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Celebratio Mathematica

Paul Baum

Twisted K-homology and Paul Baum

by Alan L. Carey

1. Preamble

It may seem hard to be­lieve but I am not ex­actly sure when I first met Paul. However the oc­ca­sion of a meet­ing or­gan­ised by Dav­id Evans in 1987 in War­wick may well qual­i­fy. Our first sub­stan­tial math­em­at­ic­al in­ter­ac­tion, however, did not oc­cur un­til an Ox­ford meet­ing or­gan­ised by John Roe in the early 90s. There Paul, Var­ghese Math­ai and I en­joyed a num­ber of en­ter­tain­ing con­ver­sa­tions and Paul made some very help­ful com­ments which played a role in se­cur­ing Math­ai a pres­ti­gi­ous Aus­trali­an Re­search Coun­cil Fel­low­ship soon after. Paul has been to Aus­tralia on sev­er­al oc­ca­sions vis­it­ing Sydney, Mel­bourne, New­castle and fi­nally Can­berra. He achieved some no­tori­ety for break­ing in­to song at the Aus­trali­an Math­em­at­ic­al So­ci­ety con­fer­ence din­ner in La Trobe Uni­versity in 2007. He was in very good spir­its after giv­ing the best plen­ary of the meet­ing a day or two be­fore­hand. Paul was awar­ded an hon­or­ary doc­tor­ate from the Aus­trali­an Na­tion­al Uni­versity in 2013 in re­cog­ni­tion of his lengthy and sig­ni­fic­ant sup­port of Aus­trali­an math­em­aticians.

Our sub­stan­tial re­search in­ter­ac­tion began in Texas Chris­ti­an Uni­versity at a meet­ing in hon­our of Jonath­an Rosen­berg. Be­fore launch­ing in­to that it is use­ful to re­call some his­tory of twis­ted K-the­ory and this will oc­cupy the next sec­tion. After that I will cov­er the ba­sics on twis­ted K-the­ory. Then in the main sec­tion of the pa­per I will de­scribe twis­ted geo­met­ric cycles from two view­points, BB (be­fore Baum) and AB (after Baum). The story ends on a happy note.

2. History

The story brings to­geth­er a num­ber of math­em­at­ic­al top­ics: gerbes, con­tinu­ous trace \( C^\ast \) al­geb­ras, K-ho­mo­logy (both ana­lyt­ic and geo­met­ric) and \( D \)-branes from string the­ory. This is a broad set of top­ics but they all fit in­to the jig­saw puzzle.

Con­tinu­ous trace al­geb­ras have their roots in work of Grothen­dieck [e3] and Dixmi­er and Douady [e1], who stud­ied bundles of al­geb­ras over a para­com­pact space \( X \) with fibre either a mat­rix al­gebra or the com­pact op­er­at­ors on an in­fin­ite-di­men­sion­al Hil­bert space. They are char­ac­ter­ised by an as­so­ci­ated in­vari­ant; a class in de­gree-3 Čech co­homo­logy of the base was iden­ti­fied (it is now known as the Dixmi­er–Douady class). From these pa­pers two in­de­pend­ent lines of de­vel­op­ment began. The first, the the­ory of gerbes, began with [e5] and was ex­ten­ded and ap­plied to prob­lems in string the­ory in [e13].

The oth­er line of de­vel­op­ment is the the­ory of con­tinu­ous trace \( C^\ast \)-al­geb­ras. These are al­geb­ras of sec­tions of the bundles in­tro­duced by Dixmi­er and Douady. Their his­tory can be found in the mono­graph of Rae­burn and Wil­li­ams [e17].

Twis­ted K-the­ory now comes in­to the pic­ture. Giv­en a para­com­pact space \( X \) with a “twist”, namely a de­gree-3 co­homo­logy class \( [\alpha] \) on \( X, \) there is a prin­cip­al bundle with fibre the pro­ject­ive unit­ary group of a sep­ar­able Hil­bert space over \( X \) char­ac­ter­ised by \( [\alpha]. \) Now re­gard the pro­ject­ive unit­ary group as the auto­morph­ism group of the com­pact op­er­at­ors on this Hil­bert space. Then, fol­low­ing Rosen­berg [e10], the twis­ted K-the­ory of \( X \) is the K-the­ory of the con­tinu­ous trace \( C^\ast \)-al­gebra of sec­tions of the bundle over \( X \) as­so­ci­ated to the giv­en prin­cip­al bundle with twist \( [\alpha]. \) The twist only spe­cifies an equi­val­ence class of con­tinu­ous trace al­geb­ras, so this means that twis­ted K-the­ory with twist \( [\alpha] \) is only defined by this meth­od up to an iso­morph­ism.

These two dis­par­ate lines formed the mo­tiv­a­tion for the work of Mur­ray [e15] on bundle gerbes. Bundle gerbes provide a dif­fer­en­tial geo­met­ric way to ap­proach twis­ted K-the­ory, as ex­plained in [e24], that is closely aligned with the geo­metry that is used in string the­ory.

The ad­di­tion­al in­gredi­ent in this pic­ture that we need from the string the­ory side is the no­tion of \( D \)-branes [e14]. They were pro­posed as a mech­an­ism for provid­ing bound­ary con­di­tions for the dy­nam­ics of open strings mov­ing in space-time. Ini­tially they were thought of as sub­man­i­folds. As \( D \)-branes them­selves can evolve over time, one needs to study equi­val­ence re­la­tions on the set of \( D \)-branes. An in­vari­ant of the equi­val­ence class is the to­po­lo­gic­al charge of the \( D \)-brane, which should be thought of as an ana­logue of the Dir­ac mono­pole charge as these \( D \)-brane charges are as­so­ci­ated with gauge fields (con­nec­tions) on vec­tor bundles over the \( D \)-brane. These vec­tor bundles are known as Chan–Paton bundles.

In [e16] Minas­i­an and Moore made the pro­pos­al that \( D \)-brane charges should take val­ues in K-groups and not in the co­homo­logy of the space-time or the \( D \)-brane. However, they pro­posed a co­homo­lo­gic­al for­mula for these charges which might be thought of as a kind of in­dex the­or­em in the sense that, in gen­er­al, in­dex the­ory as­so­ci­ates to a K-the­ory class a num­ber which is giv­en by an in­teg­ral of a closed dif­fer­en­tial form. In string the­ory there is an ad­di­tion­al field on space-time known as the \( H \)-flux which may be thought of as a glob­al closed 3-form. Loc­ally it is giv­en by a fam­ily of “2-form po­ten­tials” known as the \( B \)-field. Math­em­at­ic­ally these \( B \)-fields are in fact de­fin­ing a de­gree-3 in­teg­ral Čech class on the space-time, that is, the “twist”. Wit­ten [e18], ex­tend­ing [e16], gave a phys­ic­al ar­gu­ment for the idea that \( D \)-brane charges should be ele­ments of K-groups and, in ad­di­tion, pro­posed that the \( D \)-brane charges in the pres­ence of a twist should take val­ues in twis­ted K-the­ory (at least in the case where the twist is tor­sion). The math­em­at­ic­al ideas he re­lied on were due to Donovan and Ka­roubi [e4]. Sub­sequently Bouwknegt and [e20] ex­ten­ded Wit­ten’s pro­pos­al to the non­tor­sion case us­ing ideas from [e10]. A geo­met­ric mod­el (that is, a “string geo­metry” pic­ture) for some of these string the­ory con­struc­tions and for twis­ted K-the­ory was pro­posed in [e24] us­ing the no­tion of bundle gerbes and bundle gerbe mod­ules. Vari­ous re­fine­ments of twis­ted K-the­ory that are sug­ges­ted by these ap­plic­a­tions are also de­scribed in the art­icle of Atiyah and Segal [e26] and their ideas play a role in the next de­vel­op­ment.

2.1 Mathematical perspective

The con­tri­bu­tions of phys­i­cists raise some im­me­di­ate ques­tions. When there is no twist, the re­la­tion­ship between K-the­ory and in­dex the­ory of el­lipt­ic op­er­at­ors is now well-es­tab­lished, in­clud­ing the geo­met­ric cycle ap­proach to K-ho­mo­logy. In fact one ver­sion of the Atiyah–Sing­er in­dex the­or­em due to Baum, Hig­son, and Schick [3] es­tab­lishes a re­la­tion­ship between the ana­lyt­ic view­point provided by el­lipt­ic dif­fer­en­tial op­er­at­ors and the geo­met­ric view­point provided by the no­tion of geo­met­ric cycle in­tro­duced in the fun­da­ment­al pa­per of Baum and Douglas [1]. The view­point that geo­met­ric cycles in the sense of [1] are a mod­el for \( D \)-branes in the un­twis­ted case is ex­pounded in [e29], [e36], [e34]. Note that in this view­point \( D \)-branes are no longer sub­man­i­folds (as en­vis­aged ori­gin­ally by phys­i­cists) but the im­ages of man­i­folds un­der a smooth map.

It is thus tempt­ing to con­jec­ture that there is an ana­log­ous pic­ture of \( D \)-branes as a type of geo­met­ric cycle in the twis­ted case as well. More pre­cisely the ques­tion is wheth­er there is a way to for­mu­late the no­tion of “twis­ted geo­met­ric cycle” (in ana­logy with [2] and [1]) and to prove an in­dex the­or­em in the spir­it of [3] for twis­ted K-ho­mo­logy. An ap­proach to an­swer­ing this ques­tion was pro­posed in [e33], though there is a step in the proof that is omit­ted. Wang un­der­stood that the miss­ing step re­quires a twis­ted ver­sion of the Con­ner–Floyd split­ting the­or­em [e7] for \( \mathrm{Spin}^c \)-bor­d­ism; however a com­plete ver­sion of Bai-Ling Wang’s ar­gu­ment has not ap­peared. The good news is that there is an ar­gu­ment that is sim­il­ar in spir­it to what Wang pro­posed due to Paul and col­lab­or­at­ors that over­comes not just this one dif­fi­culty but provides a con­struc­tion of twis­ted K-ho­mo­logy for CW-com­plexes [5]. In the last sec­tion we will de­scribe briefly this new ap­proach that is in pre­par­a­tion. It is im­port­ant to em­phas­ise that string geo­metry ideas from [e19] played a key role in find­ing the cor­rect way to gen­er­al­ise [2].

In this en­ter­prise, Paul entered the pic­ture via his in­ter­ac­tion with Bai-Ling Wang over the ideas in­tro­duced in [e33]. I want to ex­plain this in de­tail in this es­say. I will also fill in many of the de­tails for the mat­ters dis­cussed pre­vi­ously in this in­tro­duc­tion. I have bor­rowed very freely from a re­view art­icle that I wrote with Bai-Ling Wang after our meet­ing with Paul in Texas Chris­ti­an Uni­versity [e37] (though wheth­er it was an epi­phany is hard to de­term­ine).

It is es­sen­tial to re­mark that none of the ex­ist­ing ar­gu­ments that are used in the geo­met­ric cycle ap­proach to in­dex the­ory (the gen­er­al in­dex prob­lem of Paul) ex­tend to cov­er the twis­ted ver­sion for CW-com­plexes. In the fi­nal sec­tion I am pleased to an­nounce that there is now a very gen­er­al ar­gu­ment that ap­plies to all in­stances.

This art­icle con­tains noth­ing ori­gin­al but does draw to­geth­er the many threads that have led us to the present un­der­stand­ing of K-ho­mo­logy in the twis­ted case.

3. Topological and analytic twisted K-theory

Though our fo­cus is twis­ted K-ho­mo­logy, in the lit­er­at­ure on this top­ic, ex­tens­ive use is made of Poin­caré du­al­ity. For this reas­on it is im­port­ant to re­view first some as­pects of twis­ted K-the­ory.

We be­gin with the no­tion of a “twist­ing”. Let \( \mathcal{H} \) be an in­fin­ite-di­men­sion­al, com­plex and sep­ar­able Hil­bert space. We shall con­sider loc­ally trivi­al prin­cip­al \( \mathrm{PU}(\mathcal{H}) \)-bundles over a para­com­pact Haus­dorff to­po­lo­gic­al space \( X; \) the struc­ture group \( \mathrm{PU}(\mathcal{H}) \) is equipped with the norm to­po­logy. The pro­ject­ive unit­ary group \( \mathrm{PU}(\mathcal{H}) \) with the to­po­logy in­duced by the norm to­po­logy on \( U(\mathcal{H}) \) (see [e2]) has the ho­mo­topy type of an Ei­len­berg–MacLane space \( K(\mathbb{Z}, 2). \) The clas­si­fy­ing space of \( \mathrm{PU}(\mathcal{H}), \) de­noted by \( \mathrm{BPU}(\mathcal{H}), \) is a \( K(\mathbb{Z}, 3). \) The set of iso­morph­ism classes of prin­cip­al \( \mathrm{PU}(\mathcal{H}) \)-bundles over \( X \) is giv­en by (Pro­pos­i­tion 2.1 in [e26]) ho­mo­topy classes of maps from \( X \) to any \( K(\mathbb{Z},3) \) and there is a ca­non­ic­al iden­ti­fic­a­tion \[ [X, \mathrm{BPU}(\mathcal{H})] \cong H^3(X, \mathbb{Z}). \]

A twist­ing of com­plex K-the­ory on \( X \) is giv­en by a con­tinu­ous map \[ \alpha: X\to K(\mathbb{Z}, 3). \] For such a twist­ing, we can as­so­ci­ate a ca­non­ic­al prin­cip­al \( \mathrm{PU}(\mathcal{H}) \)-bundle \( \mathcal{P}_\alpha \) through the usu­al pull-back con­struc­tion from the uni­ver­sal \( \mathrm{PU}(\mathcal{H}) \) bundle de­noted by \( EK(\mathbb{Z}, 2), \) as sum­mar­ised by the dia­gram \begin{equation}\label{bundle} \begin{CD} \mathcal{P}_\alpha @>{{}}>{{}}> EK(\mathbb{Z}, 2) \\ @VVV @VVV \\ X @>{{}}>{{\alpha }}> K(\mathbb{Z}, 3) \end{CD} \end{equation} We will use \( \mathrm{PU}(\mathcal{H}) \) as a group mod­el for a \( K(\mathbb{Z},2). \) We write \( \mathbf{Fred}(\mathcal{H}) \) for the con­nec­ted com­pon­ent of the iden­tity of the space of Fred­holm op­er­at­ors on \( \mathcal{H} \) equipped with the norm to­po­logy. There is a base-point-pre­serving ac­tion of \( \mathrm{PU}(\mathcal{H}) \) giv­en by the con­jug­a­tion ac­tion of \( U(\mathcal{H}) \) on \( \mathbf{Fred}(\mathcal{H}): \) \begin{equation}\label{action} \mathrm{PU}(\mathcal{H}) \times \mathbf{Fred} (\mathcal{H}) \longrightarrow \mathbf{Fred} (\mathcal{H}). \end{equation}

The ac­tion \eqref{action} defines an as­so­ci­ated bundle over \( X \) which we de­note by \[ \mathcal{P}_\alpha (\mathbf{Fred}) = \mathcal{P}_\alpha\times_{\mathrm{PU}(\mathcal{H})} {\mathbf{Fred}}(\mathcal{H}). \] We write \[ \{ \Omega^n_X \mathcal{P}_\alpha(\mathbf{Fred}) = \mathcal{P}_\alpha\times_{\mathrm{PU}(\mathcal{H})} \Omega^n \mathbf{Fred} \} \] for the fibre-wise it­er­ated loop spaces.

