aa r X i v : . [ m a t h . K T ] N ov Twisted K -homology, Geometric cyclesand T -duality Bei Liu ∗ Mathematisches Institut, Georg-August-Universit¨ a G¨ o ettingen Abstract
Twisted K -homology corresponds to D -branes in string theory . In thispaper we compare two different models of geometric twisted K -homologyand get their equivalence. Moreover, we give another description of geo-metric twisted K -homology using bundle gerbes. In the last part weconstruct T -duality transformation for geometric twisted K -homology. String theory, as a candidate for quantum gravity, abstracts interests from bothphysicians and mathematicians. Its ultimate goal is to construct quantum the-ories of the basic structures of our universe. The starting point of string theoryis that the fundamental units of our universe are 1-dimensional strings insteadof point particles.String theorists find that there are five kinds of different stringtheories, i.e type I, type IIA, type IIB, heterotic E and heterotic SO (32) stringtheories, all of which are mathematically consistent (see [18]). While there isonly one universe, therefore it becomes important to study relations betweendifferent string theories. These relations are called duality. There are threekinds of duality in string theory: S -duality, T -duality and U -duality. S -dualityis also called electric-magnetic duality [14]. T -duality, which is one of the maintopic of this paper, is a duality exchanging winding number and momentum inthe dynamic equation of D -branes. While U -duality can be seen as compositionof S -duality and T -duality. T -duality provides an equivalence between type IIAand type IIB string theory. In physics, this equivalence is reflected by two im-portant notions in string theory, D -branes and Ramond-Ramond fields livingon D -branes. Briefly speaking, D -branes can be see as a submanifold of thespacetime manifold on which strings can end. A Dp -brane is a p -dimensionalsubmanifold with some other structures on it.After Witten suggested that the Ramond-Ramond fields should be classified in(twisted) K -theory instead of de Rham cohomology in [21], twisted K -theoryhas been studied extensively (see [1] and [5]). There are mainly three differentapproaches to understand twisted K -theory:1. bundle gerbes, which only works when twists are torsion cohomologyclasses2. homotopy classes of sections of Fredlhom bundles ∗ [email protected]
1. K-theory of the C ∗ -algebra of compact operator bundles determined bythe twisting classThe dual theory of twisted K -theory i.e twisted K -homology has also beenstudied as a mathematical interpretation of D -branes. The analytic approachto twisted K -homology is the same as the third for twisted K -theory we listabove. However, geometric approaches are more useful for us to understand D -branes in string theory. In [2] and [20] they both give a construction of geometrictwisted K -homology groups. We will show that they are equivalent to each otherin this paper. After that, we will construct some properties of geometric twisted K -homology and then use the geometric twisted K -cycle in [20] to construct the T -duality transformation for spacetime compactified over S .Now we give the structure of this paper. In the next section, we review differentapproaches to twisted K -homology. In section 3 we show the equivalence ofthat two versions of geometric twisted K -homology. In section 4 we establishsome properties of geometric twisted K -homology. In section 5 we discuss thecharge map in [2] and get a positive answer to the question on the charge mapin the end of [2]. In section 6 we introduce another approach to define geo-metric twisted K -cycle using bundle gerbes. In section 7 we construct threetransformations whose composition give T -duality transformation of geomet-ric twisted K -homology. In the last section we will prove that the T -dualitytransformation is an isomorphism. Acknowledgement
The author thanks the Research Training Group 1493”Mathematical Structures in Modern Quantum Physics” for the support duringhis Ph.D period. The author deeply thanks Thomas Schick for many help-ful comments and discussions. Besides, the author also thanks Bailing Wang’swonderful lecture on geometric twisted K -homology and discussions. Let X be a locally finite CW -complex and α : X → K ( Z ,
3) be a twist over X .Denote the projective unitary group over a separable Hilbert space H by P U ( H )and denote the C ∗ -algebra of compact operators by K . Since the classifyingspace of P U ( H ) is a model of K ( Z ,
3) (see [1]), therefore α determines a principal P U ( H )-bundle P over X . We denote the associated K -bundle of P by A . Thenall of the continuous sections with compact support of A give rise to a C ∗ -algebrawhich we denote by C ∗ ( X, α ). A direct way to construct a twisted K -homologytheory for ( X, α ) is to use the K -homology of the continuous trace C ∗ -algebraof C ∗ ( X, α ). We denote this twisted K -homology group by K a ∗ ( X, α ). However,it is very difficult to see the geometric meaning K -cycles through this approach.In [2] and [20], more topological and geometric models are constructed. Let K bethe complex K -theory spectrum and P α ( K )the corresponding bundle of basedspectra over X . In [20], the topological twisted K -homology group is defined tobe K tn ( X, α ) := lim k →∞ [ S n +2 k , P α (Ω k K ) /X ] (2.1)This definition comes from the classical definition of homology theory by spectra,which is automatically a homology theory. In [20], B.L. Wang gave a geometric2wisted K -homology. Before giving his constructions, we first review the defin-ition of geometric cycles of K -homology in [4]. A geometry cycle on a pair ofspace ( X, Y )( Y ⊂ X ) is a triple ( M, f, [ E ]), such that • M is a compact spin c -manifold(probably with boundary); • f is a continuous map from M to X such that f ( ∂M ) ⊂ Y ; • [ E ] is a K -class of M The definition of twisted cases is given in [20] as follows.
Definition 2.1.
A geometric cycle for (
X, Y, α ) is a quintuple (
M, ι, υ, η, [ E ])such that • M is a α -twisted spin c -manifold,i.e M is a compact oriented manifold andthe following diagram exists M BSO
X K ( Z , ι υα W η (2.2)Here υ and W are classifying maps of the stable normal bundle of M and the third integral Stiefel-Whitney class respectively, η is a homotopybetween α ◦ ι and W ◦ υ . Moreover we require ι ( ∂M ) ⊂ Y . • [ E ] is an element class of K ( X ) which is represented by a Z -gradedvector bundle E .Let Γ( X, α ) be the collections of all geometric cycles for (
X, α ). To get geometrictwisted K -homology, we still need an equivalence relation on Γ( X, α ), which isgenerated by the following basic relations: • Direct sum - disjoint union
If (
M, ι, υ, η, [ E ]) and ( M, ι, υ, η, [ E ]) aregeometric cycles over ( X, α ), then(
M, ι, υ, η, [ E ]) ∪ ( M, ι, υ, η, [ E ]) ∼ ( M, ι, υ, η, [ E ] + [ E ]) (2.3) • Bordism
Given two geometric cycle (
M, ι, υ, η, [ E ]) and ( M, ι, υ, η, [ E ]), if there exists a α -twisted spin c -manifold ( W, ι, υ, η ) and [ E ] ∈ K ( W )such that δ ( W, ι, υ, η ) = − ( M , ι , υ , η ) ∪ ( M , ι , υ , η ) (2.4)and δ ([ E ]) = [ E ] ∪ [ E ]. Here − ( M , ι , υ , η ) means the manifold M with the opposite α -twisted Spin c structure. • Spin c vector bundle modification Given a geometric cycle (
M, ι, υ, η, E )and a spin c vector bundle V over M with even dimensional fibers, wecan choose a Riemannian metric on V ⊕ R and get the sphere bundle3 M = S ( V ⊕ R ). Then the vertical tangent bundle T v ( ˆ M ) admits a nat-ural spin c structure. Let S + V be the associated positive spinor bundle and ρ : ˆ M → M be the projection. Then( M, ι, υ, η, E ) ∼ ( ˆ M , ι ◦ ρ, υ ◦ ρ, η ◦ ρ, ρ ∗ E ⊗ S + V ) . (2.5)Here υ ′ is a classifying map of the stable normal bundle of ˆ M and η ′ is achosen homotopy between W ◦ υ ′ and α ◦ ι ◦ ρ . Definition 2.2. K g ∗ := Γ( X, α ) / ∼ . Addition is given by disjoint union rela-tion above. Let K g ( X, α )(respectively K g ( X, α )) be the subgroup of K g ∗ ( X, α )determined by all geometric cycles with even (respectively odd) dimensional α -twisted spin c -manifolds.There is an natural isomorphism µ between K g / ( X, α ) and K a / ( X, α ): µ ( M, ι, υ, η, [ E ]) = ι ∗ ◦ η ∗ ◦ I ∗ ◦ P D ([ E ]) (2.6)Here P D : K i ( M ) → K n + i ( M, W ◦ τ ) is the Poincarˇ e duality map between K -group and K -homology group, ι ∗ is the push-forward map induced by ι , η ∗ is the canonical isomorphism induced by ηη ∗ : K a / ( M, W ◦ υ ) → K a / ( M, α ◦ ι ) (2.7)and I : K a / ( M, W ◦ τ ) → K a / ( M, W ◦ υ ) is a natural isomorphism induced bythe anti-isomorphism of the associated K ( H )-bundles. The following theoremin [20] states that µ is an isomorphism. Theorem 2.3.
