Thom isomorphisms and K-orientations in KK

Written by GPT-6.1 Sol (OpenAI). Public domain (CC0).

A Euclidean vector bundle has a Bott operator in every fibre. Orthogonal changes of frame preserve its Clifford formula, so these operators form a global cycle. The fibrewise Dirac operators form its inverse. A spinor bundle then removes the Clifford coefficient algebra and gives the Thom isomorphism. We prove the two inverse identities before making that Morita reduction; this separates the analytic theorem from the choice of K-orientation.

We use the preceding bounded connection, technical-partition and associativity proofs, and the Clifford and Morita conventions in Graded C*-algebras, Clifford algebras and graded Hilbert modules. Section 0 proves the bounded-transform and unbounded-product arguments needed here. Sections 2–4 then prove the oscillator and simultaneous-rotation inverse identities for the actual bundle cycles. The relative-bundle description used below is proved in the earlier Relative difference bundles, radial pairs and the planar normalization, Theorems RK.3–RK.4. RK.6–RK.6a prove the actual planar projection and its complex-orientation sign. The exact even relative character, splitting injection and ordinary cohomological Thom inputs used in Section 8 are proved in Cohomological foundations for the complex Thom comparison, CF.1–CF.6.

For compact groups, the Haar probability measure used below is proved in Foundations for compact-group averaging and coefficient approximation, CPT-F-005. Its CPT-F-010 constructs continuous vector integrals, strict operator averages and source-localized compact integrals on arbitrary Hilbert modules.

0. The analytic product criterion used below

We prove the analytic recognition argument here. Lemma 0.0 proves the required resolvent, continuous functional-calculus and tensor-extension foundations locally. The bounded product, technical partition and essential-replacement proofs are the preceding programme lessons Kasparov's technical theorem and Connections and the existence of the Kasparov product. The following argument also treats a nonessential first representation.

Lemma 0.0 (resolvents, continuous calculus and tensor extension). Let \(E\) be a Hilbert \(B\)-module, and let \(D=D^*\) be a closed densely defined operator for which \(1+D^2\) has dense range. Then \(D\pm i\) are bijective from \(\operatorname{Dom}D\) to \(E\), with mutually adjoint inverse resolvents of norm at most one. There is a nondegenerate continuous functional calculus \(C_0(\mathbb R)\to\mathcal L(E)\); it extends to bounded continuous functions, and \[ \|(D-z)^{-1}\|\leq |\operatorname{Im}z|^{-1} \quad(\operatorname{Im}z\ne0). \] If \(Y\) is any \(B\)-\(C\) correspondence, the closure of \(D\otimes1\) on \(\operatorname{Dom}D\odot_B Y\) is self-adjoint regular on \(E\otimes_B Y\), and its bounded continuous functional calculus is \(f(D)\otimes1\). No separability or essentiality of the left action on \(Y\) is assumed.

Proof: resolvents. Symmetry cancels the mixed terms in \[ \langle(D\pm i)x,(D\pm i)x\rangle =\langle Dx,Dx\rangle+\langle x,x\rangle. \] Thus these operators are bounded below by one. Their ranges are closed: a convergent sequence of images makes the preimages Cauchy, and closedness of \(D\) then gives a preimage of the limit. The range of each contains the dense range of \(1+D^2\), since on \(\operatorname{Dom}D^2\) this is the product \((D+i)(D-i)\) in either order. Their ranges are therefore all of \(E\). Write \(R_\pm=(D\pm i)^{-1}\). For \(a,b\in E\), symmetry gives \[ \langle R_+a,b\rangle =\langle R_+a,(D-i)R_-b\rangle =\langle a,R_-b\rangle. \] Hence \(R_+^*=R_-\). The identities \(DR_\pm=1\mp iR_\pm\), first on their ranges, give \[ R_+-R_-=-2iR_+R_-=-2iR_-R_+. \] In particular they commute. Put \(U=1-2iR_+\). The preceding identities give \(UU^*=U^*U=1\), and \(1-U=2iR_+\) has dense range \(\operatorname{Dom}D\).