Defin­i­tion: The (to­po­lo­gic­al) twis­ted K-groups of \( (X, \alpha) \) are defined to be \[ K^{-n}(X, \alpha) := \pi_0( C_c(X, \Omega^n_X \mathcal{P}_\alpha(\mathbf{Fred}))), \] the set of ho­mo­topy classes of com­pactly sup­por­ted sec­tions (mean­ing they are the iden­tity op­er­at­or in \( \mathbf{Fred} \) off a com­pact set) of the bundle of \( \mathcal{P}_\alpha (\mathbf{Fred}). \)

Due to Bott peri­od­icity, we only have two dif­fer­ent twis­ted K-groups \( K^0(X, \alpha) \) and \( K^1(X, \alpha). \) Giv­en a closed sub­space \( A \) of \( X, \) \( (X, A) \) is a pair of to­po­lo­gic­al spaces, and we define re­l­at­ive twis­ted K-groups to be \[ K^{\mathrm{ev}/\mathrm{odd}}(X, A; \alpha) := K^{\mathrm{ev}/\mathrm{odd}}(X-A, \alpha). \]

Take a pair of twist­ings \[ \alpha_0, \alpha_1: X \longrightarrow K(\mathbb{Z}, 3), \] and a map \[ \eta: X\times [1, 0] \longrightarrow K(\mathbb{Z}, 3) \] which is a ho­mo­topy between \( \alpha_0 \) and \( \alpha_1, \) rep­res­en­ted dia­gram­mat­ic­ally by the fig­ure shown to the right. Then there is a ca­non­ic­al iso­morph­ism \( \mathcal{P}_{\alpha_0} \cong \mathcal{P}_{\alpha_1} \) in­duced by \( \eta. \) This ca­non­ic­al iso­morph­ism de­term­ines a ca­non­ic­al iso­morph­ism on twis­ted K-groups \begin{equation}\label{iso:eta} \eta_*: K^{\mathrm{ev}/\mathrm{odd}}(X, \alpha_0) \xrightarrow{\,\cong\ } K^{\mathrm{ev}/\mathrm{odd}}(X, \alpha_1). \end{equation} This iso­morph­ism \( \eta_* \) de­pends only on the ho­mo­topy class of \( \eta. \) The set of ho­mo­topy classes of maps between \( \alpha_0 \) and \( \alpha_1 \) is la­belled by \( [X, K(\mathbb{Z}, 2)]. \) Re­call the first Chern class iso­morph­ism \[ {\mathbf{Vect}}_1(X) \cong [X, K(\mathbb{Z}, 2)] \cong H^2(X, \mathbb{Z}) , \] where \( \mathbf{Vect}_1(X) \) is the set of equi­val­ence classes of com­plex line bundles on \( X. \) We re­mark that the iso­morph­isms in­duced by two dif­fer­ent ho­mo­top­ies between \( \alpha_0 \) and \( \alpha_1 \) are re­lated through an ac­tion of com­plex line bundles.

Let \( \mathcal{K} \) be the \( C^* \)-al­gebra of com­pact op­er­at­ors on \( \mathcal{H}. \) The iso­morph­ism \[ \mathrm{PU}(\mathcal{H}) \cong \mathrm{Aut}( \mathcal{K}) \] via the con­jug­a­tion ac­tion of the unit­ary group \( U(\mathcal{H}) \) provides an ac­tion of a \( K(\mathbb{Z}, 2) \) on the \( C^* \)-al­gebra \( \mathcal{K}. \) Hence, any \( K(\mathbb{Z}, 2) \)-prin­cip­al bundle \( \mathcal{P}_\alpha \) defines a loc­ally trivi­al bundle of com­pact op­er­at­ors, de­noted by \[ \mathcal{P}_\alpha(\mathcal{K}) = \mathcal{P}_\alpha\times_{\mathrm{PU}(\mathcal{H})} \mathcal{K}. \]

Let \( \Gamma_0(X, \mathcal{P}_\alpha(\mathcal{K})) \) be the \( C^* \)-al­gebra of sec­tions of \( \mathcal{P}_\alpha(\mathcal{K}) \) van­ish­ing at in­fin­ity. Then \( \Gamma_0(X, \mathcal{P}_\alpha(\mathcal{K}) \) is the (unique up to iso­morph­ism) stable sep­ar­able com­plex con­tinu­ous-trace \( C^* \)-al­gebra over \( X \) with Dixmi­er–Douady class \( [\alpha] \in H^3(X, \mathbb{Z}) \) (here we identi­fy the Čech co­homo­logy of \( X \) with its sin­gu­lar co­homo­logy; see [e10] and [e26]).

The­or­em 1 ([e26],[e10]): The to­po­lo­gic­al twis­ted K-groups \( K^{\mathrm{ev}/\mathrm{odd}}(X, \alpha) \) are ca­non­ic­ally iso­morph­ic to ana­lyt­ic K-the­ory of the \( C^* \)-al­gebra \( \Gamma_0(X, \mathcal{P}_\alpha(\mathcal{K})) \) \[ K^{\mathrm{ev}/\mathrm{odd}}(X, \alpha) \cong K_{\mathrm{ev}/\mathrm{odd}} (\Gamma_0(X, \mathcal{P}_\alpha(\mathcal{K}))), \] where the lat­ter group is the K-the­ory of \( \Gamma_0(X, \mathcal{P}_\alpha(\mathcal{K})), \) defined to be \[ \varinjlim_{k\to \infty} \pi_1\bigl(\mathrm{GL}_k(\Gamma_0(X, \mathcal{P}_\alpha(\mathcal{K})))\bigr). \] Note that the K-the­ory of \( \Gamma_0(X, \mathcal{P}_\alpha(\mathcal{K})) \) is iso­morph­ic to Kas­parov’s \( KK \)-the­ory [e6], [e9], [e8] \[ KK^{\mathrm{ev}/\mathrm{odd}}(\mathbb{C}, \Gamma_0(X, \mathcal{P}_\alpha(\mathcal{K})). \]

It is im­port­ant to re­cog­nise that these groups are only defined up to iso­morph­ism by the Dixmi­er–Douady class \( [\alpha] \in H^3(X, \mathbb{Z}). \) To dis­tin­guish these two defin­i­tions of twis­ted K-the­ory we will write \( K_{\mathbf{top}}^{\mathrm{ev}/\mathrm{odd}}(X, \alpha) \) and \( K_{\mathbf{an}}^{\mathrm{ev}/\mathrm{odd}}(X, \alpha) \) for the to­po­lo­gic­al and ana­lyt­ic twis­ted K-the­or­ies of \( (X, \alpha) \) re­spect­ively. Twis­ted K-the­ory is a 2-peri­od­ic gen­er­al­ised co­homo­logy the­ory: a con­trav­ari­ant func­tor on the cat­egory con­sist­ing of pairs \( (X, \alpha), \) with the twist­ing \( \alpha: X\to K(\mathbb{Z}, 3), \) to the cat­egory of \( \mathbb{Z}_2 \)-graded abeli­an groups. Note that a morph­ism between two pairs \( (X, \alpha) \) and \( (Y, \beta) \) is a con­tinu­ous map \( f: X\to Y \) such that \( \beta \circ f =\alpha. \)

3.1. Twisted K-theory for torsion twistings

There are some subtle is­sues in twis­ted K-the­ory and to handle these we have chosen to use the lan­guage of bundle gerbes, con­nec­tions and curvings as ex­plained in [e15]. We ex­plain first the so-called “lift­ing bundle gerbe” \( \mathcal{G}_\alpha \) [e15] as­so­ci­ated to the prin­cip­al \( \mathrm{PU}(\mathcal{H}) \)-bundle \( \pi: \mathcal{P}_\alpha\to X \) and the cent­ral ex­ten­sion \begin{equation}\label{cen:ext} 1\to U(1) \longrightarrow U(\mathcal{H}) \longrightarrow \mathrm{PU}(\mathcal{H}) \to 1. \end{equation} This is con­struc­ted by start­ing with \( \pi: \mathcal{P}_\alpha\to X, \) form­ing the fibre product \( \mathcal{P}_\alpha^{[2]}, \) which is a group­oid \[ \mathcal{P}_\alpha^{[2]} = \mathcal{P}_\alpha \times_X \mathcal{P}_\alpha \begin{smallmatrix} \xrightarrow{\ \pi_1\ } \\ \xrightarrow[\ \pi_2\ ]{} \end{smallmatrix} \mathcal{P}_\alpha \] with source and range maps \[ \pi_1: (y_1, y_2) \longmapsto y_1 \quad\text{and}\quad \pi_2: (y_1, y_2)\longmapsto y_2. \] There is an ob­vi­ous map from each fibre of \( \mathcal{P}_\alpha^{[2]} \) to \( \mathrm{PU}(\mathcal{H}) \) and so we can define the fibre of \( \mathcal{G}_\alpha \) over a point in \( \mathcal{P}_\alpha^{[2]} \) by pulling back the fibra­tion \eqref{cen:ext} us­ing this map. This en­dows \( \mathcal{G}_\alpha \) with a group­oid struc­ture (from the mul­ti­plic­a­tion in \( U(\mathcal{H}) \)) and in fact it is a \( U(1) \)-group­oid ex­ten­sion of \( \mathcal{P}_\alpha^{[2]}. \)

A tor­sion twist­ing \( \alpha \) is a map \[ \alpha: X\to K(\mathbb{Z}, 3) \] rep­res­ent­ing a tor­sion class in \( H^3(X, \mathbb{Z}). \) Every tor­sion twist­ing arises from a prin­cip­al \( \mathrm{PU}(n) \)-bundle \( \mathcal{P}_\alpha(n) \) with its clas­si­fy­ing map \[ X\to \mathrm{BPU}(n), \] or a prin­cip­al \( \mathrm{PU}(\mathcal{H}) \)-bundle with a re­duc­tion to \[ \mathrm{PU}(n) \subset \mathrm{PU}(\mathcal{H}). \] For a tor­sion twist­ing \[ \alpha: X\longrightarrow \mathrm{BPU}(n) \longrightarrow\mathrm{BPU}(\mathcal{H}), \] the cor­res­pond­ing lift­ing bundle gerbe \( \mathcal{G}_a \) shown to the right is defined by \[ \mathcal{P}_\alpha(n)^{[2]}\cong \mathcal{P}_\alpha(n) \rtimes \mathrm{PU}(n) \rightrightarrows \mathcal{P}_\alpha(n) \] (as a group­oid) and the cent­ral ex­ten­sion \[ 1\to U(1) \longrightarrow U(n) \longrightarrow \mathrm{PU}(n) \to 1. \]

There is an Azu­maya bundle as­so­ci­ated to \( \mathcal{P}_\alpha(n) \) arising nat­ur­ally from the \( \mathrm{PU}(n) \) ac­tion on the \( n\times n \) matrices. We de­note this as­so­ci­ated Azu­maya bundle by \( \mathcal{A}_\alpha. \) An \( \mathcal{A}_\alpha \)-mod­ule is a com­plex vec­tor bundle \( \mathcal{E} \) over \( M \) with a fibre-wise \( \mathcal{A}_\alpha \) ac­tion \[ \mathcal{A}_\sigma \times_M \mathcal{E} \longrightarrow \mathcal{E}. \] The \( C^* \)-al­gebra of con­tinu­ous sec­tions of \( \mathcal{A}_\alpha, \) van­ish­ing at in­fin­ity if \( X \) is non­com­pact, is Mor­ita equi­val­ent to a con­tinu­ous trace \( C^* \)-al­gebra \( \Gamma_0(X, \mathcal{P}_\alpha(\mathcal{K})). \) Hence there is an iso­morph­ism between \( K^0 (X, \alpha) \) and the K-the­ory of the bundle mod­ules of \( \mathcal{A}_a. \)

There is an equi­val­ent defin­i­tion of twis­ted K-the­ory us­ing bundle gerbe mod­ules (see [e24] and [e32]). A bundle gerbe mod­ule \( E \) of \( \mathcal{G}_\alpha \) is a com­plex vec­tor bundle \( E \) over \( \mathcal{P}_\alpha(n) \) with a group­oid ac­tion of \( \mathcal{G}_\alpha, \) i.e., an iso­morph­ism \[ \phi: \mathcal{G}_\alpha \times_{(\pi_2,p )} E \longrightarrow E, \] where \( \mathcal{G}_\alpha \times_{(\pi_2,\pi)} E \) is the fibre product of the source \[ \pi_2: \mathcal{G}_\alpha \to \mathcal{P}_\alpha(n) \quad\text{and}\quad p: E\to \mathcal{P}_\alpha(n) \] such that

  1. \( p\circ \phi (g, v) = \pi_1(g) \) for \( (g, v) \in \mathcal{G}_\alpha \times_{(\pi_2, p)} E, \) and \( \pi_1 \) is the tar­get map of \( \mathcal{G}_\alpha; \)

  2. \( \phi \) is com­pat­ible with the bundle gerbe mul­ti­plic­a­tion \[ m: \mathcal{G}_a \times_{(\pi_2,\pi_1)}\mathcal{G}_\alpha \to \mathcal{G}_\alpha, \] which means \[ \phi \circ (\mathrm{id} \times \phi) = \phi\circ (m\times \mathrm{id}). \]

Note that the nat­ur­al rep­res­ent­a­tion of \( U(n) \) on \( \mathbb{C}^n \) in­duces a \( \mathcal{G}_\alpha \) bundle gerbe mod­ule \[ S_n = \mathcal{P}_\alpha(n) \times \mathbb{C}^n. \] Here we use the fact that \[ \mathcal{G}_\alpha = \mathcal{P}_\alpha(n) \rtimes U(n) \rightrightarrows \mathcal{P}_\alpha(n) \] (as a group­oid). Sim­il­arly, the dual rep­res­ent­a­tion of \( U(n) \) on \( \mathbb{C}^n \) in­duces a \( \mathcal{G}_{-\alpha} \) bundle gerbe mod­ule \( S_n^* = \mathcal{P}_\alpha(n) \times \mathbb{C}^n. \) Note that \[ S^*_n \otimes S_n \cong \pi^*\mathcal{A}_\alpha \] des­cends to the Azu­maya bundle \( \mathcal{A}_\alpha. \) Giv­en a \( \mathcal{G}_\alpha \) bundle gerbe mod­ule \( E \) of rank K, as a \( \mathrm{PU}(n) \)-equivari­ant vec­tor bundle, \( S^*_n\otimes E \) des­cends to an \( \mathcal{A}_\alpha \)-bundle over \( M. \) Con­versely, giv­en an \( \mathcal{A}_\alpha \)-bundle \( \mathcal{E} \) over \( M, \) \[ S_n\otimes_{\pi^*\mathcal{A}_\alpha} \pi^* \mathcal{E} \] defines a \( \mathcal{G}_\alpha \) bundle gerbe mod­ule. These two con­struc­tions are in­verse to each oth­er due to the fact that \begin{align*} S_n^* \otimes (S_n \otimes_{\pi^*\mathcal{A}_\alpha} \pi^* \mathcal{E}) &\cong (S_n^* \otimes S_n ) \otimes_{\pi^*\mathcal{A}_\alpha} \pi^* \mathcal{E}\\ &\cong \pi^*\mathcal{A}_\alpha \otimes_{\pi^*\mathcal{A}_\alpha} \pi^* \mathcal{E}\\ &\cong \pi^* \mathcal{E} . \end{align*} There­fore, there is a nat­ur­al equi­val­ence between the cat­egory of \( \mathcal{G}_\alpha \) bundle gerbe mod­ules and the cat­egory of \( \mathcal{A}_\alpha \) bundle mod­ules, as dis­cussed in [e32]. In sum­mary, we have the fol­low­ing pro­pos­i­tion.