The assignment ( M, ι, υ, η, [ E ]) → µ ( M, ι, υ, η, [ E ]) , called theassembly map, defines a natural homomorphism µ : K g / ( X, α ) → K a / ( X, α ) which is an isomorphism for any smooth manifold X with a twisting α : X → K ( Z , . Before giving the definitions, we need to point out that there is another descrip-tion of a twist over a space here. Let X be a second countable locally compactHausdorff topological space. Then a twist on X in [6] is a locally trivial bundle A of elementary C ∗ -algebras on X , i.e each fiber of A is an elementary C ∗ -algebraand with the structure group the automorphism group of K ( H ) for some Hil-bert space H . However, these two description can be transferred from one tothe other. In [2] they used ( X , A ) as the starting point and defined topologicaltwisted K -cycles as follows. Definition 2.4. An A -twisted K -cycles on X is a triple ( M, σ, ψ ) where • M is a compact spin c -manifold without boundary. • φ : M → X is a continuous map. • σ ∈ K (Γ( M, φ ∗ A op ))Let E ( X, α ) be all of the K -cycles over ( X, A ). Then the topological A -twisted K -homology group over ( X, A ) is defined by K top ∗ ( X, α ) := E ( X, α ) / ∼ (2.8)4ere ∼ is a similar equivalence generated by disjoint unions, bordisom andvector bundle modification. Moreover, a more geometric twisted K -cycle isgiven in [2], which is called D -cycle. In order to introduce D -cycle, we still alsoneed the notion of spinor bundles. Definition 2.5.
A spinor bundle for a twisting A is a vector bundle S of Hilbertspaces on X together with a given isomorphism of twisting data A ∼ = K ( S ) (2.9)One fact we need to know about spinor bundle is that a spinor bundle for A exists if and only if DD ( A ) = 0. Now we can give the definition of D -cyclein [2]. Definition 2.6. A D -cycle for ( X, A ) is a 4-tuple ( M, E, φ, S ) such that • M is a closed oriented C ∞ -Riemannian manifold. • E is a complex vector bundle on M . • φ is a continuous map from M to X . • S is a spinor bundle for Cliff + C ( T M ) ⊗ φ ∗ A op .Two D -cycles ( M, E, φ, S ) , ( M ′ , E ′ , φ ′ , S ′ ) for ( X, A ) are isomorphic if there isan orientation preserving isometric diffeomorphism f : M → M such that thediagram M M ′ Xfφ φ ′ (2.10)commutes, and f ∗ E ′ ∼ = E, f ∗ S ′ ∼ = S .Let Γ D ( X, α ) be the collection of all D -cycles over ( X, A ). Similar, we canimpose an equivalence generated by disjoint union, bordism and vector bundlemodification on Γ D ( X, α ) and get the geometric A -twisted K -homology in [2],which we denote by K geo ∗ ( X, A ). • Direct sum - disjoint union
If (
M, E , ι, S ) and ( M, E , ι, S ) are D -cycles for ( X, α ) then(
M, E , ι, S ) ∪ ( M, E , ι, S ) ∼ ( M, E ⊕ E , ι, S ) (2.11) • Bordism
Given two D cycle ( M , E , ι , S ) and ( M , E , ι , S ) , if thereexists a 4-tuple ( W, E, Φ , S ) such that W is a compact oriented Rieman-nian manifold with boundary, E is a complex vector bundle over W , Φ isa continuous map from W to X and( δW, E | δW, Φ | δW, S + | δW ) ∼ = ( M , E , ι , S ) ∪ ( M , E , ι , S ) (2.12)Here S + | δ is the positive part of S . It is S itself when W is odd dimen-sional. 5 Spin c - vector bundle modification Given a D cycle ( M, E, ι, S ) and aspin c -vector bundle V over M with even dimensional fibers. Let S ′ bethe spinor bundle of Cliff + C ( T ˆ M ) ⊗ ( φ ◦ ρ ) ∗ ( A ) op . Use the notation in thedefinition of geometric K -cycles. Then( M, E, ι, S ) ∼ ( ˆ M , β ⊗ ρ ∗ E, ι ◦ ρ, ρ ∗ S ) . (2.13)Here β is a complex vector bundle over ˆ M whose restriction to each fiberof ρ is the Bott generator vector bundle of ˆ M .In [2] they gave a natural charge map h : K geo ∗ ( X, A ) → K top ∗ ( X, A ) as follows:Let ( M, E, φ, S ) be a D -cycle and choose a normal bundle for M with evendimensional fibers, then h ( M, E, φ, S ) := ( S ( v ⊕ R ) , φ ◦ π, σ ) (2.14)Here σ is defined as the image of E under the composition of s ! and χ .1. Let s : M → S ( υ ⊕ R ) be the canonical section of unite section on thetrivial real line bundle. For simplicity, we denote the total space of spherebundle S ( υ ⊕ R ) by M and the bundle map by ρ . Then s ! : K ( M ) → K (Γ( ˆ M , ρ ∗ ( Clif f C ( υ )))) is the Gysin homomorphism induced by s .2. χ : K (Γ( ˆ M , ρ ∗ ( Clif f C ( υ )))) → K (Γ( ˆ M , ( φ ◦ ρ ) ∗ A op )) is the isomorphisminduced by the trivialisation of T M ⊕ υ and the given spinor bundle S .They propose a question which asked that if h is an isomorphism. We will provein the paper that it is true when X is a countable CW -complex. Remark . We should note that all of these definitions of geometric twisted K -homology group cannot work for general twists. The twists should satisfythe Freed-Witten condition (see [20]), i.e, there exists a manifold M and acontinuous map l : M → X s.t l ∗ ( H ) + W ( M ) = 0 (2.15) We list the following theorem from [17] first.
Theorem 3.1.
Let E be a vector bundle over a space X and ˜ E be the Cliffordbundle of E . Let W ( E ) be the third integral Stiefel-Whitney class of E . Thenwe have: W ( E ) = ( DD ( ˜ E ) if E has even dimensionDD ( ˜ E + ) if E has odd dimension. (3.1) Remark . Through the above theorem we can get a bit of idea why thetwo versions of geometric cycle are equivalent. The existence of spinor bundleimplies the triviality of Dixmier-Douady class of DD ( Clif f + ( T M ) ⊗ φ ∗ ( A op ),so the choice of the spinor bundle S for Clif f + ( T M ) ⊗ φ ∗ A op in the definitionof D -cycle corresponds to the choice of the homotopy η in the definition ofgeometric K -cycle in [20]. 6et X be a locally finite CW -complex and α : X → K ( Z ,
3) be a twist in thesense of [20]. From the discussion in the beginning of section 2 we know that α determines an associated K -bundle A over X . While A is exactly a twist inthe sense of [2]. Therefore, we get the basic set up data of the two definitionsof geometric twisted K -cycles are equivalent.Moreover, we can consider twisting datas in different categories. Define T wist ( X )to be a category with continuous maps α : X → K ( Z ,
3) as objects. The morph-isms of
T wist ( X ) are homotopies between objects. Define T wist ( X ) to bea category with H -bundles over X as objects. The morphisms of T wist ( X )are isomorphisms of K -bundles. Obviously, we can see that T wist ( X ) and T wist ( X ) are groupoids and the sets of equivalence classes for them are both H ( X, Z ). Now we give two lemmas which is used later, whose proof are prob-ably already known and so we don’t claim that they are original. Lemma 3.3.
Denote the projection from X × I to X by p . For every K -bundle A over X × I , there exists a K -bundle A ′ over X such that A ∼ = p ∗ ( A ′ ) . The proof is obvious.
Lemma 3.4.
Let A be a K -bundle over X , then there is a canonical spinorbundle for A ⊗ A op . The proof can be found in [2].Now we give a construction which is useful in the proof of the main theorem inthis section. Assume K and K are two K -bundles over X and λ : K → K isan isomorphism. Then we can get a K -bundle over X × [0 ,
1] as follows. Firstwe can see that K × [0 , /
2] and K × [1 / ,
1] are K -bundle over X × [0 , / X × [1 / ,
1] respectively. Then we can glue K × [0 , /
2] and K × [1 / , { / } × X via λ and get a K -bundle K over X × [0 , Theorem 3.5.
