, mn ); Z ) . By Lemma 4.4, we conclude. (cid:3)
Proof of Proposition 1.4 and Theorem 1.3.
Let X = sk ( K ( Z /n, K ( Z /n, α ∈ H ( X ; Z ) the restriction of the canonical gener-ator β n ∈ H ( K ( Z /n, Z ). Recall that it suffices to work under Convention 4.2: n = 2 r , m = 2 r or 2 r +1 , r > . It follows from Theorem 1.2 that ind( α ) is either 2 r or 2 r +1 . It suffices to showind( α ) = 2 r , for which we proceed to study the homotopy lifting problem discussedin the Introduction (1.9):(4.19) B P (2 r , r ) X K ( Z / r , B P (2 r , r )[7] K ( Z / r , K ( Z / r , = By (4.1), we have the following(4.21) B P (2 r , r )[7] B P (2 r , r )[6] B P (2 r , r )[5] ≃ K ( Z / r , × K ( Z , K ( Z , K ( Z /n, r ) K ( Z / r , = κ f =Id f HE TOPOLOGICAL PERIOD-INDEX PROBLEM OVER 8-COMPLEXES, II 15 where the map f is the obvious inclusion. Therefore f ∗ annihilates all cohomol-ogy classes of K ( Z ,
4) in positive degrees, in particular, ι and Γ . Therefore, byProposition 4.8, we have f ∗ ( κ ) = f ∗ ( β r × ι + 1 × Γ ) = 0 , and the dashed arrow in (4.21) exists. Therefore, by Proposition 1.7 we haveind( α ) = 2 r +1 , and we have proved Proposition 1.4.It remains to show the divisibility relations (1.6). For an arbitrary finite CWcomplex X of dimension 8 and α ∈ H ( X ; Z ) tor of period n , take α ′ ∈ H ( X ; Z /n )such that the Bockstein homomorphism takes α ′ to α . Then α ′ , up to homotopy,gives rise to a map X → K ( Z /n, X, α ) and ( K ( Z /n, , β n ). Theorem 1.3 follows fromthe naturality of the twisted AHSSs together with Theorem 1.5. (cid:3) References [1] B. Antieau and B. Williams. The topological period–index problem over 6-complexes.
Journalof Topology , 7(3):617–640, 2013.[2] B. Antieau and B. Williams. The period-index problem for twisted topological K -theory. Geometry & Topology , 18(2):1115–1148, 2014.[3] B. Antieau and B. Williams. Prime decomposition for the index of a Brauer class.
Annalidella Scuola Normale Superiore di Pisa. Classe di scienze , 17(1):277–285, 2017.[4] M. Atiyah and G. Segal. Twisted K -theory. arXiv preprint math/0407054 , 2004.[5] M. Atiyah and G. Segal. Twisted k -theory and cohomology, arxiv: math. KT0510674 , 2005.[6] J.-L. Colliot-Th´elene. Exposant et indice d’alg`ebres simples centrales non ramifi´ees.
EN-SEIGNEMENT MATHEMATIQUE , 48(1/2):127–146, 2002.[7] D. Crowley and M. Grant. The topological period-index conjecture for spin c arXiv preprint arXiv:1802.01296 , 2018.[8] A. Grothendieck. Le groupe de Brauer. I. algebres dAzumaya et interpr´etations diverses. Dixexpos´es sur la cohomologie des sch´emas , 3:46–66, 1968.[9] X. Gu. The topological period-index problem over 8-complexes, I. to appear in Journal ofTopology, arXiv:1709.00787 , 2017.[10] J. McCleary.
A user’s guide to spectral sequences . Number 58. Cambridge University Press,2001.[11] ´E. normale sup´erieure (France) and H. Cartan.
S´eminaire Henri Cartan: ann. 7 1954/1955;Alg`ebres d’Eilenberg-Maclane et homotopie . Secretariat Mathematique, 1958.
School of Mathematics and Statistics, the University of Melbourne, Parkville VIC3010, Australia
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