Pro­pos­i­tion 2 ([e24],[e32]): For a tor­sion twist­ing \[ \alpha: X\longrightarrow \mathrm{BPU}(n) \longrightarrow \mathrm{BPU}(\mathcal{H}), \] twis­ted K-the­ory \( K^0(X, \alpha) \) has an­oth­er two equi­val­ent de­scrip­tions:
  1. the Grothen­dieck group of the cat­egory of \( \mathcal{G}_\alpha \) bundle gerbe mod­ules,

  2. the Grothen­dieck group of the cat­egory of \( \mathcal{A}_\sigma \) bundle mod­ules.

One im­port­ant ex­ample of tor­sion twist­ings comes from real ori­ented vec­tor bundles. Con­sider an ori­ented real vec­tor bundle \( E \) of even rank over \( X \) with a fixed fibre-wise in­ner product. De­note by \[ \nu_E: X\to \mathbf{BSO}(2k) \] the clas­si­fy­ing map of \( E. \) The twist­ing \[ o(E) :=W_3\circ \nu_E: X \longrightarrow \mathbf{BSO}(2k) \longrightarrow K(\mathbb{Z}, 3) \] will be called the ori­ent­a­tion twist­ing as­so­ci­ated to \( E. \) Here \( W_3 \) is the clas­si­fy­ing map of the prin­cip­al \( \mathbf{BU}(1) \)-bundle \[ \mathbf{BSpin}^c (2k) \to \mathbf{BSO} (2k). \] Note that the ori­ent­a­tion twist­ing \( o(E) \) is null-ho­mo­top­ic if and only if \( E \) is K-ori­ented.

Pro­pos­i­tion 3: Giv­en an ori­ented real vec­tor bundle \( E \) of even rank over \( X \) with an ori­ent­a­tion twist­ing \( o(E), \) there is a ca­non­ic­al iso­morph­ism \[ K^0(X, o(E)) \cong K^0(X, W_3(E)), \] where \( K^0(X, W_3(E)) \) is the K-the­ory of the Clif­ford mod­ules as­so­ci­ated to the bundle \( \mathrm{Cliff}(E) \) of Clif­ford al­geb­ras.

Proof.   De­note by \( \mathcal{F}r \) the frame bundle of \( V, \) the prin­cip­al \( \mathrm{SO}(2k) \)-bundle of pos­it­ively ori­ented or­thonor­mal frames, i.e., \[ E= \mathcal{F}r\times_{\rho_{2n}} \mathbb{R}^{2k}, \] where \( \rho_n \) is the stand­ard rep­res­ent­a­tion of \( \mathrm{SO}(2k) \) on \( \mathbb{R}^n. \) The lift­ing bundle gerbe as­so­ci­ated to the frame bundle and the cent­ral ex­ten­sion \[ 1\to U(1) \longrightarrow \mathrm{Spin}^c(2k) \longrightarrow \mathrm{SO}(2k) \to 1 \] is called the \( \mathrm{Spin}^c \) bundle gerbe \( \mathcal{G}_{W_3(E)} \) of \( E, \) whose Dixmi­er–Douady in­vari­ant is giv­en by the in­teg­ral third Stiefel–Whit­ney class \( W_3(E)\in H^3(X, \mathbb{Z}). \) The ca­non­ic­al rep­res­ent­a­tion of \( \mathrm{Spin}^c(2k) \) gives a nat­ur­al in­clu­sion \( \mathrm{Spin}^c(2k) \subset U(2^{k}) \) which in­duces a com­mut­at­ive dia­gram \[ \begin{CD} U(1) @>{{}}>{{}}> \mathrm{Spin}^c(2k) @>{{}}>{{}}> \mathrm{SO}(2k) \\ @VV=V @VVV @VVV \\ U(1) @>{{}}>{{}}> U(2^k) @>{{}}>{{}}> \mathrm{PU}(2^k) \\ @VV=V @VVV @VVV\\ U(1) @>{{}}>{{}}> U(\mathcal{H}) @>{{}}>{{}}> \mathrm{PU}(\mathcal{H}) \end{CD} \] This provides a re­duc­tion of the prin­cip­al \( \mathrm{PU}(\mathcal{H}) \)-bundle \( \mathcal{P}_{o(E)}. \) The as­so­ci­ated bundle of Azu­maya al­geb­ras is in fact the bundle of Clif­ford al­geb­ras, whose bundle mod­ules are called Clif­ford mod­ules [e12]. Hence, there ex­ists a ca­non­ic­al iso­morph­ism between \( K^0(X, o(E)) \) and the K-the­ory of the Clif­ford mod­ules as­so­ci­ated to the bundle \( \mathrm{Cliff}(E). \)

4. Topological and analytic twisted K-homology

Let \( X \) be a CW-com­plex (or para­com­pact Haus­dorff space) with a twist­ing \[ \alpha: X\to K(\mathbb{Z}, 3) \] and \( \mathcal{P}_\alpha \) be the cor­res­pond­ing prin­cip­al \( K(\mathbb{Z}, 2) \)-bundle. Any base-point-pre­serving ac­tion of a \( K(\mathbb{Z}, 2) \) on a space defines an as­so­ci­ated bundle by the stand­ard con­struc­tion. In par­tic­u­lar, as a clas­si­fy­ing space of com­plex line bundles, \( K(\mathbb{Z}, 2) \) acts on the com­plex K-the­ory spec­trum \( \mathbb{K} \) rep­res­ent­ing the tensor product by com­plex line bundles, where \[ \mathbb{K}_{\mathrm{ev}} = \mathbb{Z}\times \mathrm{BU}(\infty), \quad \mathbb{K}_{\mathrm{odd}} = U(\infty). \] De­note by \[ \mathcal{P}_\alpha (\mathbb{K}) = \mathcal{P}_\alpha\times_{K(\mathbb{Z}, 2)} \mathbb{K} \] the bundle of based K-the­ory spec­tra over \( X. \) There is a sec­tion of \[ \mathcal{P}_\alpha (\mathbb{K}) = \mathcal{P}_\alpha\times_{K(\mathbb{Z}, 2)} \mathbb{K} \] defined by tak­ing the base points of each fibre. The im­age of this sec­tion can be iden­ti­fied with \( X \) and we de­note by \( \mathcal{P}_\alpha ( \mathbb{K})/X \) the quo­tient space of \( \mathcal{P}_\alpha ( \mathbb{K}) \) ob­tained by col­lapsing the im­age of this sec­tion.

The stable ho­mo­topy groups of \( \mathcal{P}_\alpha ( \mathbb{K})/X \) by defin­i­tion give the to­po­lo­gic­al twis­ted K-ho­mo­logy groups \( K^{\mathbf{top}}_{\mathrm{ev}/\mathrm{odd}} (X, \alpha). \) (There are only two due to Bott peri­od­icity of \( \mathbb{K}. \)) Thus we have \[ K^{\mathbf{top}}_{\mathrm{ev}} (X, \alpha) = \varinjlim_{k\to\infty} \pi_{2k} ( \mathcal{P}_\alpha ( \mathrm{BU}(\infty)) /X) \] and \[ K^{\mathbf{top}}_{\mathrm{odd}} (X, \alpha) = \varinjlim_{k\to\infty} \pi_{2k+1} ( \mathcal{P}_\alpha ( \mathrm{BU}(\infty) ) /X). \] Here the dir­ect lim­its are taken by the double sus­pen­sion \[ \pi_{n+2k} ( \mathcal{P}_\alpha ( \mathrm{BU}(\infty)) /X) \longrightarrow \pi_{n+2k+2} ( \mathcal{P}_\alpha (S^2 \wedge \mathrm{BU}(\infty)) /X ) \] and then fol­lowed by the stand­ard map \[ \eqalign{ \pi_{n+2k+2} ( \mathcal{P}_\alpha (S^2 \wedge \mathrm{BU}(\infty) )/X ) &\xrightarrow{b\wedge 1} \pi_{n+2k+2} ( \mathcal{P}_\alpha (\mathrm{BU}(\infty) \wedge \mathrm{BU}(\infty))/X ) \cr &\xrightarrow{\ m\,\ } \pi_{n+2k+2} ( \mathcal{P}_\alpha ( \mathrm{BU}(\infty))/X), } \] where \( b: \mathbb{R}^2\to \mathrm{BU}(\infty) \) rep­res­ents the Bott gen­er­at­or in \( K^0(\mathbb{R}^2)\cong \mathbb{Z}, \) \( m \) is the base-point-pre­serving map in­du­cing the ring struc­ture on K-the­ory.

For a re­l­at­ive CW-com­plex \( (X, A) \) with a twist­ing \( \alpha: X\to K(\mathbb{Z}, 3), \) the re­l­at­ive ver­sion of to­po­lo­gic­al twis­ted K-ho­mo­logy, de­noted by \[ K^{\mathbf{top}}_{\mathrm{ev}/\mathrm{odd}}(X, A, \alpha), \] is defined to be \[ K^{\mathbf{top}}_{\mathrm{ev}/\mathrm{odd}}(X/A, \alpha), \] where \( X/A \) is the quo­tient space of \( X \) ob­tained by col­lapsing \( A \) to a point. Then we have the ex­act se­quence \[ \begin{CD} K^{\mathbf{top}}_{\mathrm{odd}} (X, A; \alpha) @>{{}}>{{}}> K^{\mathbf{top}}_{\mathrm{ev}} (A, \alpha|_A) @>{{}}>{{}}> K^{\mathbf{top}}_{\mathrm{ev}} (X,\alpha)\\ @AAA @. @VVV\\ K^{\mathbf{top}}_{\mathrm{odd}} (X, \alpha ) @<{{}}<{{}}< K^{\mathbf{top}}_{\mathrm{odd}} ( A, \alpha|_A) @<{{}}<{{}}< K^{\mathbf{top}}_{\mathrm{ev}} (X, A; \alpha) \end{CD} \] and the ex­cision prop­er­ties \[ K^{\mathbf{top}}_{\mathrm{ev}/\mathrm{odd}} (X, B; \alpha) \cong K^{\mathbf{top}}_{\mathrm{ev}/\mathrm{odd}} (A, A-B; \alpha|_A) \] for any CW-tri­ad \( (X; A, B) \) with a twist­ing \( \alpha: X\to K(\mathbb{Z}, 3). \) A triple \( (X; A, B) \) is a CW-tri­ad if \( X \) is a CW-com­plex, and \( A, \) \( B \) are two sub­com­plexes of \( X \) such that \( A\cup B = X. \)

Ana­lyt­ic twis­ted K-ho­mo­logy, de­noted by \( K^{\mathbf{an}}_{\mathrm{ev}/\mathrm{odd}}(X, \alpha), \) is defined to be \[ K^{\mathbf{an}}_{\mathrm{ev}/\mathrm{odd}} (X, \alpha) := KK^{\mathrm{ev}/\mathrm{odd}}(\Gamma_0(X, \mathcal{P}_\alpha(\mathcal{K})), \mathbb{C} ), \] Kas­parov’s \( \mathbb{Z}_2 \)-graded K-ho­mo­logy of the \( C^* \)-al­gebra \( \Gamma_0(X, \mathcal{P}_\alpha(\mathcal{K})). \)

For a re­l­at­ive CW-com­plex \( (X, A) \) with a twist­ing \( \alpha: X\to K(\mathbb{Z}, 3), \) the re­l­at­ive ver­sion of ana­lyt­ic twis­ted K-ho­mo­logy \( K^{\mathbf{an}}_{\mathrm{ev}/\mathrm{odd}}(X, A, \alpha) \) is defined to be \( K^{\mathbf{an}}_{\mathrm{ev}/\mathrm{odd}}(X-A, \alpha). \) Then we have the ex­act se­quence \[ \begin{CD} K^{\mathbf{an}}_{\mathrm{odd}} (X, A; \alpha) @>{{}}>{{}}> K^{\mathbf{an}}_{\mathrm{ev}} (A, \alpha|_A) @>{{}}>{{}}> K^{\mathbf{an}}_{\mathrm{ev}} (X,\alpha)\\ @AAA @. @VVV\\ K^{\mathbf{an}}_{\mathrm{odd}} (X, \alpha ) @<{{}}<{{}}< K^{\mathbf{an}}_{\mathrm{odd}} ( A, \alpha|_A) @<{{}}<{{}}< K^{\mathbf{an}}_{\mathrm{ev}} (X, A; \alpha) \end{CD} \] and the ex­cision prop­er­ties \[ K^{\mathbf{an}}_{\mathrm{ev}/\mathrm{odd}} (X, B; \alpha) \cong K^{\mathbf{an}}_{\mathrm{ev}/\mathrm{odd}} (A, A-B; \alpha|_A) \] for any CW-tri­ad \( (X; A, B) \) with a twist­ing \( \alpha: X\to K(\mathbb{Z}, 3). \)

The­or­em 4 (The­or­em 5.1 in [e33]): There is a nat­ur­al iso­morph­ism \[ \Phi: K^{\mathbf{top}}_{\mathrm{ev}/\mathrm{odd}} (X, \alpha) \longrightarrow K^{\mathbf{an}}_{\mathrm{ev}/\mathrm{odd}} (X, \alpha) \] for any smooth man­i­fold \( X \) with a twist­ing \( \alpha: X \to K(\mathbb{Z}, 3). \)

The proof of this the­or­em re­quires Poin­caré du­al­ity between twis­ted K-the­ory and twis­ted K-ho­mo­logy (we de­scribe this du­al­ity in the next the­or­em), and the iso­morph­ism (The­or­em 1) between to­po­lo­gic­al twis­ted K-the­ory and ana­lyt­ic twis­ted K-the­ory.