Let X be a locally finite CW -complex. There exists an iso-morphism F : K g ( X, α ) → K geo ( X, A ) . To make the proof easier to read, we give the basic idea first. The idea is thatwe transform spinor bundles in D -cycles to K -bundles over M × [0 , K -bundles to define homotopies in α -twisted spin c -manifolds and alsodefine F below. And we reverse the procedure to prove that F is surjective. Theinjectivity of F is essentially implied by the compatibility of F with ∼ . Now westart the proof. Proof.
Let [ x ] be a class in K geo ( X, A ) and ( M g , E, φ, S ) be a D -cycle whichrepresents [ x ]. By definition S is a spinor bundle of Cliff + C ( T M ) ⊗ φ ∗ A op . Andwe denote the chosen isomorphism between K ( S ) and Cliff + C ( T M ) ⊗ φ ∗ A op by λ . We define F ([ x ]) in K g ( X, α ) to be a class represented by (
M, φ, υ, η, E ), inwhich • M is the manifold of M g forgetting Riemiannian structure, φ and E arethe same map and bundle in the D -cycle; • υ is the classifying map of the stable normal bundle of M ; • η is a homotopy between W ◦ υ and α ◦ φ .7e only need to explain how η is constructed from ( M g , E, φ, S ). By Lemma 3.4we know that there exists a canonical Hilbert bundle V over M and a canonicalisomorphism c between K ( V ) ∼ = φ ∗ A ⊗ φ ∗ A op . Combine c and h we get anisomorphism between K ( S ) ⊗ A and Cliff + C ( T M ) ⊗ K ( V ). According to thediscussion before the theorem we can obtain a K -bundle W over M × I suchthat W M ×{ } ∼ = K ( S ) ⊗A and W M ×{ } ∼ = Clif f + C ( T M ) ⊗K ( V ). Since BP U ( H )is also a classifying space of K -bundles, therefore we can get that there existsan η : X × [0 , → K ( Z ,
3) such that ( η ◦ ( φ × Id )) ∗ ( K ) is isomorphic to W .Moreover, we get two maps α , α : X → K ( Z ,
3) by restricting η to X × { } and X × { } respectively, which gives the following isomorphisms:( α ◦ φ ) ∗ ( K ) ∼ = K ( S ) ⊗ A , ( α ◦ φ ) ∗ ( K ) ∼ = Clif f + C ( T M ) ⊗ K ( V ) (3.2)Since different choices of η are homotopic to each other, so they represent thesame class in K g ( X, α ) via bordism relation. To show that F is well defined,we still need to check that it is compatible with the relations which define ∼ . • From the definition of F we can see F ([( M g , E , φ, S )] ∪ [( M g , E , φ, S )]) = [( M, E ⊕ E , φ, S )]; F ([ M g , E , φ, S ]) ∪ F ([ M g , E , φ, S ]) = [( M, E ⊕ E , φ, S )]i.e F respects the disjoint union relation. • Let ( M g , E, φ, S ) be a bordism between ( M g , E , φ , S ) and ( M g , E , φ ,S ). Denote the associated isomorphisms of the spinor bundles by h , h and h . Denote the chosen representing cycles of the image of their homo-logy classes under F by ( M, φ, υ, η, E ), ( M , φ , υ , η , E ) and ( M , φ , υ ,η , E ) respectively. Choosing a tubular neighborhood of M in M we canget that T M | M ∼ = T M ⊗ R , so we get the stable normal bundle of M agrees with the restriction of the stable normal bundle of M to M i.e υ | M is homotopic to υ . Similarly we can get υ M is homotopic to υ .Let W , W and W be the three K -bundles over M × [0 , M × [0 , M × [0 ,
1] respectively. The isomorphism c in Lemma 3.4 is naturaland h | K ( S i ) = h i ( i = 0, 1), so we get that W| M i × [0 , is isomorphic to W i ( i = 0, 1). This implies that η | M ×{ i } is homotopic to η i ( i = 0, 1). So F is compatible with bordism. • Use the notion in Section 2 . F ([( ˆ M g , S + V ⊗ ρ ∗ E, φ ◦ ρ, ρ ∗ S ) by[ ˆ M , S + V ⊗ ρ ∗ E, φ ◦ ρ, υ ′ , η ′ ]. We only need to prove that η ′ is homotopic the η ◦ π . The is implied by the observation that the corresponding K -bundles V ′ and V ′′ over ˆ M × [0 ,
1] are isomorphic.Obviously F maps a trivial D -cycle to a trivial geometric twisted K -cycle, so weget that F is a well defined injective homomorphism. The left is to show that F issurjective. For any class [ y ] ∈ K g ( X, α ), choose a geometric cycle (
M, φ, υ, η, E )to represent it. η induces a K -bundle over M × [0 , X . Bythe definition of η we have X | M ×{ } ∼ = Cliff + C ( T M ) ⊗ K , X | M ×{ } ∼ = ( α ◦ φ ) ∗ ( K ).Let l be an isomorphism between Cliff + C ( T M ) ⊗ K and ( α ◦ φ ) ∗ ( K ) and L be a K -bundle constructed via gluing Cliff + C ( T M ) ⊗K× [0 , /
2] and ( α ◦ φ ) ∗ ( K ) × [1 / , l at M × { / } . Then we have that L is isomorphic X since they have thesame Dixmier-Dourady class. Combining l with the canonical isomorphism in8emma 3.4, we get a spinor bundle S for Cliff + C ( T M ) ⊗ φ ∗ A op . The D -cycle( M g , E, φ, S ) satisfies that F ([ M g , E, φ, S ]) = [ y ]. Remark . The above proposition shows that geometric K -cycles in [2] and D -cycles in [20] are equivalent to each other. Therefore we can choose one ofthem to construct the topological T -duality for geometric twisted K -homology.Moreover, K g ( X, α ) ∼ = K a ( X, α ) imples that K geo ( X, α ) is also isomorphic to K a ( X, α ). In this section we prove that geometric twisted K -homology satisfies Elienberg-Steenrod axioms of homology theory. We first give a simple lemma. Lemma 4.1. If f : Y → X is a continuous map, then f induces a homomorph-ism f ∗ : K g ( Y, α ◦ f ) → K g ( X, α ) .Proof. Given a twisted geometric K -cycle ( M, φ, υ, η, E ) over (
Y, α ◦ f ), wedefine f ∗ by f ∗ ( M, φ, υ, η, E ) = (
M, f ◦ φ, υ, η, E ) (4.1)We need to check that f ∗ is compatible with disjoint union, bordism and spin c -vector bundle modification. • Given two geometric K -cycles ( M, φ, υ, η, E i ) ( i = 1 , f ∗ (( M, φ, υ, η, E ) ∪ ( M, φ, υ, η, E )) = ( M, φ ◦ f, υ, η, E ⊕ E ) • If (
M, φ, υ, η, E ) is a bordism between ( M , φ , υ , η , E ) and ( M , φ , υ ,η , E ), then clearly ( M, f ◦ φ, υ, η, E ) gives a bordism between ( M , f ◦ φ , υ , η , E ) and ( M , f ◦ φ , υ , η , E ). • Since f (( ˆ M , φ ◦ ρ, υ ◦ ρ, η ◦ ( ρ × Id ) , ρ ∗ E ⊗ S + V )) is ( ˆ M , f ◦ φ ◦ ρ, υ ◦ ρ, η ◦ ( ρ × Id ) , ρ ∗ E ⊗ S + V ), which is exactly a spin c -vector bundle modificationof ( M, f ◦ φ, υ, η, E ), so we get that f ∗ respects the spin c -vector bundlemodification relation. Theorem 4.2 ( Homotopy ) . If f : Y → X is a homotopy equivalence, thenthe induced map f ∗ : K g ∗ ( Y, α ◦ f ) → K g ∗ ( X, α ) is an isomorphism.Proof. We first show that if g : Y → X is homotopic to f , then K g ∗ ( Y, α ◦ f ) ∼ = K g ∗ ( Y, α ◦ g ). Let H : Y × [0 , → X be a homotopy from f to g i.e H ( y,
0) = f ( y ) and H ( y,
1) = g ( y ). Given a twisted geometric K -cycle ( M, φ, υ, η, E )over (
Y, α ◦ f ), we get a twisted geometric K -cycle over ( Y, α ◦ g ) as follows:( M, φ, υ, η ′ , E ). Here η ′ : M × [0 , → K ( Z ,
3) is defined by η ′ ( m, t ) = ( η ( m, t ) 0 ≤ t ≤ / α ◦ H ( m, t − ◦ ( φ × Id ) 1 / ≤ t ≤
19t is not hard to check that the above map is compatible with the disjointunion, bordism and spin c -vector bundle modification. We skip the details heresince they are similar to the proof of the above lemma. Therefore we get ahomomorphism H ∗ from K g ∗ ( Y, α ◦ f ) to K g ∗ ( Y, α ◦ g ). Similarly we can get theinverse of H ∗ by using H (1 − t, y ) as a homotopy from g to f . So we get that H ∗ is an isomorphism. Clearly, we have that f ∗ = g ∗ ◦ H ∗ . Let q : X → Y bea homotopy inverse of f : Y → X i.e f ◦ q is homotopic to id X and q ◦ f ishomotopic to id Y . Denote the associated homotopies by H and H respectively.Then we have that ( f ◦ q ) ∗ = ( H ) ∗ and ( q ◦ f ) ∗ = ( H ) ∗ . Since ( H ) ∗ and( H ) ∗ are both isomorphisms, we get that f ∗ is an isomorphism as well. Theorem 4.3 ( Excision ) . Let ( X, Y ) be a pair of locally finite CW -spaces, U is an open set of X such that U ∈ Y . Then we the inclusion i : ( X − U, Y − U ) ֒ → ( X, Y ) induces an isomorphism K g ∗ ( X − U, Y − U ; α ◦ i ) ∼ = K g ∗ ( X, Y ; α ) (4.2) Proof.
