The topological K-theory of crystallographic groups with holonomy Z/2
aa r X i v : . [ m a t h . K T ] S e p THE TOPOLOGICAL K-THEORY OF CRYSTALLOGRAPHICGROUPS WITH HOLONOMY Z / MARIO VEL ´ASQUEZ
Abstract.
In this note we present a complete computation of the topologicalK-theory of the reduced C*-algebra of a semidirect product of the form Γ = Z n ⋊ ρ Z / ρ . Forthis, we use some results for Z / ρ is freeoutside the origin. Introduction
Let Γ be a crystallographic group with holonomy Z /
2, it means that Γ is definedby an extension(1.1) 0 → Z n → Γ → Z / → . We denote by C ∗ r (Γ) to the reduced C*-algebra of Γ. In [DL13] the topologicalK-theory of C ∗ r (Γ) is computed, under the assumption of free conjugacy action on Z n − { } . In this note, we give a complete computation of K ∗ ( C ∗ r (Γ)) avoidingthe above assumption. We use the Baum-Connes conjecture for Γ, reducing theproblem, to the computation of the Γ-equivariant K-homology of the classifyingspace for proper actions EΓ. For details on the Baum-Connes conjecture consult[Val02] and for a proof of the Baum-Conjecture for Γ see [HK01].The computations are obtained following a very simple idea, firstly, Γ can bedecomposed as a pullback Γ × Z / Γ over Z /
2, where Γ has trivial conjugacyaction and Γ has free conjugacy action outside the origin. A model for EΓ is R n with the natural Γ-action, as Z n acts freely we have a natural isomorphism K Γ ∗ (EΓ) ∼ = K Z / ∗ (( S ) n ) . The pullback decomposition of Γ gives a decomposition of ( S ) n as a cartesianproduct ( S ) r × ( S ) n − r , where Z / Z / S ) n as is defined in [Seg68], weapply a Kunneth formula for Z / K Γ ∗ (EΓ).A similar procedure to compute the topological K-theory of crystallographicgroups with others holonomy groups is not possible at the moment, because thereis no generalizations of results in [Ros13] for finite groups with order greater than2, mainly because the irreducible real representations of Z /n , ( n >
2) are actuallycomplex.
Date : September 23, 2020.
Key words and phrases.
Equivariant K-Theory, Baum-Connes Conjecture, Equivariant K-homology, Crystallographic groups. 2010 Math Subject classification: Primary: 19L64, Secondary:19L50, 19K33, 19L47. Kunneth Theorem for Z / -equivariant K-theory Throughout this note, K-theory or equivariant K-theory means complex topo-logical K-theory with compact supports for locally compact Hausdorff spaces. Bottperiodicity implies that we will regard this theory as being Z / R = R ( Z /
2) be the representation ring of Z /
2, which is isomorphic to Z [ t ] / ( t − t representing the 1-dimensional sign representation, this ring isthe coefficients for Z / I = ( t −
1) be the augmentationideal and J = ( t + 1), each prime ideal p of R contains either I or J , and these arethe unique minimal prime ideals of R . We denote by R p the localization of R atthe prime ideal p .The group Z / R and by R − respectively. From the sign representation wecan define some kind of equivariant twisted K-theory groups K ∗ Z / , − ( X ) = K ∗ Z / ( X × R − ) . The coefficients of this theory are K ∗ Z / . − ( {•} ) ∼ = R/J ∼ = I concentrated in evendegrees.Let us consider the inclusion { } → R − , it induces for every Z / X ahomomorphism of R -modules ϕ : K ∗ Z / , − ( X ) → K ∗ Z / ( X ) , in the other hand, consider the following composition: K ∗ Z / ( X ) Bott −−−→ K ∗ Z / ( X × R − × R − ) → K ∗ Z / ( X × R − × { } ) . It defines a homomorphism of R -modules ψ : K ∗ Z / ( X ) → K ∗ Z / , − ( X ) . Proposition 2.1.
For any Z / -space X we have a natural diagram K ∗ Z / ( X ) : K ∗ Z / ( X ) ψ . . K ∗ Z / , − ( X ) . ϕ m m Where the maps ϕ and ψ preserves the Z / -grading and the composite in any orderis giving by multiplication by (1 − t ) . Moreover if p ⊆ R is a prime ideal containing I , then ψ and ϕ vanish after localizing at p .Proof. [Ros13]. (cid:3) We have the following version of the Kunneth formula for Z / Theorem 2.2.