Fix an iso­morph­ism \( \mathcal{H} \otimes \mathcal{H} \cong \mathcal{H} \) which in­duces a group ho­mo­morph­ism \[ U(\mathcal{H}) \times U(\mathcal{H}) \rightarrow U(\mathcal{H}) \] whose re­stric­tion to the centre is the group mul­ti­plic­a­tion on \( U(1). \) So we have a group ho­mo­morph­ism \[ \mathrm{PU}(\mathcal{H}) \times \mathrm{PU}(\mathcal{H}) \longrightarrow \mathrm{PU}(\mathcal{H}) \] which defines a con­tinu­ous map, de­noted by \( m_\ast, \) of CW-com­plexes \[ \mathrm{BPU}(\mathcal{H}) \times B PU(\mathcal{H}) \longrightarrow \mathrm{BPU}(\mathcal{H}). \] As \( \mathrm{BPU}(\mathcal{H}) \) is iden­ti­fied as \( K(\mathbb{Z}, 3), \) we may think of this as a con­tinu­ous map tak­ing \[ K(\mathbb{Z}, 3) \times K(\mathbb{Z}, 3) \quad\text{to}\quad K(\mathbb{Z}, 3), \] which can be used to define \( \alpha + o_X. \)

There are nat­ur­al iso­morph­isms from twis­ted K-ho­mo­logy (to­po­lo­gic­al, resp. ana­lyt­ic) to twis­ted K-the­ory (to­po­lo­gic­al, resp. ana­lyt­ic) of a smooth man­i­fold \( X \) where the twist­ing is shif­ted by \( \alpha \mapsto \alpha + o_X, \) where \( \tau: X \to \mathbf{BSO} \) is the clas­si­fy­ing map of the stable tan­gent space and \( \alpha + o_X \) de­notes the map \( X \to K(\mathbb{Z}, 3), \) rep­res­ent­ing the class \[ [\alpha]+ W_3(X) \quad\text{in}\quad H^3(X, \mathbb{Z}). \]

The­or­em 5: Let \( X \) be a smooth man­i­fold with a twist­ing \( \alpha: X\to K(\mathbb{Z}, 3). \) There ex­ist iso­morph­isms \[ K^{\mathbf{top}}_{\mathrm{ev}/\mathrm{odd}} (X, \alpha) \cong K_{\mathbf{top}}^{\mathrm{ev}/\mathrm{odd}} (X, \alpha +o_X ) \] and \[ K^{\mathbf{an}}_{\mathrm{ev}/\mathrm{odd}} (X, \alpha) \cong K_{\mathbf{an}}^{\mathrm{ev}/\mathrm{odd}} (X, \alpha +o_X ), \] with the de­gree shif­ted by \( \dim X \pmod 2. \)

Ana­lyt­ic Poin­caré du­al­ity was es­tab­lished in [e31] and [e35], and to­po­lo­gic­al Poin­caré du­al­ity was es­tab­lished in [e33]. The­or­em 4 and the ex­act se­quences for a pair \( (X, A) \) im­ply the fol­low­ing co­rol­lary.

Co­rol­lary 6: There is a nat­ur­al iso­morph­ism \[ \Phi: K^{\mathbf{top}}_{\mathrm{ev}/\mathrm{odd}} (X, A, \alpha) \longrightarrow K^{\mathbf{an}}_{\mathrm{ev}/\mathrm{odd}} (X, A, \alpha) \] for any smooth man­i­fold \( X \) with a twist­ing \( \alpha: X \to K(\mathbb{Z}, 3) \) and a closed sub­man­i­fold \( A\subset X. \)

Re­mark: In fact, Poin­caré du­al­ity as in The­or­em 5 holds for any com­pact Rieman­ni­an man­i­fold \( W \) with bound­ary \( \partial W \) and a twist­ing \( \alpha:W\to K(\mathbb{Z}, 3). \) This du­al­ity takes the form \[ K^{\mathbf{top}}_{\mathrm{ev}/\mathrm{odd}} (W, \alpha) \cong K_{\mathbf{top}}^{\mathrm{ev}/\mathrm{odd}} (W, \partial W, \alpha +o_W ) \] and \[ K^{\mathbf{an}}_{\mathrm{ev}/\mathrm{odd}} (W, \alpha) \cong K_{\mathbf{an}}^{\mathrm{ev}/\mathrm{odd}} (X, \partial X, \alpha +o_W ), \] with the de­gree shif­ted by \( \dim W \pmod 2. \) From this, we have a nat­ur­al iso­morph­ism \[ \Phi: K^{\mathbf{top}}_{\mathrm{ev}/\mathrm{odd}} (X, A, \alpha) \longrightarrow K^{\mathbf{an}}_{\mathrm{ev}/\mathrm{odd}} (X, A, \alpha) \] for any CW pair \( (X, A) \) with a twist­ing \( \alpha: X \to K(\mathbb{Z}, 3) \) us­ing the five lemma.

5. Geometric cycles and geometric twisted K-homology

This is the main sec­tion of this es­say. Here we out­line both the ideas due to Bai-Ling Wang and the ideas due mostly to Paul. One may eas­ily move between these points of view.

As usu­al \( X \) is a para­com­pact Haus­dorff space and \( \alpha: X \rightarrow K(\mathbb{Z}, 3) \) is a twist­ing over \( X. \)

Defin­i­tion: Giv­en a smooth ori­ented man­i­fold \( M \) with a clas­si­fy­ing map \( \nu \) of its stable nor­mal bundle, we say that \( M \) is an \( \alpha \)-twis­ted \( \mathrm{Spin}^c \) man­i­fold over \( X \) if \( M \) is equipped with an \( \alpha \)-twis­ted \( \mathrm{Spin}^c \) struc­ture, that means, a con­tinu­ous map \( \iota: M\to X \) such that the dia­gram shown to the right com­mutes up to a fixed ho­mo­topy \( \eta \) from \( W_3\circ \nu \) and \( \alpha \circ \iota. \) Such an \( \alpha \)-twis­ted \( \mathrm{Spin}^c \) man­i­fold over \( X \) will be de­noted by \( (M, \nu, \iota, \eta). \)

Pro­pos­i­tion 7: \( M \) ad­mits an \( \alpha \)-twis­ted \( \mathrm{Spin}^c \) struc­ture if and only if there is a con­tinu­ous map \( \iota: M\to X \) such that \[ \iota^*([\alpha]) + W_3(M)=0. \] If \( \iota \) is an em­bed­ding, this is the an­om­aly can­cel­la­tion con­di­tion ob­tained by Freed and Wit­ten in [e19].

As shown in [e33], this way of think­ing about twis­ted \( \mathrm{Spin}^c \) struc­tures gen­er­al­ises to cov­er high­er-de­gree twists. The lat­ter are im­port­ant in string to­po­logy.

A morph­ism between \( \alpha \)-twis­ted \( \mathrm{Spin}^c \) man­i­folds \[ (M_1, \nu_1, \iota_1, \eta_1) \quad\text{and}\quad (M_2, \nu_2, \iota_2, \eta_2) \] is a con­tinu­ous map \( f: M_1 \to M_2 \) where the dia­gram shown to the right is a ho­mo­topy com­mut­at­ive dia­gram such that

  1. \( \nu_1 \) is ho­mo­top­ic to \( \nu_2 \circ f \) through a con­tinu­ous map \( \nu: M_1 \times [0, 1] \to \mathbf{BSO}; \)

  2. \( \iota_2 \circ f \) is ho­mo­top­ic to \( \iota_1 \) through a con­tinu­ous map \( \iota : M_1 \times [0, 1] \to X; \)

  3. the com­pos­i­tion of ho­mo­top­ies \[ ( \alpha \circ \iota ) * (\eta_2 \circ (f\times \mathrm{id}) ) * (W_3 \circ \nu) \] is ho­mo­top­ic to \( \eta_1. \)

Two \( \alpha \)-twis­ted \( \mathrm{Spin}^c \) man­i­folds \( (M_1, \nu_1, \iota_1, \eta_1) \) and \( (M_2, \nu_2, \iota_2, \eta_2) \) are called iso­morph­ic if there ex­ists a dif­feo­morph­ism \( f: M_1 \to M_2 \) such that the above holds. If the iden­tity map on \( M \) in­duces an iso­morph­ism between \( (M, \nu_1, \) \( \iota_1, \eta_1) \) and \( (M, \nu_2, \) \( \iota_2, \eta_2), \) then these two \( \alpha \)-twis­ted \( \mathrm{Spin}^c \) struc­tures are called equi­val­ent.

Ori­ent­a­tion re­versal in the Grass­man­ni­an mod­el defines an in­vol­u­tion \[ r: \mathbf{BSO} \rightarrow \mathbf{BSO}. \] Us­ing this, one may de­term­ine a unique equi­val­ence class of \( \alpha \)-twis­ted \( \mathrm{Spin}^c \) struc­tures on \( -M, \) called the op­pos­ite \( \alpha \)-twis­ted \( \mathrm{Spin}^c \) struc­ture, simply de­noted by \( -(M, \nu, \iota, \eta). \)

5.1. Bai-Ling Wang’s approach to twisted geometric cycles or BB

I will first re­view the ori­gin­al ap­proach to twis­ted geo­met­ric cycles and then re­view the ideas ori­gin­at­ing with Paul.

Defin­i­tion: A geo­met­ric cycle for \( (X, \alpha) \) is a quin­tuple \( (M, \iota, \nu, \eta, [E]), \) where \( [E] \) is a K-class in \( K^0(M) \) and \( M \) is a smooth closed man­i­fold equipped with an \( \alpha \)-twis­ted \( \mathrm{Spin}^c \) struc­ture \( (M, \iota, \nu, \eta). \)

Two geo­met­ric cycles \( (M_1, \iota_1, \nu_1, \eta_1, [E_1]) \) and \( (M_2, \iota,_2 \nu_2, \eta_2, [E_2]) \) are iso­morph­ic if there is an iso­morph­ism \[ f: (M_1, \iota_1, \nu_1, \eta_1) \longrightarrow (M_2, \iota_2, \nu_2, \eta_2), \] as \( \alpha \)-twis­ted \( \mathrm{Spin}^c \) man­i­folds over \( X, \) such that \( f_! ([E_1]) = [E_2]. \)

Let \( \Gamma (X, \alpha) \) be the col­lec­tion of all geo­met­ric cycles for \( (X, \alpha). \) We now im­pose an equi­val­ence re­la­tion \( \sim \) on \( \Gamma (X, \alpha) \) gen­er­ated by the fol­low­ing three ele­ment­ary re­la­tions:

  1. Dir­ect sum–dis­joint uni­on: If \( (M , \iota , \nu , \eta , [E_1]) \) and \( (M , \iota, \nu , \eta , [E_2]) \) are two geo­met­ric cycles with the same \( \alpha \)-twis­ted \( \mathrm{Spin}^c \) struc­ture, then \[ (M , \iota , \nu , \eta , [E_1]) \cup ( M , \iota , \nu , \eta , [E_2]) \sim (M , \iota , \nu , \eta , [E_1]+ [E_2]). \]

  2. Bor­d­ism: Giv­en two geo­met­ric cycles \( (M_1, \iota_1, \nu_1, \eta_1, [E_1]) \) and \( (M_2, \iota_2, \) \( \nu_2, \) \( \eta_2, [E_2]), \) we call them bord­ant if there ex­ists an \( \alpha \)-twis­ted \( \mathrm{Spin}^c \) man­i­fold \( (W, \iota, \nu, \eta) \) and \( [E]\in K^0(W) \) such that \[ \partial (W, \iota, \nu, \eta) = -(M_1, \iota_1, \nu_1, \eta_1) \cup (M_2, \iota_2, \nu_2, \eta_2) \] and \( \partial ([E]) = [E_1] \cup [E_2]. \) Here \( -(M_1, \iota_1, \nu_1, \eta_1) \) de­notes the man­i­fold \( M_1 \) with the op­pos­ite \( \alpha \)-twis­ted \( \mathrm{Spin}^c \) struc­ture.

  3. \( \mathrm{Spin}^{c} \) vec­tor bundle modi­fic­a­tion: Sup­pose we are giv­en a geo­met­ric cycle \( (M, \iota, \nu, \eta, [E]) \) and a \( \mathrm{Spin}^c \) vec­tor bundle \( V \) over \( M \) with even-di­men­sion­al fibres. De­note by \( \underline{\mathbb{R}} \) the trivi­al rank-1 real vec­tor bundle. Choose a Rieman­ni­an met­ric on \( V\oplus \underline{\mathbb{R}}, \) and let \( \hat{M}= S(V\oplus \underline{\mathbb{R}}) \) be the sphere bundle of \( V\oplus \underline{\mathbb{R}}. \) Then the ver­tic­al tan­gent bundle \( T^v(\hat{M}) \) of \( \hat{M} \) ad­mits a nat­ur­al \( \mathrm{Spin}^c \) struc­ture with an as­so­ci­ated \( \mathbb{Z}_2 \)-graded spinor bundle \( S^+_V\oplus S^-_V \) . De­note by \( \rho: \hat{M} \to M \) the pro­jec­tion which is K-ori­ented. Then \[ (M, \iota, \nu, \eta, [E]) \sim (\hat{M}, \iota\circ \rho , \nu \circ \rho, \eta \circ \rho, [\rho^*E\otimes S^+_V]). \]

Defin­i­tion: De­note by \( K^{\mathbf{\mathrm{geo}}}_*(X, \alpha) = \Gamma (X, \alpha)/\sim \) the geo­met­ric twis­ted K-ho­mo­logy. Ad­di­tion is giv­en by the dir­ect sum - dis­joint uni­on re­la­tion. Note that the equi­val­ence re­la­tion \( \sim \) pre­serves the par­ity of the di­men­sion of the un­der­ly­ing \( \alpha \)-twis­ted \( \mathrm{Spin}^c \) man­i­fold. Now let \( K^{\mathbf{\mathrm{geo}}}_{0}(X, \alpha) \) (resp. \( K^{\mathbf{\mathrm{geo}}}_1(X, \alpha) \)) be the sub­group of \( K^{\mathbf{\mathrm{geo}}}_*(X, \alpha) \) de­term­ined by all geo­met­ric cycles with even-di­men­sion­al (resp. odd-di­men­sion­al) \( \alpha \)-twis­ted \( \mathrm{Spin}^c \) man­i­folds.