By Lemma 4.1 the inclusion i induces a homomorphism i ∗ . And moreover i ∗ is injective by its definition. The remainder is to prove that i ∗ is surjective.For any y ∈ K g ∗ ( X, Y ; α ), we choose a geometric cycle ( M, φ, υ, η, E ) to repres-ent it. We choose a point p ∈ U . By the Urysohn’s Lemma, there exists acontinuous function f : X → [0 ,
1] such that f ( p ) = 0 and f ( x ) = 1 for any x ∈ X − U . Then f ◦ φ is a continuous function from M to [0 , W be asub-manifold of M × [0 ,
1] defined by W = { ( t, ( f ◦ φ ) − ([ t, | t ∈ [0 , } . Let π : W → M be the canonical projection. For each slice W t = ( f ◦ φ ) − ([ t, W , we denote the inclusion of W t to M by i t . Define η ′ : W × [0 ,
1] as follows η ′ ( m, s, t ) = η ◦ ( i s × id )( m, t )Here ( m, s ) ∈ W and t ∈ [0 ,
1] Then the α -twisted spin c -manifold ( W, φ ◦ π, υ ◦ π, η ′ ), which gives a bordism between ( M, φ, υ, η, E ) and ( f − ( X − U ) , φ ◦ j, υ ◦ j, η ◦ ( j × Id ) , j ∗ E ). Here j : f − ( X − U ) ֒ → M is the inclusion. While( f − ( X − U ) , φ ◦ j, υ ◦ j, η ◦ ( j × Id ) , j ∗ E ) is also a twisted geometric K -cycle over( X − U, Y − U ; α ◦ i ), whose image under i ∗ is equivalent to ( M, φ, υ, η, E ). Remark . In the above proof we assume each slice W t to be a sub-manifoldof M , this is not true in general. But here we assume that both X and U arehomotopic to finite CW -complexes, so we can always find a smooth manifoldwhich is homotopy equivalent to W t . This makes the proof still works for generalcases.Another important axiom in Eilenberg-Steenrod axioms is the long exact se-quence. Before moving on to the long exact sequence of geometric twisted K -homology, we first introduce a notion. Definition 4.5.
A twist α : X → K ( Z ,
3) is called representable if there existsan oriented real vector bundle V over X such that W ( V ) = [ α ]. Here [ α ] is thepullback of the generator of H ( K ( Z ) ,
3) along α .10 heorem 4.6 ( Six-term exact sequence ) . Let Y be a sub-space of X and i be the inclusion map from Y to X . Let α be a representable twist over X . Thenwe have a six-term exact sequence: K g ( Y, α ◦ i ) K g ( X, α ) K g ( X, Y ; α ) K g ( X, Y ; α ) K g ( X, α ) K g ( Y, α ◦ i ) i ∗ j ∗ δi ∗ j ∗ δ Here the boundary operator is given by δ ([ M, φ, υ, η, E ]) = [( ∂M, φ | ∂M , υ | ∂M , η | ∂M × [0 , , E | ∂M )] (4.3)To prove this theorem, we first prove two lemmas as a preparation. Lemma 4.7.
Let θ = ( M, ι, υ, η, E ) be a geometric K -cycle over ( X, α ) and E i ( i = 1 , ) be spin c -vector bundles over M with even dimensional fibers. Denotethe vector bundle modification of θ with a spin c -vector bundle F by θ F . Thenwe have that θ E ⊕ E is bordant to ( θ E ) p ∗ E , in which p is the projection fromthe sphere bundle S ( E ⊗ R ) to M .Proof. Assume the dimension of the fiber of E i is n i and write θ E ⊕ E and( θ E ) p ∗ E explicitly as ( V, ι V , υ V , η V , E V ) and ( W, ι W , υ W , η W , E W ) respect-ively. Then the fibers of V and W are S n + n and S n × S n respectively. Wecan embed both of the two bundles over M into the vector bundle E ⊕ E ⊕ R as follows. First we choose a Riemannian metric over E ⊕ E ⊕ R and embed V into E ⊕ E ⊕ R as the standard unit sphere bundle. We embed W into E ⊕ E ⊕ R such that in each fiber over p ∈ M it is embedded as follows:(( x, s ) , ( y, t )) ((5 − t )( x, s ) , y )in which x ∈ ( E ) p , y ∈ ( E ) p and s, t ∈ ( R ) p . A careful tells us that this indeedinduces an embedding of W into E ⊕ E ⊕ R and we still denote the image ofthe embedding by W . We embed V into E ⊕ E ⊕ R with a scaling such thatthe radius of each fiber is 10. Denote the standard disk bundle with radius 10of E ⊕ E ⊕ R by D n + n +1 (10) and the solid torus bundle bounds by V by¯ V . Then we have that D n + n +1 (10) − ¯ V (which we denote by Z ) gives riseto a bordism between V and W . And ( Z, ι ◦ p Z , υ Z , η ◦ ( p Z × id ) , E Z ) gives abordism between ( V, ι V , υ V , η V , E V ) and ( W, ι W , υ W , η W , E W ). Lemma 4.8.
Let α be a representable twist over X and ( M, ι, υ, η, E M ) be ageometric cycle over ( X, α ) . Let ( ∂M, ι ∂M ,υ ∂M , η ∂M , E ∂M ) be the restriction to the boundary of M . If a spin c -vector bundlemodification with vector bundle E of ( ∂M, ι ∂M , υ ∂M , η ∂M , E ∂M ) is bordant totrivial cycle, then there exists a spin c -vector bundle V over ∂M such that thespin c -vector bundle modification with V is bordant to the trivial cycle and V canbe extended to a spin c -vector bundle over M . roof. Denote the spin c -vector bundle modification of ( ∂M, ι ∂M , υ ∂M , η ∂M , E ∂M )with vector bundle E by ( Q, ι ∂M ◦ π, υ Q , η Q , E Q ), which is bordant to the trivialcycle via a bordism of ( W, ι W , υ W , η W , E W ). There exists a normal bundle F over W , whose restriction to Q is also a normal bundle of T Q . On the otherhand, by the construction of spin c -vector bundle modification we can observethat there exists a normal bundle of T Q such that it is isomorphic to the pull-back (along π ) of the direct sum of a normal bundle of T ∂M (which we denoteby N ( T ∂M )) and a normal bundle of E (which we denote by N ( E )). Con-sider the spin c -modification of ( W, ι W , υ W , η W , E W ) with F ⊕ ι ∗ W ( V ) (here V is the vector bundle over X with W ( V ) = [ α ]). It gives a bordism from thespin c -modification of ( Q, ι ∂M ◦ π, υ Q , η Q , E Q ) with ( F ⊕ ι ∗ W ( V )) | Q and the trivialcycle. According to Lemma 4.7 and the observation before we can see that thespin c -modification of ( Q, ι ∂M ◦ π, υ Q , η Q , E Q ) with ( F ⊕ ι ∗ W ( V )) | Q is bordantto the spin c -modification of ( ∂M, ι ∂M , υ ∂M , η ∂M , E ∂M ) with E ⊕ N ( T ∂M ) ⊕ N ( E ) ⊕ ι ∗ W ( V ) | Q . While E ⊕ N ( E ) is trivial, we get that the spin c -modificationof ( ∂M, ι ∂M , υ ∂M , η ∂M , E ∂M ) with N ( T ∂M ) ⊕ ι ∗ ∂M ( V ) is bordant to a trivialcycle. The normal bundle on the boundary can be extended to the whole man-ifold obviously and ι ∗ ∂M ( V ) can be extended to a vector bundle ι ∗ M ( V ) So weget our statement.Now we start the proof of Theorem 4.6. Proof.
