# Celebratio Mathematica

## Paul Baum

### Twisted K-homology and Paul Baum

#### 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 $$\label{bundle} \begin{CD} \mathcal{P}_\alpha @>{{}}>{{}}> EK(\mathbb{Z}, 2) \\ @VVV @VVV \\ X @>{{}}>{{\alpha }}> K(\mathbb{Z}, 3) \end{CD}$$ 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 $\mathbf{Fred}(\mathcal{H}):$ $$\label{action} \mathrm{PU}(\mathcal{H}) \times \mathbf{Fred} (\mathcal{H}) \longrightarrow \mathbf{Fred} (\mathcal{H}).$$

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 $$\label{iso:eta} \eta_*: K^{\mathrm{ev}/\mathrm{odd}}(X, \alpha_0) \xrightarrow{\,\cong\ } K^{\mathrm{ev}/\mathrm{odd}}(X, \alpha_1).$$ 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 $$\label{cen:ext} 1\to U(1) \longrightarrow U(\mathcal{H}) \longrightarrow \mathrm{PU}(\mathcal{H}) \to 1.$$ 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]) $$\label{cond} \iota^*([\alpha]) + W_3(M)=0,$$ 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

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[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