Re­mark:

  1. If \( M, \) in a geo­met­ric cycle \( (M, \iota, \nu, \eta, [E]) \) for \( (X, \alpha), \) is a com­pact man­i­fold with bound­ary, then \( [E] \) has to be a class in \( K^0(M, \partial M). \)

  2. If \( f: X\to Y \) is a con­tinu­ous map and \( \alpha: Y\to K(\mathbb{Z}, 3) \) is a twist­ing, then there is a nat­ur­al ho­mo­morph­ism of abeli­an groups \[ f_*: K^{\mathbf{\mathrm{geo}}}_{\mathrm{ev}/\mathrm{odd}}(X, \alpha \circ f ) \longrightarrow K^{\mathbf{\mathrm{geo}}}_{\mathrm{ev}/\mathrm{odd}}(Y, \alpha) \] send­ing \( [M, \iota, \nu, \eta, E ] \) to \( [M,f \circ \iota , \nu, \eta, E]. \)

  3. Let \( A \) be a closed sub­space of \( X \) and \( \alpha \) be a twist­ing on \( X. \) A re­l­at­ive geo­met­ric cycle for \( (X, A; \alpha) \) is a quin­tuple \( (M, \iota, \nu, \eta, [E]) \) such that

    1. \( M \) is a smooth man­i­fold (pos­sibly with bound­ary), equipped with an \( \alpha \)-twis­ted \( \mathrm{Spin}^c \) struc­ture \( (M, \iota, \nu, \eta); \)

    2. if \( M \) has a nonempty bound­ary, then \( \iota (\partial M) \subset A; \)

    3. \( [E] \) is a K-class in \( K^0(M) \) rep­res­en­ted by a \( \mathbb{Z}_2 \)-graded vec­tor bundle \( E \) over \( M, \) or a con­tinu­ous map \( M \to \mathrm{BU}(\infty). \)

The re­la­tion \( \sim \) gen­er­ated by dir­ect sum - dis­joint uni­on, bor­d­ism and \( \mathrm{Spin}^c \) vec­tor bundle modi­fic­a­tion is an equi­val­ence re­la­tion. The col­lec­tion of re­l­at­ive geo­met­ric cycles, mod­ulo the equi­val­ence re­la­tion is de­noted by \( K^{\mathbf{\mathrm{geo}}}_{\mathrm{ev}/\mathrm{odd}}(X, A; \alpha ). \)

5.2. \( D \)-cycles or AB

The dif­fi­culty with the ori­gin­al ap­proach to these twis­ted geo­met­ric cycles out­lined in the pre­vi­ous sub­sec­tion is that it is not clear how to con­struct them (that is, they do not solve the “gen­er­al twis­ted in­dex prob­lem” that we will an­nounce be­low). A new ap­proach was sug­ges­ted by Paul and ex­plained in our joint pa­per [4]. The main dif­fer­ence between this new ap­proach and the ap­proach de­scribed earli­er in this art­icle is that we make much heav­ier use of the the­ory of con­tinu­ous trace \( C^\ast \)-al­geb­ras. We em­phas­ise that in this new ap­proach the pro­ject­ive unit­ary group is equipped with the strong op­er­at­or to­po­logy, not the norm to­po­logy as was used in earli­er sec­tions.

We be­gin by re­call­ing a few con­cepts.

  1. Giv­en an ori­ented real Eu­c­lidean vec­tor bundle \( F \) of rank K over a para­com­pact Haus­dorff to­po­lo­gic­al space \( Y, \) a \( \mathrm{Spin}^c \) struc­ture on \( F \) is a lift of the ori­ented frame bundle \( P_{\mathrm{SO}}(F) \) to a prin­cip­al \( \mathrm{Spin}^c(k) \)-bundle \( P_{\mathrm{Spin}^c}(F), \) where \[ 1\to U(1) \longrightarrow \mathrm{Spin}^c(k) \longrightarrow \mathrm{SO}(k) \to 1 \] is the unique (for \( k > 2 \)) non­trivi­al cent­ral ex­ten­sion of \( \mathrm{SO}(k) \) by \( U(1). \) See Ap­pendix D in [e11] for an equi­val­ent defin­i­tion of \( \mathrm{Spin}^c \) struc­tures (and note that there is a well-known modi­fic­a­tion needed for \( k\leq 2 \)). A real vec­tor bundle with a \( \mathrm{Spin}^c \) struc­ture is called a \( \mathrm{Spin}^c \) vec­tor bundle. \( \mathrm{Spin}^c \) struc­tures are ori­ent­a­tion con­di­tions for com­plex K-the­ory in the sense that a \( \mathrm{Spin}^c \) vec­tor bundle is a real vec­tor vec­tor bundle with a giv­en com­plex spinor bundle or a K-the­ory Thom class. See Sec­tion 4 in [3] and The­or­em C.12 in [e11] for more dis­cus­sions of this. In par­tic­u­lar, a spinor bundle for \( F \) de­term­ines an ori­ent­a­tion of \( F. \)

  2. (two-out-of-three prin­ciple) Let \[ 0\rightarrow F^{\prime}\longrightarrow F \longrightarrow F^{\prime\prime}\rightarrow 0 \] be a short ex­act se­quence of ori­ented real vec­tor bundles on a para­com­pact Haus­dorff to­po­lo­gic­al space \( Y. \) Then \( \mathrm{Spin}^c \) struc­tures for any two of the vec­tor bundles de­term­ine a \( \mathrm{Spin}^c \) struc­ture for the third vec­tor bundle.

  3. A \( \mathrm{Spin}^c \) Rieman­ni­an man­i­fold is a Rieman­ni­an man­i­fold \( W \) (per­haps with bound­ary) whose tan­gent bundle \( TW \) is a \( \mathrm{Spin}^c \) vec­tor bundle. If \( W \) has a bound­ary \( \partial W, \) at each bound­ary point, the out­ward nor­mal vec­tor defines a trivi­al rank-1 real vec­tor bundle \( N_{\partial W} \) over \( \partial W \) and \[ 0 \to T(\partial W) \longrightarrow TW|_{\partial W} \longrightarrow N_{\partial W} \to 0 \] is an ex­act se­quence of real Eu­c­lidean vec­tor bundles over \( \partial W. \) There­fore, the two-out-of-three prin­ciple im­plies that if a \( \mathrm{Spin}^c \) man­i­fold \( W \) has a bound­ary, then this bound­ary, \( \partial W, \) is a \( \mathrm{Spin}^c \) man­i­fold in a ca­non­ic­al way.

5.2.1. Twisting data

If \( H \) is a com­plex Hil­bert space, \( \mathcal{K}(H) \) will de­note the \( C^* \)-al­gebra of all com­pact op­er­at­ors on \( H. \) Hil­bert spaces will be as­sumed to be sep­ar­able. Re­call that a \( C^* \)-al­gebra \( A \) is an ele­ment­ary \( C^* \)-al­gebra if there ex­ists a com­plex Hil­bert space \( \mathcal{H} \) and an iso­morph­ism of \( C^* \)-al­geb­ras \( A\cong\mathcal{K}(\mathcal{H}). \)

Defin­i­tion: Let \( X \) be a second-count­able loc­ally com­pact Haus­dorff to­po­lo­gic­al space. A twist­ing datum on \( X \) is a loc­ally trivi­al bundle \( \mathcal{A} \) of ele­ment­ary \( C^* \)-al­geb­ras on \( X, \) that is, each fibre of \( \mathcal{A} \) is an ele­ment­ary \( C^* \)-al­gebra with struc­ture group the auto­morph­ism group of \( \mathcal{K}(\mathcal{H}) \) for some com­plex Hil­bert space.

If \( E \) is a (loc­ally trivi­al) bundle of Hil­bert spaces on \( X, \) then \( \mathcal{K}(E) \) is the twist­ing datum defined by \[ \mathcal{K}(E)_x = \mathcal{K}(E_x),\quad x\in X. \] Any twist­ing datum \( \mathcal{A} \) is loc­ally of the form \( \mathcal{K}(E); \) i.e., for any \( x \in X \) there ex­ists an open set \( U \) in \( X \) with \( x \in U \) and a (loc­ally trivi­al) Hil­bert space vec­tor bundle \( E \) on \( U \) with \( \mathcal{A}|U \cong \mathcal{K}(E). \)

For a \( C^* \)-al­gebra \( A, \) \( A^{\mathrm{op}} \) de­notes the op­pos­ite \( C^* \)-al­gebra. As Banach spaces \( A = A^{\mathrm{op}}, \) and \( \ast \) re­mains un­changed. Thus \( ab \) in \( A^{\mathrm{op}} \) is \( ba \) in \( A. \) If \( \mathcal{A} \) is a twist­ing datum on \( X, \) then \( \mathcal{A}^{\mathrm{op}} \) is the twist­ing datum ob­tained by re­pla­cing each fibre \( \mathcal{A}_x \) by \( \mathcal{A}_x^{\mathrm{op}}. \) If \( \mathcal{A} \) and \( \mathcal{B} \) are twist­ing data on \( X, \) then \( \mathcal{A} \otimes \mathcal{B} \) is the twist­ing datum on \( X \) whose fibre at \( x \in X \) is the \( C^* \)-al­gebra \( \mathcal{A}_x \otimes \mathcal{B}_x. \)

Let \( \mathcal{A} \) be a twist­ing datum on \( X, \) and as­sume that the fibre of \( \mathcal{A} \) is in­fin­ite-di­men­sion­al. As be­fore \( \mathcal{P} \) de­notes the prin­cip­al \( P\mathcal{U}(\mathcal{H}) \) bundle on \( X \) whose fibre at \( x \in X \) is \[ \mathcal{P}_x = \{ C^*\text{-algebra isomorphisms}: \mathcal{K}(\mathcal H) \to \mathcal{A}_x\}. \] There is then the ca­non­ic­al iso­morph­ism of twist­ing data on \( X \) \[ \mathcal{A} \cong \mathcal{P}\times_{\mathcal{U}(H)}\mathcal{K}(\mathcal{H}). \] Note that the prin­cip­al \( P\mathcal{U}(\mathcal{H}) \) bundle \( \mathcal{P} \) is clas­si­fied by a con­tinu­ous map \( X\rightarrow BP\mathcal{U}(\mathcal{H}). \) Let \( \mathrm{DD}(\mathcal{A}) \) de­note the Dixmi­er–Douady in­vari­ant of \( \mathcal{A} \) (in \( H^3(X,\,\mathbb{Z}) \)).

On the oth­er hand if \( \mathcal{A} \) has fi­nite-di­men­sion­al fibres, we let \( \mathcal{B} \) be any twist­ing datum on \( X \) with every fibre of \( \mathcal{B} \) in­fin­ite-di­men­sion­al and then \( \mathrm{DD}(\mathcal{A}) \) is defined by \[ \mathrm{DD}(\mathcal{A}) =: \mathrm{DD}(\mathcal{A} \otimes \mathcal{B}) - \mathrm{DD}(\mathcal{B}). \] For any two twist­ing data \( \mathcal{A},\,\mathcal{B} \) on \( X \) we have \[ \mathrm{DD}(\mathcal{A} \otimes \mathcal{B}) = \mathrm{DD}(\mathcal{A}) + \mathrm{DD}(\mathcal{B}) \] and \[ \mathrm{DD}(\mathcal{A}^{\mathrm{op}}) = -\mathrm{DD}(\mathcal{A}). \]

Defin­i­tion: A spinor bundle for \( \mathcal{A} \) is a vec­tor bundle \( S \) of Hil­bert spaces on \( X \) to­geth­er with a giv­en iso­morph­ism of twist­ing data \( \mathcal{A} \cong \mathcal{K}(\mathcal{S}). \) A spinor bundle for \( \mathcal{A} \) ex­ists if and only if \( \mathrm{DD}(\mathcal{A}) = 0. \)

Lemma 8: Let \( \mathcal{A} \) be any twist­ing datum on \( X. \) Then there is a ca­non­ic­al spinor bundle for \( \mathcal{A}\otimes\mathcal{A}^{\mathrm{op}}. \)

Proof.  The set of Hil­bert–Schmidt op­er­at­ors on \( \mathcal{H}, \) de­noted by \( \mathcal{L}_{\text{H-S}}(\mathcal{H}), \) is an ideal in \( \mathcal{K}(\mathcal{H}). \) The \( \mathbb{C} \)-val­ued in­ner product \[ \langle T_1, T_2 \rangle = \mathrm{Trace}(T_1T_2^*) \] makes \( \mathcal{L}_{\text{H-S}}(\mathcal{H}) \) in­to a Hil­bert space. Now let \( A \) be an ele­ment­ary \( C^* \)-al­gebra. Choose an iso­morph­ism of \( C^* \)-al­geb­ras \( \psi : A \rightarrow \mathcal{K}(\mathcal{H}). \) Now, \( \psi^{-1}(\mathcal{L}_{\text{H-S}}(\mathcal{H})) \) is an ideal in \( A \) and is in­de­pend­ent of the choice of \( \psi \) be­cause the Hil­bert–Schmidt op­er­at­ors are in­vari­ant un­der \[ \mathrm{Aut}( {\mathcal K}(\mathcal{H}))= P{\mathcal U}(\mathcal{H}). \] De­note this ideal by \( A_{\text{H-S}}. \) The left-mul­ti­plic­a­tion and right-mul­ti­plic­a­tion ac­tions of \( A \) on \( A_{\text{H-S}} \) com­bine to give a left ac­tion of \( A \otimes A^{\mathrm{op}} \) on \( A_{\text{H-S}} \) which iden­ti­fies \( A \otimes A^{\mathrm{op}} \) with the com­pact op­er­at­ors on the Hil­bert space \( A_{\text{H-S}}: \) \[ A \otimes A^{\mathrm{op}} \cong \mathcal{K}(A_{\text{H-S}}). \] If \( \mathcal{A} \) is a twist­ing datum on \( X, \) let \( \mathcal{S} \) be the vec­tor bundle of Hil­bert spaces on \( X \) whose fibre at \( x \in X \) is \( (\mathcal{A}_{x})_{\text{H-S}}. \) Then \( \mathcal{S} \) is a spinor bundle for \( \mathcal{A} \otimes \mathcal{A}^{\mathrm{op}} \) and is well-defined as our con­struc­tion is in­de­pend­ent of \( \psi. \)

We im­port our pre­vi­ous nota­tion: if \( \mathcal{A} \) is a twist­ing datum on \( X, \) then \( \Gamma_0(X, \mathcal{A}) \) de­notes the \( C^* \)-al­gebra of all con­tinu­ous van­ish­ing-at-in­fin­ity sec­tions of \( \mathcal{A}. \) Re­call that the com­pactly sup­por­ted Kas­parov group \( KK_c^*(\Gamma_0(X, \mathcal{A}), \mathbb{C}) \) is \[ KK_c^j(\Gamma_0(X, \mathcal{A}), \mathbb{C}) := \lim_{\substack{\longrightarrow\\ \Delta\subset X\\ \Delta \text{ compact}}} KK^j(\Gamma(\Delta, \mathcal{A}), \mathbb{C}), \quad j= 0, 1, \] where \( \Gamma(\Delta, \mathcal{A}) \) is the \( C^* \)-al­gebra of all con­tinu­ous sec­tions of \( \mathcal{A} \) re­stric­ted to \( \Delta. \) Since \( X \) is a CW-com­plex, this is equal to the dir­ect lim­it over the fi­nite sub-CW-com­plexes of \( X. \) We will refer to \( KK_c^*(\Gamma_0(X, \mathcal{A}), \mathbb{C}) \) as the twis­ted Kas­parov K-ho­mo­logy of \( X. \)

We now ex­pound some ele­ment­ary res­ults with proofs to give the fla­vour of the Kas­parov the­ory we used.