We will show the exactness at K g ( X, α ), K g ( X, Y ; α ) and K g ( Y, α ). Theproof of the rest part is similarly. • For any [ y ] ∈ K g ( Y, α ), we choose a twisted geometric K -cycle ( M , φ , υ ,η , E ) to represent it. Its image under j ∗ ◦ i ∗ is ( M , i ◦ φ , υ , η , E ),which is cobordant to trivial K -cycle relative Y in X . Therefore we have j ∗ ◦ i ∗ ([ y ]) is trivial. Assume that [ x ] ∈ K g ( X, α ) and j ∗ ([ x ]) = 0. We stillchoose a twisted geometric K -cycle ( M , φ , υ , η , E ) to represent [ x ].Since j ∗ ([ x ]) is trivial, we obtain that if we do several times of spin c -vectorbundle modification for ( M , j ◦ φ , υ , η , E ) relative to Y in X i.e if wedenote the results of Spin c -vector bundle modifications by ( ˆ M , j ◦ φ ◦ ρ, υ ′ , η ′ , E ′ ), then ( ˆ M , j ◦ φ ◦ ρ, υ ′ , η ′ , E ′ ) satisfies that ( j ◦ φ ◦ ρ ( ˆ M )) ⊂ Y , which also implies that j ◦ φ ( M ) ⊂ Y . Therefore we get that [ x ] ∈ im i ∗ . • First of all we need to point out that δ is well defined i.e it is com-patible with disjoint union, bordism and spin c -vector bundle modific-ation. It is a tedious check from the definition of δ , which we skiphere. By the definition of δ we can see that δ ◦ j ∗ = 0. To show that kerδ ⊂ imj ∗ , we choose a twisted geometric K -cycle ( M , φ , υ , η , E )such that δ [( M , φ , υ , η , E )] is trivial i.e several times of spin c -vectorbundle modification for ( ∂M , φ | ∂M , υ | ∂M , η | ∂M , E | ∂M ) is cobordant totrivial K -cycle over Y . By Lemma 4.8 we can assume that each spin c -vector bundle over the boundary of a manifold can be extended to a spin c -vector bundle over the whole manifold, therefore we get that if we do thespin c -vector bundle modifications for ( M , φ , υ , η , E ), then the result-ing twisted spin c -manifold is bordant to a twisted spin c -manifold withoutboundary over X . Finally we obtain that ( M , φ , υ , η , E ) is equivalentto a twisted geometric K -cycle whose underling twisted spin c -manifold is12losed, which implies that [( M , φ , υ , η ,E )] lies in the image of j ∗ . • Let ( M , φ , υ , η , E ) be a geometric α -twisted K -cycle over ( X, Y ; α ).Then [( ∂M , φ | ∂M , υ | ∂M , η | ∂M , E | ∂M ) is clearly bordant to a trivialtwisted geometric K -cycle over X i.e i ∗ ◦ δ = 0. Let [( M , φ , υ , η , E )] ∈ K g ( Y, α ◦ i ) be a class which lies in the kernel of i ∗ . A similar strategyleads us to get that the underling twisted spin c -manifold of several timesof spin c -vector bundle modification of [( M , φ , υ , η , E )] is a boundaryof a twisted spin c -manifold over X , from which we can easily get that[( M , φ , υ , η , E )] belongs to the image of δ . Remark . The condition of representability of the twist is essential for theproof here. In general, a twist is not always representable. We leave the six-termexact sequence of geometric twisted K -homology for general twists as a furtherquestion to be investigated. Theorem 4.10 ( Additivity ) . Let ( X i ) i ∈ I be a family of locally finite CW -complexes and α i : X i → K ( Z , be a twist over X i for each i . Moreover, werequire that there exists an α i -twisted spin c -manifold over X i for each i . Denote X to be the disjoint union of X i and α is a twist over X such that the restrictionof X to each X i is α i . Then we have the following isomorphism: K g ∗ ( X, α ) ∼ = ⊕ i K g ∗ ( X i , α i ) (4.4)The proof is not difficult from the definition, so we skip it here for simplicity.The above theorem implies the Mayer-Vietoris sequence and the Milnor’s lim ←− -exact sequence of geometric twisted K -homology. Theorem 4.11 ( Mayer-Vietoris sequence ) . Assume two open set U and V of X satisfies X = U S V and the twist α is representable, we have the Mayer-Vietoris sequence of twisted K -homology: K g ( X, α ) K g ( U T V, α U T V ) K g ( U, α U ) ⊕ K g ( V, α V ) K g ( U, α U ) ⊕ K g ( V, α V ) K g ( U T V, α U T V ) K g ( X, α ) δ j U ∗ ⊕ j V ∗ i U ∗ − i V ∗ δj U ∗ ⊕ j V ∗ i U ∗ − i V ∗ Proof.
Let Z be the disjoint union of U and V , Y be U ∩ V . Then consider thesix-term exact sequence for the pair ( Z, Y ) and use the excision isomorphism K g ∗ ( Z, Y ; α ) ∼ = K g ∗ ( X, α ) we can get the Mayer-Vietoris sequence for twistedgeometric K -homology groups. Theorem 4.12 ( lim ←− - exact sequence ) . Given a countable CW -complex X and a representable twist α over X such that there exists an α -twisted spin c -manifold over X . We denote the n -skeleton of X to be X n and the inclusion X n ֒ → X to be i n . Let α n = α ◦ i n . Then we have the following exact sequence → lim ←− K g ∗− ( X n , α n ) → K g ∗ ( X, α ) → lim ←− K g ∗ ( X n , α n ) → lim ←− -sequence ofhomology theory, which can be found in chapter 19 of [15]. In Section 8 of [2] they propose a question:Is the charge map h (see (2.14)) an isomorphism?In this section we show this is true for countable CW -complexes by consideringthe following diagram. K geo ( X, A ) K top ( X, A ) K g ( X, α ) K a ( X, α ) F hµ η (5.1)Here µ is the analytic index map in [20] and η : K top ∗ ( X, A ) → K a ∗ ( X, A ) is thenatural map in [2] , which is defined as follows: η ( M, φ, σ ) = φ ∗ ( P D ( σ )) (5.2)Moreover, we know that µ is an isomorphisms for compact manifolds and η isan isomorphism for locally finite CW -complexes. And we proved that F is anisomorphism for any locally finite CW -complexes. If we can show that diagram(5.1) is commutative, then we will get that h must also be an isomorphism. Proposition 5.1.
For any locally finite CW -complex X , the diagram (5.1) iscommutative.Proof. If we write the formula of the map in diagram (5.1) for a D -cycle( M, E, ι, S ) over (
X, α ) using the notation before, we get η ◦ h ( M, E, ι, S ) = ι ∗ ◦ ρ ∗ ◦ P D ◦ χ ◦ s ! ([ E ]) (5.3) µ ◦ F ( M, E, ι, S ) = ι ∗ ◦ η ∗ ◦ I ∗ ◦ P D ([ E ]) (5.4)Therefore the commutativity of diagram (5.1) is equivalent to ι ∗ ◦ ρ ∗ ◦ P D ◦ χ ◦ s ! = ι ∗ ◦ η ∗ ◦ I ∗ ◦ P D (5.5)14.e equivalent to the commutativity of the following diagram: K ( E ) K ∗ ( M, W ◦ τ ) K ∗ ( ˆ M , W ◦ υ ◦ ̺ )) K ∗ ( M, W ◦ υ ) K ∗ ( ˆ M , − α ◦ ι ◦ ρ ) K ∗ ( ˆ M , α ◦ ι ◦ ρ ) K ∗ ( M, α ◦ ι ) s ! P DχP D I ∗ η ∗ ρ ∗ (5.6)Here P D : K ∗ ( ˆ M , ( ι ◦ ρ ) ∗ ( A op ) → K ∗ ( ˆ M , ( ι ◦ ρ )( A )) is the P oincar ´ e dualitymap between twisted K -theory and twisted K -homology in [9] and [19]. By thenaturality of Poincarˇ e duality (one can see Corollary 3 . K ( M ) K ∗ ( M, W ◦ τ ) K ( ˆ M , W ◦ υ ◦ ρ ) K ∗ ( ˆ M , W ◦ τ ◦ ρ ) s ∗ ρ ∗ s ! P DP D (5.7)The twist of the lower right K -homology group is W ◦ τ ◦ ρ since ˆ M admits aspin c -structure. Since ρ ◦ s = Id , we have ρ ∗ ◦ s ∗ = id , therefore we can get P D = ρ ∗ ◦ s ∗ ◦ P D = ρ ∗ ◦ P D ◦ s ! (5.8)To prove the whole proposition, we first give the following lemma. Lemma 5.2.