Let X and Y be Z / -spaces, let p ⊆ R be a prime ideal containing J , with p = ( J, there is short exact sequence of Z / -graded, R p -modules → K nG ( X ) p ⊗ R p K mG ( Y ) p ω p −−→ K m + nG ( X × Y ) p → Tor R p ( K nG ( X ) p , K n +1 G ( Y ) p ) → . Remark 2.3.
When p contains I we have a Kunneth formula taking K ∗ Z / ( − )instead of K ∗ Z / ( − ) on Thm. 2.2, this is a result in [Ros13]. On the other hand, asis observed in [Ros13], when p contains I , Thm. 2.2 is not true for K ∗ Z / ( − ) (as canbe observed taking X = Y = Z / K ∗ Z / ( − ). HE TOPOLOGICAL K-THEORY OF CRYSTALLOGRAPHIC GROUPS. 3 Crystallographic groups
Let Γ be a group defined by the extension 1.1, with the conjugation action of Z / ρ : Z / → GL( n, Z ) . From now on we will supposethat the conjugacy action is not free outside the origin.Let H be the subgroup of Z n where Z / r ≥ { v , . . . , v n } of Z n such that { v , · · · v r } is a base of H , in this base, thehomomorphism ρ : Z / → GL ( n, Z ) is determined by a matrix (the image of thegenerator of Z /
2) with the following form: (cid:18) I r A n − r (cid:19) . Where the conjugation action on the last n − r coordinates (denoted by ρ n − r : Z / → GL ( n − r, Z )) is free outside the origin.Let us consider the canonical action of Γ over R n , as Z n acts freely, we have K ∗ Γ ( R n ) ∼ = K ∗ Z / (( S ) n ) . Now we will use Theorem 2.2, considering X = ( S ) r with the trivial Z / Y = ( S ) n − r with the Z / ρ n − r . Note that K ∗ Z / ( X ) canbe computed easily (being X a trivial Z / K ∗ Z / ( Y ) was computed in[DL13].As the Z / X is trivial, we have an isomorphism of R -modules K ∗ Z / ( X ) ∼ = R ⊗ Z K ∗ ( X ). On the other hand K ∗ ( X ) ∼ = ( Z r − ∗ = 0 Z r − ∗ = 1 . Then we have an isomorphism of R -modules K ∗ Z / ( X ) ∼ = ( R r − ∗ = 0 R r − ∗ = 1 . We need to recall the following result from [Ros13].
Proposition 3.1.
Let X be a locally compact Z / -space. Then there is a natural6-term exact sequence K ( X ) / / K Z / , − ( X ) ϕ / / K Z / ( X ) f (cid:15) (cid:15) K Z / ( X ) f O O K Z / , − ( X ) ϕ o o K ( X ) o o where the vertical arrows denoted by f on the left and right are the forgetful mapsfrom equivariant to non-equivariant K-theory. Now we have to determine the R -module structure of K ∗ Z / ( Y ), first note that K ∗ ( Y ) ∼ = ( Z n − r − ∗ = 0 Z n − r − ∗ = 1 . Now we need to recall Thm. 7.1 in [DL13].
Proposition 3.2. (i) There is a split short exact sequence of R -modules → K ( Y / ( Z / → K Z / ( Y ) → I n − r → MARIO VEL´ASQUEZ (ii) K Z / ( Y ) = 0 .Where the R -module structure in K ( Y / ( Z / is determined by the augmenta-tion map R ǫ −→ Z .Proof. It is proved in Thm. 7.1 in [DL13], we only have to remark that the map K ( Y / ( Z / → K Z / ( Y )is the pullback of the quotient Y → Y / ( Z /
2) and is a homomorphism of R -modules,in a similar way the map K Z / ( Y ) → I n − r is induced by the inclusion of repre-sentatives of the conjugacy classes of finite non-trivial subgroups of Γ, then it is ahomomorphism of R -modules, then we have a short exact sequence of R -modules.A splitting can be defined identifying the R -module K ( Y / ( Z / K ( Y ) Z / (it happens because both groups are torsion free as abelian groups), by the forgetfulmap K Z / ( Y ) → K ( Y ) Z / . (cid:3) Identifying the R -modules Z with J (it can be done because t ∈ R acts trivially)and applying the above proposition, we obtain an isomorphism of R -modules K Z / ( Y ) ∼ = J n − r − ⊕ I n − r . Now we will use Thm. 2.2 to compute K ∗ Z / ( X × Y ) p for every prime ideal p ⊆ R with p ⊇ J and p = ( I, → K ∗ Z / ( X ) p ⊗ R p K ∗ Z / ( Y ) p → K ∗ Z / ( X × Y ) p → Tor R p ( K ∗ Z / ( X ) p , K ∗ +1 Z / ( Y ) p ) → . In this specific case as K ∗ Z / ( X ) p is a free R p -module, we obtain thatTor R p ( K ∗ Z / ( X ) p , K ∗ +1 Z / ( Y ) p ) = 0 , then we have K ∗ Z / ( X × Y ) p ∼ = K ∗ Z / ( X ) p ⊗ R p K ∗ Z / ( Y ) p ∼ = ( ( I p ) n − ∗ = 0( I p ) n − ∗ = 1 . In particular K ∗ Z / ( X × Y ) p is torsion free as abelian group. Now suppose p ⊇ I ,combining Prop. 3.1 and Prop. 2.1 applied to X × Y we obtain short exact sequences0 → K Z / ( X × Y ) p f −→ K ( X × Y ) p → K Z / , − ( X × Y ) p → → K Z / ( X × Y ) p f −→ K ( X × Y ) p → K Z / , − ( X × Y ) p → , where we are considering K ∗ ( X × Y ) as a R -module via the augmentation map ǫ : R → Z , in particular, K ∗ Z / ( X × Y ) p can be considered as a submodule of K ∗ ( X × Y ) p .On the other hand, as X × Y is the n -torus we have an isomorphism of R p -modules K ∗ ( X × Y ) p ∼ = ( J n − p ∗ = 0 J n − p ∗ = 1 . Then K ∗ Z / ( X × Y ) p ⊆ K ∗ ( X × Y ) p is torsion free as abelian group.As our final goal is to obtain the structure as abelian group we only need toprove that the above information implies that K ∗ Z / ( X × Y ) is torsion free (asabelian group), it can be done in the following way.Let M be a R -module, consider the Z -torsion module of M defined as Z T ( M ) = { x ∈ M | there is n ∈ Z − { } , n · x = 0 } . HE TOPOLOGICAL K-THEORY OF CRYSTALLOGRAPHIC GROUPS. 5
Note that Z T ( M ) is a R -submodule of M , and moreover Z T ( M ) p can be consideredas a submodule of Z T ( M p ) , but the above computations imply that Z T ( K ∗ Z / ( X × Y ) p ) = 0for every prime ideal p ⊆ R , then we have Z T ( K ∗ Z / ( X × Y )) = 0 , and then K ∗ Z / ( X × Y ) is a torsion free Z / K ∗ Z / ( X × Y ), it can be done by the wellknow formula proved for example [AS89] or [LO01]rk( K ∗ Z / ( X × Y )) = X g ∈ Z / rk( K ∗ ( X g × Y g ) C G ( g ) )= rk( K ∗ ( X × Y ) Z / ) + rk( K ∗ ( X Z / × Y Z / ))= rk( K ∗ ( X )) rk( K ∗ Z / ( Y ))= ( · n − ∗ = 03 · n − ∗ = 1 . Then we obtain.
Theorem 3.3.
Let Γ a group defined by an extension 1.1, where the action of Z / is not free outside the origin, then we have an isomorphism of abelian groups K ∗ Γ ( E Γ) ∼ = ( Z · n − ∗ = 0 Z · n − ∗ = 1 . Topological K-theory of the reduced group C*-algebra
Now we can compute Γ-equivariant K-homology groups of EΓ as is defined forexample in [JL13].As K ∗ Γ (EΓ) is torsion free, the universal coefficient theorem for equivariant K-theory (Thm. 0.3 in [JL13]) reduces to K Γ ∗ (EΓ) ∼ = hom Z ( K ∗ Γ (EΓ) , Z ) ∼ = ( Z · n − ∗ = 0 Z · n − ∗ = 1 . Finally by the Baum-Connes conjecture and previous results in [DL13] we obtaina complete computation of the reduced group C*-algebra of the group Z n ⋊ Z / Theorem 4.1.
Let Γ a group defined by an extension 1.1. If the action of Z / is not free outside the origin, then we have an isomorphism of abelian groups K ∗ ( C ∗ r ( Z n ⋊ Z / ∼ = ( Z · n − ∗ = 0 Z · n − ∗ = 1 , if the action of Z / is free outside the origin K ∗ ( C ∗ r ( Z n ⋊ Z / ∼ = ( Z . n − ∗ = 00 ∗ = 1 . MARIO VEL´ASQUEZ
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Departamento de Matem´aticas, Pontificia Universidad Javeriana, Cra. 7 No. 43-82 -Edificio Carlos Ortz 6to piso, Bogot´a D.C, Colombia
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