Lemma 9: A spinor bundle \( \mathcal{S} \) for \( \mathcal{A}\otimes\mathcal{B}^{\mathrm{op}} \) de­term­ines a stable iso­morph­ism between \( \Gamma_0(X, \mathcal{A}) \) and \( \Gamma_0(X, \mathcal{B}), \) and thus gives an iso­morph­ism \[ \Psi_\mathcal{S}: KK_c^j(\Gamma_0(X, \mathcal{A}), \mathbb{C}) \cong KK_c^j(\Gamma_0(X, \mathcal{B}), \mathbb{C}), \quad j=0, 1. \]

Proof.  Let \( \mathcal{B}_{\text{H-S}} \) be the spinor bundle for \( \mathcal{B}^{\mathrm{op}}\otimes\mathcal{B}. \) Lemma 8 im­plies \[ \mathcal{A} \otimes \mathcal{B}^{\mathrm{op}} \otimes \mathcal{B} \cong \mathcal{A} \otimes \mathcal{K}(\mathcal{B}_{\text{H-S}}). \] If \( \mathcal{S} \) is a spinor bundle for \( \mathcal{A} \otimes \mathcal{B}^{\mathrm{op}}, \) then \[ \mathcal{A} \otimes \mathcal{B}^{\mathrm{op}} \otimes \mathcal{B} \cong \mathcal{K}(\mathcal{S}) \otimes \mathcal{B}. \] There­fore \[ \mathcal{A} \otimes \mathcal{K}(\mathcal{B}_{\text{H-S}}) \cong \mathcal{K}(\mathcal{S}) \otimes \mathcal{B}. \] Note that the Dixmi­er–Douady in­vari­ants of \( \mathcal{K}(\mathcal{B}_{\text{H-S}}) \) and \( \mathcal{K}(\mathcal{S}) \) are zero. For any com­pact sub­space \( \Delta \subset X, \) we have \[ \begin{array}{lll} KK^j(\Gamma(\Delta, \mathcal{A}), \mathbb{C}) &\cong & KK^j(\Gamma(\Delta, \mathcal{A} \otimes \mathcal{K}(\mathcal{B}_{\text{H-S}})),\mathbb{C})\\ &\cong & KK^j(\Gamma(\Delta,\mathcal{K}(\mathcal{S}) \otimes \mathcal{B}), \mathbb{C} )\\ &\cong & KK^j(\Gamma(\Delta, \mathcal{B}), \mathbb{C}). \end{array} \] Here the first and the third iso­morph­isms are provided by Mor­ita equi­val­ence bimod­ules \( \mathcal{B}_{\text{H-S}} \) and \( \mathcal{S} \) re­spect­ively. Passing to the dir­ect lim­it, we get the de­sired iso­morph­ism \( \Psi_\mathcal{S}. \)

5.2.2. K-cycles for twisted K-homology

As above, \( X \) is a loc­ally fi­nite CW-com­plex and \( \mathcal{A} \) is a twist­ing datum on \( X. \)

Defin­i­tion: A twis­ted-by-\( \mathcal{A} \) K-cycle on \( X \) is a triple \( (M, \sigma, \varphi) \) where

  • \( M \) is a com­pact \( \mathrm{Spin}^c \) man­i­fold without bound­ary,

  • \( \varphi : M \rightarrow X \) is a con­tinu­ous map,

  • \( \sigma\in K_0(\Gamma(M, \varphi^*\mathcal{A}^{\mathrm{op}})) \) (the \( K_0 \)-group of the \( C^* \)-al­gebra \( \Gamma(M, \varphi^*\mathcal{A}^{\mathrm{op}})). \)

Re­mark: The twis­ted K-cycles defined here are closely re­lated to the cycles in the ori­gin­al for­mu­la­tion of the Baum–Connes con­jec­ture. Later we will define the no­tion of \( D \)-cycles, mo­tiv­ated by \( D \)-branes in string the­ory, which may be re­garded as rep­res­ent­ing geo­met­ric cycles for \( (X, \mathcal{A}). \) In fact the two res­ult­ing twis­ted K-ho­mo­lo­gies are iso­morph­ic.

Keep­ing \( X, \mathcal{A} \) fixed, de­note by \( \{(M, \varphi, \sigma)\} \) the col­lec­tion of all twis­ted-by-\( \mathcal{A} \) K-cycles on \( X. \) On this col­lec­tion im­pose the equi­val­ence re­la­tion \( \sim \) gen­er­ated by the two ele­ment­ary steps

  • bor­d­ism,

  • vec­tor bundle modi­fic­a­tion.

Next, we elab­or­ate on these two steps for the case of twis­ted K-cycles.

Iso­morph­ism: We say \( (M, \varphi, \sigma) \) is iso­morph­ic to \( (M^{\prime}, \varphi ^{\prime}, \sigma^{\prime}) \) if and only if there ex­ists a dif­feo­morph­ism \( \psi : M\to M^{\prime} \) pre­serving the \( \mathrm{Spin}^c \)-struc­tures and with com­mut­ativ­ity in the dia­gram shown to the right and in ad­di­tion \( \psi^*(\sigma^{\prime}) = \sigma , \) where \[ \psi^*: K_0\Gamma(M^{\prime}, \varphi^{\prime *}\mathcal{A})\longrightarrow K_0\Gamma(M, \varphi^*\mathcal{A}) \] is the iso­morph­ism of K-the­ory de­term­ined by the iso­morph­ism of \( C^* \)-al­geb­ras \[ \Gamma(M^{\prime}, \varphi^{\prime *}\mathcal{A}^{\mathrm{op}})\cong\Gamma(M, \varphi^*\mathcal{A}^{\mathrm{op}}). \]

Bor­d­ism: We say \( (M_0, \varphi_0, \sigma_0) \) is bord­ant to \( (M_1,\varphi_1, \sigma_1) \) if and only if there ex­ists \( (W, \varphi, \sigma) \) such that

  1. \( W \) is a com­pact \( \mathrm{Spin}^c \) man­i­fold with bound­ary;

  2. \( \varphi \) is a con­tinu­ous map from \( W \) to \( X; \)

  3. \( \sigma\in K_0(\Gamma(W, \varphi^*\mathcal{A}^{\mathrm{op}})); \)

  4. \( (\partial W, \varphi |_{\partial W},\sigma |_{\partial W})\cong (M_0, \varphi_0, \sigma_0)\sqcup (-M_1, \varphi_1, \sigma_1). \)

Vec­tor bundle modi­fic­a­tion: Let \( (M, \varphi, \sigma) \) be a twis­ted-by-\( \mathcal{A} \) K-cycle on \( X, \) and let \( F \) be a \( \mathrm{Spin}^c \) vec­tor bundle on \( M \) of even rank. As in the un­twis­ted case, \( \mathbf{1}_{\mathbb{R}} \) de­notes the trivi­al real line bundle on \( M, \) \( S(F\oplus \mathbf{1}_{\mathbb{R}}) \) is the unit sphere bundle of \( F\oplus \mathbf{1}_{\mathbb{R}} \) and \[ \pi: S(F\oplus \mathbf{1}_{\mathbb{R}}) \to M \] is a fibra­tion. Let \( s: M \to S(F\oplus 1_\mathbb{R}) \) be the ca­non­ic­al unit sec­tion of \( \mathbf{1}_{\mathbb{R}}. \) Then the giv­en \( \mathrm{Spin}^c \) struc­ture for \( F \) de­term­ines a Gys­in ho­mo­morph­ism (see [e32]) \[ s_*: K_0( \Gamma(M, \varphi^*\mathcal{A}^{\mathrm{op}})) \longrightarrow K_0(\Gamma(S(F\oplus \mathbf{1}_{\mathbb{R}}) , (\varphi\circ\pi)^*\mathcal{A}^{\mathrm{op}})). \] Here we use the fact that \[ s^*( \varphi\circ\pi)^*\mathcal{A}^{\mathrm{op}} = \varphi^*\mathcal{A}^{\mathrm{op}}. \] Then \[ (M, \varphi, \sigma) \sim (S(F\oplus\mathbf{1}_{\mathbb{R}}), \varphi\circ\pi, s_*\sigma). \] In the fol­low­ing \( (S(F\oplus\mathbf{1}_{\mathbb{R}}), \varphi\circ\pi, s_*\sigma) \) will be de­noted by \( F\#(M, \varphi, \sigma) \) and will be re­ferred to as the modi­fic­a­tion of \( (M, \varphi, \sigma) \) by \( F. \)

Com­pos­i­tion Lemma: Let \( (M, \varphi, \sigma) \) be a twis­ted-by-\( \mathcal{A} \) K-cycle on \( X, \) and let \( F \) be an even-rank \( \mathrm{Spin}^c \) vec­tor bundle on \( M. \) Let \( F_1 \) be an even-rank \( \mathrm{Spin}^c \) vec­tor bundle on \( S(F \oplus \mathbf{1}_{\mathbb{R}}). \) Then \( F_1\#(F \# (M, \varphi, \sigma)) \) is (in a ca­non­ic­al way) bord­ant to \( (s^*F_1\oplus F )\# (M, \varphi, \sigma). \)

Proof.  We first con­struct a bord­ant man­i­fold \( W \) between \[ S(F_1\oplus \mathbf{1}_{\mathbb{R}}) \qquad\text{and}\qquad S(s^*F_1\oplus F \oplus \mathbf{1}_{\mathbb{R}}). \] Let \( D(F_1\oplus \mathbf{1}_{\mathbb{R}}) \) be the unit ball bundle of \( F_1\oplus \mathbf{1}_{\mathbb{R}}; \) then \[ S(F_1\oplus \mathbf{1}_{\mathbb{R}}) = \partial ( D(F_1\oplus \mathbf{1}_{\mathbb{R}})) \] is the un­der­ly­ing man­i­fold for \( F_1\# (F \# (M, \varphi, \sigma)). \) Note that \( s^*F \oplus F\oplus\mathbf{1}_{\mathbb{R}} \) is iso­morph­ic to the nor­mal bundle \( \nu_\iota \) for the in­clu­sion map \[ \iota: M \rightarrow D(F_1\oplus \mathbf{1}_{\mathbb{R}}), \] where \( \iota \) is defined by the com­pos­i­tion of \[ s: M \to S(F \oplus \mathbf{1}_{\mathbb{R}}) \] and the zero sec­tion of the bundle \( F_1\oplus \mathbf{1}_{\mathbb{R}} \) over \( S(F \oplus \mathbf{1}_{\mathbb{R}}). \) This en­sures that we can identi­fy the sphere bundle of \( s^*F_1 \oplus F\oplus\mathbf{1}_{\mathbb{R}} \) with the bound­ary of the ball bundle of ra­di­us \( \frac14 \) in the nor­mal bundle \( \nu_\iota. \) Define \[ \widetilde W = D(F_1\oplus \mathbf{1}_{\mathbb{R}}) - D_{1/4}(\nu_\iota). \] Then \[ \partial \widetilde W \cong S(F_1\oplus \mathbf{1}_{\mathbb{R}}) \sqcup - S(s^*F_1 \oplus F\oplus\mathbf{1}_{\mathbb{R}}). \] Let \[ \tilde{\varphi}: \widetilde W \subset D(F_1\oplus \mathbf{1}_{\mathbb{R}}) \xrightarrow{\ \pi\ } S(F\oplus \mathbf{1}_{\mathbb{R}}) \xrightarrow{\pi_F} M \] be the ob­vi­ous pro­jec­tion and \( \tilde{s} \) be the com­pos­i­tion of the ca­non­ic­al unit sec­tions \( s \) and \( s_1 \) for \( F \oplus \mathbf{1}_{\mathbb{R}} \) and \( F_1 \oplus \mathbf{1}_{\mathbb{R}} \) re­spect­ively. Then \( (\widetilde W, \tilde{\varphi}, \tilde{s}_*\sigma ) \) provides the bor­d­ism between \( F_1\#(F \# (M, \varphi, \sigma)) \) and \( (s^*F_1\oplus F )\#(M, \varphi, \sigma). \) Here we ap­plied the facts that the push­for­ward map \( \tilde{s}_* \) is func­tori­al and is also a ho­mo­topy in­vari­ant. ☐

Two twis­ted-by-\( \mathcal{A} \) K-cycles \( (M, \varphi, \sigma) \) and \( (M^{\prime}, \varphi^{\prime}, \sigma^{\prime}) \) on \( X \) are equi­val­ent if and only if it is pos­sible to pass from \( (M, \varphi, \sigma) \) to \( (M^{\prime}, \varphi^{\prime}, \sigma^{\prime}) \) by a fi­nite se­quence of the two ele­ment­ary steps. The K-cycle (or to­po­lo­gic­al) twis­ted-by-\( \mathcal{A} \) K-ho­mo­logy of \( X, \) de­noted by \( K_*^{\mathrm{top}}(X, \mathcal{A}), \) is the set of equi­val­ence classes of twis­ted-by-\( \mathcal{A} \) K-cycles: \[ K_*^{\mathrm{top}}(X, \mathcal{A}) := \{(M, \varphi, \sigma)\}/\sim. \] Ad­di­tion in \( K_*^{\mathrm{top}}(X, \mathcal{A}) \) is defined by dis­joint uni­on of twis­ted-by-\( \mathcal{A} \) K-cycles: \[ (M, \varphi, \sigma) + (M^{\prime}, \varphi^{\prime}, \sigma^{\prime}) = (M\sqcup M^{\prime}, \varphi \sqcup \varphi^{\prime}, \sigma\oplus\sigma^{\prime}). \]

Defin­i­tion: Let \( (M, \varphi, \sigma) \) be a twis­ted-by-\( \mathcal{A} \) K-cycle on \( X. \) We say \( (M, \varphi, \sigma) \) bounds if and only if there ex­ists \( (W, \tilde{\varphi}, \tilde{\sigma}), \) where

  1. \( W \) is a com­pact \( \mathrm{Spin}^c \) man­i­fold with bound­ary,

  2. \( \tilde{\varphi}: W \rightarrow X \) is a con­tinu­ous map,

  3. \( \tilde{\sigma}\in K_0(\Gamma(W, \tilde{\varphi}^*\mathcal{A}^{\mathrm{op}})), \)

  4. \( (\partial W, \tilde{\varphi}|_{\partial W}, \tilde{\sigma}|_{\partial W})\cong (M, \varphi, \sigma). \)