Denote the map on twisted K -homology groups induced by chan-ging twist by ˜ χ : K ∗ ( ˆ M , W ◦ τ ◦ ρ ) → K ∗ ( ˆ M , α ◦ ι ◦ ρ ) . Then the followingdiagram is commutative K ∗ ( ˆ M , W ◦ υ ◦ ρ ) K ∗ ( ˆ M , W ◦ τ ◦ ρ ) K ∗ ( M, W ◦ τ ) K ∗ ( ˆ M , − α ◦ ι ◦ ρ ) K ∗ ( ˆ M , α ◦ ι ◦ ρ ) K ∗ ( M, α ◦ ι ) χ P DP D ˜ χ ρ ∗ ρ ∗ η ∗ ◦ I ∗ (5.9)If we combine the above lemma and diagram (5.7), we can get that diagram(5.1) is commutative. 15 orollary 5.3. If X is an smooth manifold and A is a twisting on X , then thecharge map h : K geo ∗ ( X, α ) → K top ∗ ( X, α ) is an isomorphism.Proof of (5.2) . • Step 1
We prove the first square in diagram ((5.9)) iscommutative. First of all, we review the definition of Poincar´ e duality P D : K ∗ ( ˆ M , A ) → K ∗ ( ˆ M , A op ) for twisted K -theory in Lemma 2 . P D ( x ) = σ C ( ˆ M, A op ) ( x ) (5.10)Here σ C ( ˆ M, A op ) : KK ( C ( ˆ M , A ) ˆ ⊗ A, B ) → KK ( A, C ( ˆ
M , A op ) ˆ ⊗ B ) is a ca-nonical isomorphism for any C ∗ -algebra A and B , which is given by tensor-ing with C ( ˆ M , A op ). If we choose A and B both to be C and C ( ˆ M , A ) tobe C ( ˆ M , ( W ◦ υ ◦ ρ ) ∗ ( K )), then we get the Poincar´ e duality P D on thetop of the first square . If we choose A and B to be C and C ( ˆ M , A ) tobe C ( ˆ M , ( − α ◦ ι ◦ ρ ) ∗ ( K )), then we get the Poincar´ e duality P D on thebottom of the first square. Then the commutativity of the first squarefollows from that
P D is natural over C ( ˆ M , A op ). • Step 2
From the definition of ˜ χ , we know that it is induced by changingthe twist from W ◦ τ ◦ ρ to α ◦ ι ◦ ρ using the trivialization given by thespinor bundle ρ ∗ S and the canonical trivialization of T M ⊕ υ . On the otherhand, we know that I ∗ is the changing twist map induced by the canonicaltrivialization of T M ⊕ υ and η ∗ is the map induced by the homotopy η ,which is induced by the spinor bundle S . So the commutativity of thesecond square follows from that the changing twist map is natural.In section 4, we proved that the geometric twisted K -homology group definedin [2] is homotopy invariant and each finite CW -complex is homotopy equivalentto a smooth manifold. Therefore we have that the charge map is an isomorphismfor any finite CW -complex. In this section we give another definition of geometric twisted K -homology viabundle gerbes. First of all, we give the definition of bundle gerbes first. Definition 6.1.
Given a CW -complex B , a bundle gerbe over B is a pair( P, Y ), where π : Y → B is a locally split map and P is a principal U (1)-bundleover Y × M Y with an associative product, i.e for every point ( y , y ) , ( y , y ) ∈ Y × M Y , there is an isomorphism P ( y ,y ) ⊗ C P ( y ,y ) → P ( y ,y ) (6.1)16nd the following diagram commutes P ( y ,y ) ⊗ P ( y ,y ) ⊗ P ( y ,y ) P ( y ,y ) ⊗ P ( y ,y ) P ( y ,y ) ⊗ P ( y ,y ) P ( y ,y ) (6.2)For every principal U (1)-bundle J over B we can define a bundle gerbe δ ( J ) by δ ( J ) ( y ,y ) = J y ⊗ J ∗ y . The product is induced by the pairing between J ∗ and J . A bundle gerbe ( P, Y ) is called trivial if there is a hermitian line bundle J such that P ∼ = δ ( J ). In this case, J is called a trivialization of ( P, Y ). For everybundle gerbe (
P, Y ) over M we can associate a third integer cohomology class d ( P ) ∈ H ( M, Z ) to describe the non-triviality of the bundle gerbe which wecall Dixmier-Douady class (see [16]). Definition 6.2.
Two bundle gerbes (
P, Y ) and (
Q, Z ) are stable isomorphic toeach other if there is a trivialization of p ∗ ( P ) ⊗ p ∗ ( Q ) ∗ . Here p : Y × B Z → Y and p : Y × B Z → Z are the natural projections. And the trivialization iscalled a stable isomorphism between ( P, Y ) and (
Q, Z ).The following theorem gives the relation between stable isomorphism classesand Dixmier-Douady classes.
Theorem 6.3.
Two bundle gerbes are stable isomorphic iff their Dixmier-Douady classes are equal to each other. Moreover, the Dixmier-Douady classdefines a bijection between stable isomorphic classes of bundle gerbes over M and H ( M, Z ) . Now we give another definition which is important in our construction of geo-metric twisted K -homology. Definition 6.4.
Let (
P, Y ) be a bundle gerbe over B . A finite dimensionalhermitian bundle E over Y is called a ( P, Y )-module if there is a complex vectorbundle isomorphism φ : P ⊗ π ∗ ( E ) ∼ = π ∗ ( E )which is compatible with the bundle gerbe product, i.e the following diagram iscommutative P ( y ,y ) ⊗ P ( y ,y ) ⊗ E y P ( y ,y ) ⊗ E y P ( y ,y ) ⊗ E y E y where π i ( i = 1 ,
2) are two projections from Y × B Y to Y . Moreover, theGrothendieck group of isomorphism classes of bundle gerbe module over ( P, Y )is called the K -group of ( P, Y ).Now we give the construction of geometric twisted K -cycles17 efinition 6.5. Let B be a space , H ∈ H tor ( X, Z ) and ( P, Y ) be a bundlegerbe over B with − H as Dixmier-Douady class. A geometric twisted K -cycleis a triple ( M, f, E ) where • M is a compact spin c -manifold; • f : M → B is continuous; • [ E ] is an isomorphism class of f ∗ ( P, Y )-module.We denote the whole twisted K -cycles over ( B, H ) by Γ ( P,Y ) ( B )To give twisted K -homology group, we also need to define an equivalence ∼ onthese geometric cycles as follows • Direct sum-disjoint union
For any two cycles (
M, f, E ) and ( M, f, E )over ( B, ( P, Y )), then we have(
M, f, [ E ]) ∪ ( M, f, [ E ]) ∼ ( M, f, [ E ] + [ E ]) (6.3) • Bordism
Given two cycles ( M , f , E ) and ( M , f , E ) over ( B, ( P, Y )),if there exists a cycle (
M, f, E ) over ( B, ( P, Y )) such that ∂M = − M ∪ M (6.4)and E ∂M = − [ E ] ∪ [ E ], then( M , f , E ) ∼ ( M , f , E ) (6.5) • Spin c -vector bundle modification Let (
M, f, E ) be a geometric twisted K -cycle over ( B, ( P, Y )) and an even dimensional spin c -vector bundle V over M . Let ˆ M to be the sphere bundle of V ⊕ R . Denote the bundlemap by ρ : ˆ M → M and the positive spinor bundle of T v ( ˆ M ) by S + V . Thevector bundle S + V ⊗ ρ ∗ ( E ) over ˆ M is a ( ρ ◦ f ) ∗ ( P, Y ) module. Then(
M, f, [ E ]) ∼ ( ˆ M , ρ ◦ f, [ S + V ⊗ ρ ∗ ( E )]) (6.6) Definition 6.6.
For any space B and bundle gerbe ( P, H ) over B . We define K ggi, ( P,Y ) ( B, H ) = Γ i ( B, ( P, Y )) / ∼ ( i =0, 1). The parity depends on the dimen-sion of the spin c -manifold in a twisted K -cycle. Proposition 6.7. If ( P, Y ) and ( Q, Z ) are two bundle gerbes over B with thesame Dixmier-Douady class − H , then we have K ggi, ( P,Y ) ( B, H ) is isomorphic to K ggi, ( Q,Z ) ( B, H ) .Proof. Let R be a stable isomorphism between ( P, Y ) and (
Q, Z ) i.e a trivi-alization of p ∗ ( P ) ⊗ p ∗ ( Q ). Without loss of generality we can just assumethat Z = Y . Otherwise we can consider the bundle gerbe ( p ∗ P, Y × B Z ) and( p ∗ Q, Y × B Z ) instead. Let ( M, f, [ E ]) ∈ Γ i ( B, ( P, Y )). Since Q ∼ = P ⊗ R ,therefore ( M, f, [ E ] ⊗ L R ) (here L R is the natural associated line bundle of R ) isa twisted geometric K -cycle over ( B, ( Q, Y )). So we get a homomorphism fromΓ( B, ( P, Y )) to Γ( B, ( Q, Z )), which we denote by r . A tedious check tells usthat r respects disjoint union, bordism and spin c -bundle modification. There-fore r induces a homomorphism from K ggi ( B, ( P, Y )) to K ggi ( B, ( Q, Y )). If we18hange the roles of (
P, Y ) and (
Q, Z ) in the above construction, then we get aninverse of r . Therefore r is an isomorphism.Let ( P, Y ) be a bundle gerbe over B with Dixmier-Douady class H . Accordingto Proposition 6.4 in [5], the i -th K -group of bundle gerbe ( P, Y ) is isomorphicto the i -th twisted K -group K i ( X, − H ). Then the definitions of K gg ∗ ( X, H ) and K top ∗ ( X, α ) implies the following proposition:
Proposition 6.8.
Let X be a finite CW -complex and H ∈ H torsion ( X, Z ) .Then we have K ggi ( X, ( P, Y )) ∼ = K topi ( X, α ) ∼ = K geoi ( X, α ) (6.7)
First of all we review notions about topological T -duality in [7]. Let π : P → B and ˆ π : ˆ P → B be principal S -bundles over B and H ∈ H ( P, Z ) , ˆ H ∈ H ( ˆ P , Z ). Denote the associated line bundles of P and ˆ P by E and ˆ E re-spectively. Let V = E L ˆ E and r : S ( V ) → B be the unit sphere bundle of V . Definition 7.1.
A class Th ∈ H ( S ( V ) , Z ) is called Thom class if r ! ( Th ) =1 ∈ H ( B, Z ).Let i : P → S ( V ) and ˆ i : ˆ P → S ( V ) be inclusion of principal S -bundle into S -bundle. Definition 7.2.
We say that ((
P, H ) , ( ˆ P , ˆ H )) is a T -dual pair over B or ( P, H )and ( ˆ
P , ˆ H ) are T -dual to each other if there exists a Thom class Th ∈ H ( S ( V ) , Z )such that H = i ∗ ( Th ) , ˆ H = ˆ i ∗ ( Th ) (7.1)Given a T -dual pair (( P, H ) , ( ˆ P , ˆ H )) over B , there exists a T -duality isomorph-ism between the associated twisted K -groups, which is given by T = ˆ p ! ◦ u ◦ p ∗ : K i ( P, H ) → K i − ( ˆ P , ˆ H ) (7.2)Here p ∗ and ˆ p ! are the pullback and pushforward maps respectively. u is achanging twist map from K i ( P × B ˆ P , p ∗ ( H )) to K i ( P × B ˆ P , ˆ p ∗ ( ˆ H )). In orderto establish the T -duality transformation of geometric twisted K -homology wefirst give the construction of analogue maps of induced map , wrong way mapand changing twist map for twisted K -homology.1. Induced map
Assume f : ( X , Y ) → ( X , Y ) is a continuous mapbetween two pairs of topological spaces. Then the induced map f ∗ : K gi ( X , Y ; α ◦ f ) → K gi ( X , Y ; α ) is defined as follows: f ∗ ([ M, φ, υ, η, E ]) = [(
M, φ ◦ f, υ, η, E )] (7.3)19. Wrong way map
Let f : P → N be a K -oriented map between smoothmanifolds and α : N → K ( Z ,
3) be a twist over N . we define the wrongway map f ! : K gi − ( N, α ) → K gi ( P, α ◦ f ) to be π ! ([ M, φ, υ, η, E ]) = [( ˜
M , ˜ φ, ˜ υ, ˜ η, ˜ π ∗ ( E ))] (7.4)Here ˜ M is the fiber product M × N M , ˜ υ is the stable normal bundle of˜ M and ˜ η is induced by η . Remark . As f is K -oriented, therefore W (˜ υ ⊕ f ∗ ( υ )) is trivial, whichimplies that W ◦ ˜ υ is homotopic to α ◦ f ◦ ˜ φ via a homotopy λ . ˜ η :˜ M × [0 , → K ( Z ,
3) is given by the combination of λ and η as follows:˜ η ( x, t ) = ( λ ( x, t ) , ≤ t ≤ / η ◦ ( f ′ × id )( x, t − , / ≤ t ≤ . in which f ′ : ˜ M → M is the canonical projection to M . Therefore weget that ( ˜ M , ˜ φ, ˜ υ, ˜ η, ˜ π ∗ ( E )) is a twisted geometric cycle over ( P, α ◦ f ). Inparticular, when f is the bundle map for a principal S -bundle, ˜ M is thepullback S -bundle along f .3. Changing twist map
Let P → B be a principal T -bundle over B . Ifwe have two twist α , α : P → K ( Z ,
3) and their associated cohomologyclasses are the same, then α and α are homotopic. Choose a homotopy h such that the restriction of h to each fiber of P corresponds to thecohomology class θ ∪ ˆ θ ∈ H ( P b , Z ). Here θ and ˆ θ are generators ofthe first cohomology group of the two copies of S of a fiber. Then fora geometric cycle δ of ( X, α ) one can define the changing twist map u : K gi ( P, α ) → K gi ( P, α ) as follows: u ([ M, φ, υ, η, E ]) = [(
M, φ, υ, ˆ η, E )] (7.5)Here ˆ η is induced by the following diagram: W ◦ υ α ◦ φ α ◦ φη h ◦ ( φ × id )More explicitly, the ˆ η is given by the composition of η and h ◦ ( φ × id ),which we denote by ( h ◦ ( φ × id )) ∗ η ( h ◦ ( φ × id ))( η )( x, t ) = ( η ( x, t ) , ≤ t ≤ / h ◦ ( φ × id )( x, t − , / ≤ t ≤ . Lemma 7.4.
The induced map, wrong way map and changing twist map aboveare all compatible with disjoint union, bordism and spin c -vector bundle modific-ation.Proof. We have proved the induced map part in Lemma 4.1 and it is not hardto see that they all respect the disjoint union. We do the rest here. • Let (
M, φ, υ, η, E ) be a bordism between ( M , φ , υ , η , E ) and ( M , φ ,υ , η , E ) over X . Denote p ! ( M i , φ i , υ i , η i , E i ) by ( ˜ M i , ˜ φ i , ˜ υ i , ˜ η i , ˜ π ∗ E i ).20ince the boundary of a pullback space is the pullback of the originalboundary and the stable normal bundle of a principal S -bundle is iso-morphic to the stable normal bundle of the base space, we get that ( ˜ M , ˜ φ, ˜ υ, ˜ η, ˜ π ∗ E ) gives a bordism between ( ˜ M , ˜ φ , ˜ υ , ˜ η , ˜ π ∗ E ) and ( ˜ M , ˜ φ , ˜ υ , ˜ η , ˜ π ∗ E ).Use the notation above. p ! ( ˆ M , φ ◦ ρ, υ ′ , η ′ , S + V ⊗ ρ ∗ E ) is ( ˜ˆ M, φ ◦ ρ ◦ ˜ˆ π, ˜ υ ′ , ˜ η ′ , ˜ π ∗ ( S + V ⊗ ρ ∗ E )). On the other hand, ˜ π ∗ V is a spin c -vector bundle over˜ M . The associated spin c -vector bundle modification of ( ˜ M , ˜ φ, ˜ υ, ˜ η, ˜ π ∗ E )is that ( ˜ˆ M, ˜ φ ◦ ˜ ρ, υ ′′ , ˜ η ′ , ˜ π ∗ ( S + V ⊗ ρ ∗ E )). The maps in the above twistedgeometric K -cycles are shown in the following diagram:˜ˆ M ˜ M ˆ MM ˜ ρ ˜ˆ π ˜ π ρ (7.6)By the commutativity, we have that φ ◦ ρ ◦ ˜ˆ π = ˜ φ ◦ ˜ ρ . Moreover, ˜ υ ′ and υ ′′ are homotopic because they are both classifying maps of the stablenormal bundle of ˜ˆ M . The coincidence of ˜ υ ′ and υ ′′ implies that ˜ η ′ and ˜ η ′ are homotopic to each other. So the wrong way map respects the spin c -vector bundle modification relation. • Use the above notions. u (( M, φ, υ, η, E )) still gives a bordism between( M , φ , υ , ˆ η , E ) and ( M , φ , υ , ˆ η , E ) i.e u respects bordism equival-ence. Since the composition of homotopies are associative up to homotopy,we get that u respects the spin c -vector bundle modification relation.21 T-duality for twisted geometric K-homology
Theorem 8.1.