As in the un­twis­ted case, the ad­dit­ive in­verse of \( (M, \varphi, \sigma) \) is \( (-M, \varphi, \sigma). \) The equi­val­ence re­la­tion \( \sim \) on twis­ted-by-\( \mathcal{A} \) K-cycles \( (M, \varphi, \sigma) \) pre­serves the di­men­sion of \( M \) mod­ulo 2. There­fore, as an abeli­an group, \( K_*^{\mathrm{top}}(X, \mathcal{A}) \) is the dir­ect sum \[ K_*^{\mathrm{top}}(X, \mathcal{A}) = K_0^{\mathrm{top}}(X, \mathcal{A}) \oplus K_1^{\mathrm{top}}(X, \mathcal{A}), \] where \( K_j^{\mathrm{top}}(X, \mathcal{A}) \) is the sub­group of \( K_*^{\mathrm{top}}(X,\mathcal{A}) \) gen­er­ated by those twis­ted-by-\( \mathcal{A} \) K-cycles \( (M, \varphi, \sigma) \) such that every con­nec­ted com­pon­ent of \( M \) has di­men­sion \( \equiv j\thinspace \mathrm{modulo}\thinspace 2, \) \( j = 0, 1. \)

There is a nat­ur­al map \[ \eta_X : K_j^{\mathrm{top}}(X, \mathcal{A}) \longrightarrow KK_c^j(\Gamma_0(X, \mathcal{A}), \mathbb{C}) \] defined as fol­lows. Giv­en a twis­ted-by-\( \mathcal{A} \) K-cycle \( (M, \varphi, \sigma) \) on \( X, \) by ap­ply­ing Poin­caré du­al­ity in twis­ted K-the­ory (see [e31] and [e35]) \[ KK_0(\mathbb{C}, \Gamma (M, \varphi^*\mathcal{A}^{\mathrm{op}} )) \cong KK^j( \Gamma (M, \varphi^*\mathcal{A} ), \mathbb{C}), \] we have \[ PD ( \sigma ) \in KK^j( \Gamma (M, \varphi^*\mathcal{A} ), \mathbb{C}). \] De­note by \[ \varphi_* : KK^*(\Gamma(M, \varphi^*\mathcal{A}), \mathbb{C} ) \longrightarrow KK_c^*(\Gamma_0(X, \mathcal{A}), \mathbb{C}) \] the map of abeli­an groups in­duced by \( \varphi : M \rightarrow X. \) Then the nat­ur­al map \( \eta_X \) is giv­en by \[ \eta_X (M, \varphi, \sigma) = \varphi_* (PD ( \sigma ) ). \] It is routine to check that \( \eta_X \) is well-defined on \( K_j^{\mathrm{top}}(X, \mathcal{A}) \) and is func­tori­al in the fol­low­ing sense. Let \( f: Y\to X \) be a con­tinu­ous map and \( X \) be equipped with a twist­ing datum \( \mathcal{A}; \) then the fol­low­ing dia­gram com­mutes: \[ \begin{CD} K_j^{\mathrm{top}}(Y,f^* \mathcal{A}) @>{{f_* }}>{{}}> K_j^{\mathrm{top}}(X, \mathcal{A})\\ @VV{\eta_Y}V @VV{\eta_X}V \\ KK_c^j(\Gamma_0(Y,f^* \mathcal{A}), \mathbb{C}) @>{{f_* }}>{{}}> KK_c^j(\Gamma_0(X, \mathcal{A}), \mathbb{C}). \end{CD} \] Here \[ f_*: K_j^{\mathrm{top}}(Y, f^*\mathcal{A}) \longrightarrow K_j^{\mathrm{top}}(X, \mathcal{A}) \] is defined by \[ (M, \varphi, \sigma) \longmapsto (M, f \circ \varphi, \sigma) \] on the level of K-cycles.

Paul has of­ten enun­ci­ated the un­twis­ted ver­sion of the fol­low­ing prob­lem.

The Gen­er­al Twis­ted In­dex Prob­lem: When \( X \) is a loc­ally fi­nite CW-com­plex and \( \mathcal{A} \) is a twist­ing datum on \( X \) then the gen­er­al twis­ted in­dex prob­lem is: giv­en an ele­ment \[ \xi\in KK_c^*(\Gamma_0(X, \mathcal{A}), \mathbb{C}), \] ex­pli­citly con­struct a twis­ted-by-\( \mathcal{A} \) K-cycle \( (M, \varphi, \sigma ) \) for \( ( X, \mathcal{A}) \) with \( \eta(M, \varphi, \sigma) = \xi. \)

The is­sue in this prob­lem is con­struct­ib­il­ity of the K-cycle.

In [4] the fol­low­ing is proved us­ing the no­tion of nor­mal bor­d­ism (in­tro­duced next) to es­tab­lish the six-term ex­act se­quence in Kas­parov K-ho­mo­logy of \( X. \)

The­or­em 10: Let \( X \) be a loc­ally fi­nite CW-com­plex with a giv­en twist­ing datum \( \mathcal{A}. \) Then the nat­ur­al map \[ \eta_X : K_j^{\mathrm{top}}(X, \mathcal{A}) \longrightarrow KK_c^j(\Gamma_0(X, \mathcal{A}),\mathbb{C}), \quad j=0, 1, \] is an iso­morph­ism of abeli­an groups.

5.2.3. Normal bordism

One of the main in­nov­a­tions in [4] was the elu­cid­a­tion of the fun­da­ment­al role of nor­mal bor­d­ism. Let \( (M, \varphi, \sigma) \) be a twis­ted-by-\( \mathcal{A} \) K-cycle on \( X. \) A nor­mal bundle for \( M \) is an real vec­tor bundle \( \nu \) on \( M \) to­geth­er with a giv­en short ex­act se­quence \[ 0\rightarrow TM\longrightarrow M\times\mathbb{R}^q\longrightarrow\nu\rightarrow 0 \] of real vec­tor bundles on \( M. \) Here \( q \) is a pos­it­ive in­teger and \( M\times\mathbb{R}^q \) is the trivi­al real vec­tor bundle with fibre \( \mathbb{R}^q. \) The two-out-of-three prin­ciple im­plies that \( \nu \) is a \( \mathrm{Spin}^c \) vec­tor bundle. If \( \nu \) has even-di­men­sion­al fibres, then the modi­fic­a­tion of \( (M, \varphi, \sigma) \) by \( \nu \) will be de­noted by \( \nu\# (M, \varphi, \sigma). \)

Defin­i­tion: Two twis­ted-by-\( \mathcal{A} \) K-cycles \( (M, \varphi, \sigma) \) and \( (M^{\prime}, \varphi^{\prime}, \sigma^{\prime}) \) are nor­mally bord­ant, de­noted by \[(M, \varphi, \sigma) \sim_N (M^{\prime}, \varphi^{\prime}, \sigma^{\prime}), \] if there ex­ist nor­mal bundles with even-di­men­sion­al fibres \( \nu \) and \( \nu^{\prime} \) for \( M \) and \( M^{\prime} \) re­spect­ively such that \( \nu\#(M, \varphi, \sigma) \) is bord­ant to \( \nu^{\prime}\#(M^{\prime}, \varphi^{\prime}, \sigma^{\prime}). \)

Lemma 11: Nor­mal bor­d­ism is an equi­val­ence re­la­tion.

Re­mark: The con­tent of the next lemma is that for nor­mal bor­d­ism there is no need to use ele­ment­ary steps etc. The idea of nor­mal bor­d­ism was first con­struc­ted by Jakob in [e22] for an al­tern­at­ive defin­i­tion of gen­er­al­ised ho­mo­logy the­ory, and fur­ther ap­plied in [e28] and [e38] in the study of vari­ous ver­sions of geo­met­ric K-ho­mo­logy.

Lemma 12: Two twis­ted-by-\( \mathcal{A} \) K-cycles \( (M, \varphi, \sigma) \) and \( (M^{\prime}, \varphi^{\prime}, \sigma^{\prime}) \) are equi­val­ent if and only if they are nor­mally bord­ant.

A twis­ted-by-\( \mathcal{A} \) K-cycle \( (M, \varphi, \sigma) \) is zero in \( K_*^{\mathrm{top}}(X, \mathcal{A}) \) if and only if \( (M, \varphi, \sigma) \) is nor­mally bord­ant to a twis­ted-by-\( \mathcal{A} \) K-cycle which bounds.

5.2.4. The group of \( D \)-cycles

In this sec­tion, we in­tro­duce an­oth­er no­tion of K-cycles for \( (X, \mathcal{A}), \) called \( D \)-cycles, which are closely re­lated to the no­tion of \( D \)-branes in string the­ory. We be­gin with the fol­low­ing:

Defin­i­tion: A \( D \)-cycle for \( (X, \mathcal{A}) \) is a 4-tuple \( (M, E, \varphi, \mathcal{S}) \) such that

  1. \( M \) is a closed ori­ented \( C^{\infty} \) Rieman­ni­an man­i­fold;

  2. \( E \) is a com­plex vec­tor bundle on \( M; \)

  3. \( \varphi \) is a con­tinu­ous map from \( M \) to \( X; \)

  4. \( \mathcal{S} \) is a spinor bundle for \( \mathrm{Cliff}^+_\mathbb{C}(TM)\otimes \varphi^* \mathcal{A}^{\mathrm{op}}. \)

Re­mark:

  1. If \( M \) is even-di­men­sion­al, then \( \mathrm{Cliff}^+_\mathbb{C}(TM) \) is the twist­ing datum on \( M \) whose fibre at \( p \in M \) is the com­plexi­fied Clif­ford al­gebra \( \mathbb{C} \otimes_{\mathbb{R}} \mathrm{Cliff}(T_pM). \) As usu­al, \( T_pM \) is the tan­gent space to \( M \) at \( p. \) On the oth­er hand, if \( M \) is odd-di­men­sion­al, giv­en \( p \in M, \) choose a pos­it­ively ori­ented or­thonor­mal basis \( e_1, e_2, \dots, e_n \) for \( T_pM. \) Set \( n = 2r + 1 \) and define \( \omega(p) \in \mathbb{C} \otimes_{\mathbb{R}} \mathrm{Cliff}(T_pM) \) by \[ \omega(p) = i^{r+1}e_1e_2 \cdots e_n. \] Then \( \omega(p) \) does not de­pend on the choice of pos­it­ively ori­ented or­thonor­mal basis for \( T_pM. \) Also \( \omega(p) \) is in the centre of \[ \mathrm{Cliff}_{\mathbb{C}}(T_pM) = \mathbb{C} \otimes_{\mathbb{R}} \mathrm{Cliff}(T_pM) \] and \( \omega(p)^2 = 1. \)

    Now set \[ \mathrm{Cliff}^+_{\mathbb{C}}(T_pM) = \{a \in \mathrm{Cliff}_{\mathbb{C}}(T_pM)\mid \omega(p)a = a\} . \] Then \( \mathrm{Cliff}^+_\mathbb{C}(TM) \) is the twist­ing datum on \( M \) whose fibre at \( p \in M \) is \( \mathrm{Cliff}^+_\mathbb{C}(T_pM). \)

  2. The ex­ist­ence of a spinor bundle \( \mathcal{S} \) for \( \mathrm{Cliff}^+_\mathbb{C}(TM)\otimes \varphi^* \mathcal{A}^{\mathrm{op}} \) im­plies \[ \mathrm{DD}(\mathrm{Cliff}^+_\mathbb{C}(TM)) = \varphi^* (\mathrm{DD}(\mathcal{A})). \] By stand­ard al­geb­ra­ic to­po­logy, \( \mathrm{DD}(\mathrm{Cliff}^+_\mathbb{C}(TM)) \) is the third (in­teg­ral) Stiefel–Whit­ney class of \( M, \) so the ex­ist­ence of \( \mathcal{S} \) im­plies \[ W_3(M) = \varphi^* (\mathrm{DD}(\mathcal{A})), \] which is the Freed–Wit­ten an­om­aly can­cel­la­tion con­di­tion for Type IIB \( D \)-branes as ex­plained in [e33].

Defin­i­tion: Keep­ing \( (X, \mathcal{A}) \) fixed, de­note by \( \{(M, E, \varphi, \mathcal{S})\} \) the col­lec­tion of all \( D \)-cycles for \( (X, \mathcal{A}). \) On this col­lec­tion im­pose the equi­val­ence re­la­tion \( \sim \) gen­er­ated by the three ele­ment­ary steps

  • bor­d­ism,

  • dir­ect sum - dis­joint uni­on,

  • vec­tor bundle modi­fic­a­tion.

These three ele­ment­ary moves can be pre­cisely defined as fol­lows.

Iso­morph­ism: Two \( D \)-cycles \( (M, E, \varphi, \mathcal{S}), \) \( (M^{\prime}, E^{\prime}, \varphi^{\prime}, \mathcal{S}^{\prime}) \) for \( (X, \mathcal{A}) \) are iso­morph­ic if there is an ori­ent­a­tion-pre­serving iso­met­ric dif­feo­morph­ism \( f : M \to M^{\prime} \) such that the dia­gram shown to the right com­mutes, and \( f^*E^{\prime}\cong E, \) \( f^*\mathcal{S}^{\prime}\cong \mathcal{S}. \)

Bor­d­ism: Two \( D \)-cycles \( (M_0, E_0, \varphi_0, \mathcal{S}_0), \) \( (M_1, E_1, \varphi_1, \mathcal{S}_1) \) for \( (X, \mathcal{A}) \) are bord­ant if there ex­ists a 4-tuple \( (W, E, \Phi, \mathcal{S}) \) such that \( W \) is a com­pact ori­ented Rieman­ni­an man­i­fold with bound­ary, \( E \) is a com­plex vec­tor bundle on \( W, \) \( \Phi \) is a con­tinu­ous map from \( W\to X \) and \[ (\partial W, E | \partial W , \Phi | \partial W, \mathcal{S}^{(+)}|\partial W) \cong (M_0, E_0, \varphi_0, \mathcal{S}_0) \sqcup (- M_1, E_1, \varphi_1, \mathcal{S}_1). \] When \( W \) is of odd di­men­sion, \[ \mathrm{Cliff}_\mathbb{C}^{(+)} (TW)|\partial W \cong \mathrm{Cliff}_\mathbb{C} (T(\partial W). \] Then \( \mathcal{S}^{(+)}= \mathcal{S}. \) When \( W \) is of even di­men­sion, \( \mathcal{S}^{(+)} \) is the pos­it­ive part of \( \mathcal{S}, \) the \( (+1) \)-ei­gen­bundle of the chir­al­ity sec­tion of \( \mathrm{Cliff}_\mathbb{C}^{(+)} (TW). \)

Dir­ect sum-dis­joint uni­on: Let \( (M, E, \varphi, \mathcal{S}) \) be a \( D \)-cycle for \( (X, \mathcal{A}) \) and let \( E^{\prime} \) be a com­plex vec­tor bundle on \( M; \) then \[ (M, E, \varphi, \mathcal{S}) \sqcup (M, E^{\prime}, \varphi, \mathcal{S}) \sim (M, E\oplus E^{\prime}, \varphi, \mathcal{S}). \]