Let B be a finite CW -complex and (( P, H ) , ( ˆ P , ˆ H )) be a T -dualpair over B . P × B ˆ PP ˆ PBp ˆ pπ ˆ π Moreover, we assume that α : P → K ( Z , and ˆ α : ˆ P → K ( Z , satisfythat α ∗ ([Θ]) = H and ˆ α ∗ ([Θ]) = ˆ H (Here [Θ] is the positive generator of H ( K ( Z , , Z ) ). Moreover we assume that both α and [ ˆ α ] is representable.Then the map T = ˆ p ∗ ◦ u ◦ p ! : K g ∗ ( P, α ) K g ∗ +1 ( ˆ P , ˆ α ) is an isomorphism. To prove the theorem (8.1) we need the following lemmas.
Lemma 8.2. T is compatible with the boundary operator and the induced mapin the Mayer-Vietoris sequence.Proof. We first prove the compatibility with the induced map. Assume we havea map f : X Y and we have the associated T -duality diagrams over Y and pullback it to X . f induces maps by F : P X P Y , ˆ F : ˆ P X → ˆ P Y and G : P X × X ˆ P X → P Y × Y ˆ P Y . Then we have the following identities:ˆ F ∗ ◦ T X = ˆ F ∗ ◦ (ˆ p X ) ∗ ◦ u X ◦ p ! X = (ˆ p Y ) ∗ ◦ G ∗ ◦ u X ◦ p ! X = (ˆ p Y ) ∗ ◦ u Y ◦ ( G ◦ p Y ) ! ◦ F ∗ = T Y ◦ F ∗ (8.1)Now we turn to the compatibility with the boundary map, in the Mayer-Vietorissequence of the boundary operator δ : K g ∗ ( X, α ) → K g ∗ +1 ( U ∩ V, α ◦ i U ∩ V ) is givenas follows: Choose a continuous map f : X → [0 ,
1] such that f U − U ∩ V is 0 and f V − U ∩ V is 1. Without loss of generality we assume that f ◦ φ : M → [0 ,
1] is asmooth function and 1 / f ◦ φ . For any twisted geometric K -cycle x = ( M, φ, υ, η, E ), define δx = ( f − (1 / , φ ◦ i, υ ◦ i, η ◦ ( i × id ) , i ∗ E ). Bythis formula, we get that δ is compatible with induced map. Also the homotopies( h ◦ ( φ ◦ i × id )) ∗ ( η ◦ ( i × id )) and ( η ∗ ( h ◦ ( φ × id )) ◦ ( i × id ) are homotopicto each other, which implies that u ◦ δ = δ ◦ u . The remainder is to show thatˆ p ! ◦ δ = δ ◦ ˆ p ! . We write both sides explicitly first:Given a principal S -bundle π : P → B and a twisted geometric cycle ( M, φ, υ, η, E )(which we denote by x ) over P ˆ p ! ◦ δx = ( ˜( f ◦ φ ) − (1 / , φ ◦ ˜ π ◦ i, ˜ υ ◦ i, η ◦ ((˜ π ◦ i ) × id ) , ( i ◦ ˜ π ) ∗ E ); δ ◦ ˆ p ! x = (( f ◦ φ ◦ ˜ π ) − (1 / , φ ◦ ˜ π ◦ i, ˜ υ ◦ i, η ◦ ((˜ π ◦ i ) × id ) , ( i ◦ ˜ π ) ∗ E )22ince ˜ f ◦ φ − (1 /
2) is exactly ( f ◦ φ ◦ ˜ π ) − (1 / p ! ◦ δ = δ ◦ ˆ p ! .Finally, we have that T ◦ δ = (ˆ p ! ◦ u ◦ p ∗ ) ◦ δ = δ ◦ (ˆ p ! ◦ u ◦ p ∗ ) = δ ◦ T Lemma 8.3.
Proof of Theorem 8.1.
We do the proof by induction on the number of cells.Assume X is a point, then P and ˆ P are both S and the correspondence spaceis S × S . Denote p ∗ ◦ t − ◦ ˆ p by T ′ . We can see that for any geometric cycle( M, ι, υ, η, [ E ]), the image of this cycle under the map T ′ ◦ T is ( M × S × S , ι ◦ ˘ p, υ ◦ ˘ p, ˘ η, ˘ p ∗ ([ E ])). Here ˘ p is the projection from M × S × S to M .One can see that this is cobordant to a trivial S -bundle as spin c -manifolds.The cobordism can be given by M × B , where B is a solid torus with a soliddisk cut from it. As all of the bundles involved are trivial, the geometric cycle( M × B, ι ◦ ´ p, υ ◦ ´ p, ´ η, ´ p ∗ ([ E ]) gives the cobordism between the image of T and thespin c -modification i.e T ′ ◦ T is equal to identity in this case. As a consequencewe also get that T is an isomorphism. Assume T is an isomorphism when thenumber of cells is n ,then we adjoin another cell σ n +1 to X and we choose openset U = X ∪ σ n +1 − pt, V = σ n +1 − ¯ pt and we can get the Mayer-Vietorissequence of geometric twisted K -homology groups. Then the conclusion of thistheorem follows from induction and the Five-Lemma. Remark . The construction of T -duality transformation of geometric twisted K -homology can be easily generalized to T -dual pairs of higher dimensionaltorus bundles.We end up with this paper with an interesting question. Can we release thecondition in Remark 2.7 and construct geometric twisted K -theory for generaltwists? An idea is to replace twisted spin c -manifolds in the construction ofgeometric K -cycles by noncommutative analogue objects. We leave this forfurther investigation. References [1] M. Atiyah and G. Segal, Twisted K -theory, arXiv:math/0407054v2 (2005)[2] P. Baum, A. Carey and B.L. Wang, K -cycles for twisted K -homology. J. K -Theory , (2013), 69-98[3] P. Baum, N. Higson and T. Schick, On the equivalence of Geometric andAnalytic K -homology, Pure and Applied Mathematics Quarterly 3 , 1-24(2009)[4] P. Baum, N. Higson and T. Schick, A geometric description of equivariant K -homology for proper actions. Quanta of Maths (2009)[5] P. Bouwknegt, A. Carey, V. Mathai, W. Murray and D. Stevenson, twisted K -theory and K -theory of bundle gerbes, Comm. Math. Phys . , 17-45(2002) 236] P. Bouwknegt, J. Evslin and V. Mathai. T-duality: Topological chargefrom H-flux. Comm. Math. Phys . (2004),383[7] U. Bunke and T. Schick, on the topology of T-duality. Rev. Math. Phys . (2005)[8] A. Connes and G. Skandalis, The longitudinal index theorem for foliations. Publ. RIMS, Kyoto Univ . (1984), 1139-1183[9] S. Echterhoff, H. Emerson, and H. J. Kim, KK -theoretic duality for propertwisted actions, Mann. Ann . (2008) 839-873[10] B. Lawson and M.L. Michelsohn, Spin Geometry, Princeton UniversityPress (1989)[11] N. Higson and J. Roe, Analytic K -homology, Oxford Science Publications [12] M. Karoubi, Lectures on K -theory.[13] G.G. Kasparov, Equivariant KK -theory and the Novikov conjectures, In-vent.Math . (1988)[14] A. Kapustin and E. Witten, Electric-Magnetic Duality and the GeometricLanglands Program, axXiv: hep-th/0604151v3 (2007)[15] J. P. May, A Concise Course in Algebraic Topology, University of ChicagoPress (1999)[16] M.K. Murray, Bundle Gerbes, arXiv: dg-ga/94070151v1 (1994)[17] R. J. Plymen, Strong Morita equivalence, spinors and symplectic spinors,
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