Vec­tor bundle modi­fic­a­tion: Let \( (M, E, \varphi, \mathcal{S}) \) be a \( D \)-cycle for \( (X, \mathcal{A}) \) and let \( F \) be a \( \mathrm{Spin}^c \) vec­tor bundle on \( M \) with even-di­men­sion­al fibres. Then, with \( \mathcal{S}_{F} \) be­ing the Spinor bundle for the \( \mathrm{Spin}^c \) vec­tor bundle \( F, \) \[ (M, E, \varphi, \mathcal{S}) \sim (S(F\oplus 1_\mathbb{R}), \beta\otimes \pi^*E , \varphi\circ \pi, \pi^*\mathcal{S}_{F}\otimes \pi^*\mathcal{S}), \] with the oth­er nota­tion in this equi­val­ence be­ing the same as that in­tro­duced pre­vi­ously. The group of \( D \)-cycles for \( (X, \mathcal{A}), \) de­noted by \( K^{\mathrm{geo}}_*(X, \mathcal{A}), \) is the set of equi­val­ence classes of these \( D \)-cycles un­der the equi­val­ence defined above in terms of the three re­la­tions. Ad­di­tion in \( K^{\mathrm{geo}}_*(X, \mathcal{A}) \) is dis­joint uni­on of \( D \)-cycles: \[ (M, E, \varphi, \mathcal{S}) + (M^{\prime}, E^{\prime}, \varphi^{\prime}, \mathcal{S}^{\prime}) = (M\sqcup M^{\prime}, E\sqcup E^{\prime}, \varphi \sqcup \varphi^{\prime}, \mathcal{S} \sqcup \mathcal{S}^{\prime}). \] The equi­val­ence re­la­tion \( \sim \) on \( D \)-cycles \( (M, E, \varphi, \mathcal{S}) \) pre­serves the di­men­sion of \( M \) mod­ulo 2 so that we have \[ K^{\mathrm{geo}}_*(X, \mathcal{A}) = K^{\mathrm{geo}}_0(X, \mathcal{A}) \oplus K^{\mathrm{geo}}_1(X, \mathcal{A}), \] where \( K^{\mathrm{geo}}_j(X, \mathcal{A}) \) is the sub­group of \( K^{\mathrm{geo}}_*(X, \mathcal{A}) \) gen­er­ated by those \( (X, \mathcal{A}) \) \( D \)-cycles \( (M, E, \varphi, \mathcal{S}) \) such that every con­nec­ted com­pon­ent of \( M \) has di­men­sion \( \equiv j \) mod­ulo 2, \( j = 0, 1. \)

Let \( (M, E, \varphi, \mathcal{S}) \) be a \( D \)-cycle on \( (X, \mathcal{A}) \) such that every con­nec­ted com­pon­ent of \( M \) has its di­men­sion con­gru­ent to \( j \) mod­ulo \( 2. \) Let \( D_E \) de­note the Dir­ac ele­ment of \( M \) tensored with \( E. \) It can be de­scribed in terms of the \( \frac12 \)-sig­na­ture op­er­at­or on \( M. \) Then as in [e8], \( D_E \) yields an ele­ment in the Kas­parov K-ho­mo­logy group \[ [D_E] \in KK^j(C(M, \mathrm{Cliff}^{(+)}_\mathbb{C} (M)), \mathbb{C}). \] The iso­morph­ism \( \Psi_\mathcal{S} \) defined in Lemma 9 in Sec­tion 5 reads as \[ \Psi: KK^j(C(M, \mathrm{Cliff}^{(+)}_\mathbb{C} (M)), \mathbb{C}) \longrightarrow KK^j(\Gamma(M, \varphi^*\mathcal{A}), \mathbb{C}). \] The map of \( C^* \)-al­geb­ras \( \Gamma (X, \mathcal{A}) \to \Gamma (M, \varphi^*\mathcal{A}) \) in­duces a ho­mo­morph­ism of abeli­an groups \[ \varphi_*: KK^j(\Gamma(M, \varphi^*\mathcal{A}), \mathbb{C}) \longrightarrow KK^j(\Gamma(X, \mathcal{A}), \mathbb{C}). \] Then \( (M, E, \varphi, \mathcal{S}) \mapsto \varphi_*[D_E] \) yields a ho­mo­morph­ism of abeli­an groups, de­noted by \[ \mu: K^{\mathrm{geo}}_*(X, \mathcal{A}) \longrightarrow KK^*(\Gamma (X, \mathcal{A}), \mathbb{C}). \] We call this the twis­ted in­dex map in twis­ted K-ho­mo­logy of \( (X, \mathcal{A}). \)

5.2.5. The charge map \( K^{\mathrm{geo}}_*(X, \mathcal{A})\rightarrow K_*^{\mathrm{top}}(X, \mathcal{A})\)
We use nor­mal bundle modi­fic­a­tion to define a ho­mo­morph­ism of abeli­an groups \[ h: K^{\mathrm{geo}}_*(X, \mathcal{A})\longrightarrow K_*^{\mathrm{top}}(X, \mathcal{A}). \] Here \( h \) should be viewed as the map which sends a \( D \)-cycle \( (M, E, \varphi, \mathcal{S}) \) to its charge.

Defin­i­tion: Giv­en a \( D \)-cycle \( (M, E, \varphi, \mathcal{S}) \) choose a nor­mal bundle \( \nu \) for \( M, \) with even-di­men­sion­al fibres. Then \[ h(M, E, \varphi, \mathcal{S}):= \nu\#(M, E, \varphi, \mathcal{S}) = (S(\nu\oplus 1_{\mathbb{R}}), \varphi\circ\pi, \sigma). \] Note that \( S(\nu\oplus 1_{\mathbb{R}}) \) is a \( \mathrm{Spin}^c \) man­i­fold be­cause its tan­gent bundle is stably trivi­al­ized. Here \( \sigma \) is the ele­ment in \[ K_0(\Gamma(S(\nu\oplus\mathbf{1}_{\mathbb{R}}, (\varphi\circ\pi)^*\mathcal{A}^{\mathrm{op}})) \] ob­tained from \( E \) as fol­lows:

  1. De­note by \[ s_*: K^0(M)\longrightarrow K_0\bigl(\Gamma(S(\nu\oplus\mathbf{1}_{\mathbb{R}}), \pi^*\mathrm{Cliff}_{\mathbb{C}}(\nu))\bigr) \] the Gys­in ho­mo­morph­ism as­so­ci­ated to the ca­non­ic­al sec­tion \( s: M\rightarrow S(\nu\oplus\mathbf{1}_{\mathbb{R}}). \)

  2. Ob­serve that, as twist­ing data on \( M, \) a trivi­al­isa­tion of \( TM\oplus \nu \) gives an equi­val­ence \[ \mathrm{Cliff}_{\mathbb{C}}^+(TM)\otimes\varphi^*\mathcal{A}^{\mathrm{op}} \sim \mathrm{Cliff}_{\mathbb{C}}(\nu)^{\mathrm{op}}\otimes\varphi^*\mathcal{A}^{\mathrm{op}}, \] a stable iso­morph­ism of bundles of ele­ment­ary \( C^* \)-al­geb­ras. Hence the giv­en spinor bundle \( \mathcal{S} \) for \[ \mathrm{Cliff}_{\mathbb{C}}^+(TM)\otimes\varphi^*\mathcal{A}^{\mathrm{op}} \] de­term­ines a spinor bundle \( \widetilde{\mathcal{S}} \) for \[ \mathrm{Cliff}_{\mathbb{C}}(\nu)^{\mathrm{op}}\otimes\varphi^*\mathcal{A}^{\mathrm{op}}. \] Then \( \pi^*\widetilde{\mathcal{S}} \) is a spinor bundle for \[ \pi^*\mathrm{Cliff}_{\mathbb{C}}(\nu)^{\mathrm{op}}\otimes(\varphi\circ\pi)^*\mathcal{A}^{\mathrm{op}} \] and there­fore yields an iso­morph­ism of abeli­an groups \[ \chi: K_0\bigl(\Gamma(S(\nu\oplus\mathbf{1}_{\mathbb{R}}), \pi^*\mathrm{Cliff}_{\mathbb{C}}(\nu))\bigr) \longrightarrow K_0\bigl(\Gamma(S(\nu\oplus\mathbf{1}_{\mathbb{R}}), (\varphi\circ\pi)^*\mathcal{A}^{\mathrm{op}}) \bigr). \]

  3. \( \sigma :=\chi (s_*[E]). \)

Ques­tion: Is the map \[ h: K^{\mathrm{geo}}_j(X, \mathcal{A})\longrightarrow K_j^{\mathrm{top}}(X, \mathcal{A}) \] an iso­morph­ism for any loc­ally fi­nite CW-com­plex?

Giv­en a pos­it­ive an­swer to this ques­tion then we would have the fol­low­ing co­rol­lary:

Con­sequence: Let \( X \) be a loc­ally fi­nite CW-com­plex with a twist­ing datum \( \mathcal{A} \) defined by a prin­cip­al \( \mathrm{PU}(H) \)-bundle \( \mathcal{P}, \) and \( K^{\mathrm{geo}}_*(X, \mathcal{P}) \) be the twis­ted geo­met­ric K-ho­mo­logy of \( (X, \mathcal P) \) in [e33]. Then the twis­ted in­dex map \[ \mu: K^{\mathrm{geo}}_*(X, \mathcal{P}) \longrightarrow KK^*_c(\Gamma(X, \mathcal{A} ), \mathbb{C}) \] is an iso­morph­ism.

6. The latest chapter in the story

The au­thors of [4] have, both sep­ar­ately and in col­lab­or­a­tion, come up with vari­ous pro­pos­als for an­swer­ing the ques­tion posed above but a com­plete ar­gu­ment eluded us. Re­cently Paul and col­lab­or­at­ors [5] answered the ques­tion in the af­firm­at­ive, thus ex­plain­ing how the ori­gin­al ideas of Bai-Ling Wang, in un­der­stand­ing the Freed–Wit­ten an­om­aly can­cel­la­tion con­di­tion, form the key as­sump­tion for cre­at­ing twis­ted geo­met­ric cycles.

Moreover this very latest proof is the most gen­er­al so far. It al­lows by one meth­od to es­tab­lish all pre­vi­ous ver­sions of the gen­er­al in­dex prob­lem in both the twis­ted and un­twis­ted cases.

In this new ar­gu­ment Wang’s ap­peal to a twis­ted ver­sion of the Con­ner–Floyd split­ting the­or­em is re­placed by a twis­ted ana­logue of a the­or­em of Hop­kins and Hovey. They proved that \( \mathrm{Spin}^c \)-bor­d­ism of a CW-com­plex de­term­ines its K-ho­mo­logy by a simple al­geb­ra­ic tensor product. That is, the af­firm­at­ive an­swer to the ques­tion posed above that is es­tab­lished in [5] rests on a twis­ted ver­sion of this the­or­em of Hop­kins and Hovey.

Re­call that giv­en a CW-com­plex \( X \) with a twist­ing \( \alpha: X\to K(\mathbb{Z}, 3), \) there are two ap­proaches to define to twis­ted \( \mathrm{Spin}^c \)-bor­d­ism the­ory of \( (X, \alpha) \) in [e33]. The first ap­proach is the geo­met­ric defin­i­tion us­ing a smooth man­i­fold \( M \) with a con­tinu­ous map \( \iota: M \to X \) and a ho­mo­topy real­ising the Freed–Wit­ten con­di­tion (see [e19]) \begin{equation}\label{cond} \iota^*([\alpha]) + W_3(M)=0, \end{equation} where \( W_3(M) \) is the third in­teg­ral Stiefel–Whit­ney class of \( M \) and \( [\alpha] \) de­notes the ho­mo­topy class of \( \alpha \) in \[ [X, K(\mathbb{Z}, 3)] \cong H^3(X, \mathbb{Z}). \] The \( \alpha \)-twis­ted \( \mathrm{Spin}^c \)-bor­d­ism group of \( X, \) de­noted by \( \Omega^{\mathrm{Spin}^c}_*(X, \alpha), \) is defined to be the set of all iso­morph­ism classes of closed \( \alpha \)-twis­ted \( \mathrm{Spin}^c \) man­i­folds over \( X \) mod­ulo null-bor­d­ism, with the sum giv­en by the dis­joint uni­on. The twis­ted Hop­kins–Hovey the­or­em in [5] gives a pre­cise re­la­tion­ship between this former group and the \( D \)-cycle mod­el for twis­ted K-ho­mo­logy.

This res­ult is the key to prov­ing that the geo­met­ric mod­el for twis­ted K-ho­mo­logy in terms of \( D \)-cycles, (stud­ied in one form by Bai-Ling Wang and re­for­mu­lated by us as ex­plained in [4]) is in fact iso­morph­ic to Kas­parov’s ana­lyt­ic twis­ted K-ho­mo­logy. Moreover these \( D \)-cycles are a math­em­at­ic­al ver­sion of the phys­i­cists’ \( D \)-branes.

The up­shot for the present es­say is that all of the mod­els for the group of twis­ted K-ho­mo­logy classes dis­cussed pre­vi­ously are iso­morph­ic. I will not dis­cuss the de­tails of [5] here. Paul is pre­par­ing an ex­pos­it­ory ac­count which will provide a use­ful over­view.

Works

[1] P. Baum and R. G. Douglas: “In­dex the­ory, bor­d­ism, and \( K \)-ho­mo­logy,” pp. 1–​31 in Op­er­at­or al­geb­ras and \( K \)-the­ory (San Fran­cisco, 7–8 Janu­ary 1981). Edi­ted by R. G. Douglas and C. Schochet. Con­tem­por­ary Math­em­at­ics 10. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 1982. MR 658506 Zbl 0507.​55004 incollection

[2] P. Baum and R. G. Douglas: “\( K \) ho­mo­logy and in­dex the­ory,” pp. 117–​173 in Op­er­at­or al­geb­ras and ap­plic­a­tions (King­ston, ON, 14 Ju­ly–2 Au­gust 1980), part 1. Edi­ted by R. V. Kadis­on. Pro­ceed­ings of Sym­po­sia in Pure Math­em­at­ics 38. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 1982. MR 679698 Zbl 0532.​55004 incollection

[3] P. Baum, N. Hig­son, and T. Schick: “On the equi­val­ence of geo­met­ric and ana­lyt­ic \( K \)-ho­mo­logy,” pp. 1–​24 in Spe­cial is­sue: In hon­or of Robert D. MacPh­er­son, Part 3, published as Pure Ap­pl. Math. Q. 3 : 1. In­ter­na­tion­al Press (Som­merville, MA), 2007. MR 2330153 Zbl 1146.​19004 incollection

[4] P. Baum, A. Carey, and B.-L. Wang: “\( K \)-cycles for twis­ted \( K \)-ho­mo­logy,” pp. 69–​98 in Nanjing spe­cial is­sue on K-the­ory, num­ber the­ory and geo­metry, published as J. K-The­ory 12 : 1. Issue edi­ted by X. Guo, H. Qin, and G. Tang. Cam­bridge Uni­versity Press, August 2013. MR 3126635 Zbl 1300.​19003 incollection

[5] P. Baum, M. Joachim, M. Khorami, and T. Schick: Twis­ted \( spin^c \)-bor­d­ism, D-branes, and twis­ted K-ho­mo­logy. In pre­par­a­tion. techreport