Proof: functional calculus. We first prove the continuous calculus of a unitary locally. For a Laurent polynomial \(p(z)=\sum_{j=-m}^m a_jz^j\), put \(M=\sup_{|z|=1}|p(z)|\). Given \(\xi\in E\) and \(N>2m\), the continuous \(E\)-valued polynomial \[ w_N(z)=N^{-1/2}\sum_{k=0}^{N-1}z^kU^{-k}\xi \] has \(\int_{\mathbb T}\langle w_N,w_N\rangle\,d\theta/(2\pi)=\langle\xi,\xi\rangle\). For \(m\leq l\leq N-1-m\), the coefficient of \(z^l\) in \(p(z)w_N(z)\) is \(N^{-1/2}U^{-l}p(U)\xi\). Integrating the inner product cancels distinct monomials, and every omitted coefficient contributes a positive element. Since pointwise \(|p(z)|^2\leq M^2\), this gives the positive-order inequality \[ \frac{N-2m}{N}\langle p(U)\xi,p(U)\xi\rangle \leq \int_{\mathbb T}\langle p w_N,p w_N\rangle\,\frac{d\theta}{2\pi} \leq M^2\langle\xi,\xi\rangle. \] Take norms and then \(N\to\infty\). Thus \(\|p(U)\|\leq\|p\|_\infty\), for every Laurent polynomial, on a Hilbert module as well as a Hilbert space.

Laurent polynomials are uniformly dense in \(C(\mathbb T)\). Indeed the kernels \(K_N(\theta)=N^{-1}|\sum_{k=0}^{N-1}e^{ik\theta}|^2\) are positive with normalized integral one. The geometric-series formula bounds their integral outside any fixed neighborhood of zero by a constant times \(N^{-1}\). Uniform continuity then makes convolution with \(K_N\) converge uniformly to a continuous circle function; each convolution is a Laurent polynomial. The polynomial norm bound therefore extends evaluation at \(U\) to a unital contractive *-homomorphism \(C(\mathbb T)\to\mathcal L(E)\). Multiplication and adjoints pass through uniform polynomial approximation. No earlier normal-calculus or spectral-representation assertion is needed for this construction. The map \[ u(t)=(t-i)/(t+i) \] identifies \(\mathbb R\) with the circle minus \(1\). For \(f\in C_0(\mathbb R)\), its transported function extends by zero at \(1\); evaluate it at \(U\) to define \(f(D)\). This is a contractive *-homomorphism. It is nondegenerate because the function \(1-u(t)=2i/(t+i)\) has operator value \(1-U\), whose range is dense.

For completeness, its extension to \(C_b(\mathbb R)\) is also constructive. Let \(e_n\in C_c(\mathbb R)\) be positive cutoffs tending uniformly to one on compact sets, with \(0\leq e_n\leq1\). For bounded continuous \(h\), the uniformly bounded operators \((he_n)(D)\) converge on every vector \(f(D)\xi\), because \(he_nf\to hf\) uniformly. Such vectors have dense span, so the operators converge on every vector to a bounded operator \(h(D)\). The same argument for \(\overline h\) gives its adjoint. Products and adjoints pass to these uniformly bounded vector limits, proving the extended *-homomorphism. Its value does not depend on the chosen cutoffs, by the same dense-span argument.

The identity \(D=i(1+U)(1-U)^{-1}\) holds on the dense domain \(\operatorname{ran}(1-U)\). For nonreal \(z\), the bounded scalar function \((t+i)/(t-z)\) therefore shows that the value of \((t-z)^{-1}\) has range in \(\operatorname{Dom}D\) and is a two-sided inverse of \(D-z\). Its norm is bounded by the scalar supremum \(|\operatorname{Im}z|^{-1}\). Polynomial identities on \(U\) and their uniform limits show that this calculus agrees with the two original resolvents and with the usual products of their functions. In particular the value of \((1+t^2)^{-1}\) is \((1+D^2)^{-1}=R_+R_-\).

We also need the converse Cayley construction. If \(V\) is unitary and \(1-V\) has dense range, then it is injective: its kernel equals the kernel of \(1-V^*\), which is orthogonal to that dense range. On \(\operatorname{ran}(1-V)\) define \[ S((1-V)x)=i(1+V)x. \] Expanding both inner products and using \(V^*V=1\) proves symmetry. The operators \(S+i\) and \(S-i\) have bounded, everywhere-defined, mutually adjoint inverses \[ Q_+=(1-V)/(2i),\qquad Q_-=(V^*-1)/(2i). \] These formulas prove that \(S\) is closed, since the inverse of an injective bounded operator is closed on its range. If \(y\in\operatorname{Dom}S^*\), choose \(x\in\operatorname{Dom}S\) with \((S-i)x=(S^*-i)y\). Then \(y-x\in\ker(S^*-i)\), which is orthogonal to the surjective range of \(S+i\). Hence \(y=x\), proving \(S=S^*\). The commuting product \(Q_+Q_-\) has range in \(\operatorname{Dom}S^2\) and is the inverse of \(1+S^2\), by the two resolvent identities. Thus \(S\) is regular. This proves the converse without a graph-complement theorem.

Proof: tensor extension. Lemma 6.0a of Lesson 05 supplies the unital *-homomorphism \(T\mapsto T\otimes1\) on adjointable operators. Thus \(V=U\otimes1\) is unitary. The range of \(1-V\) is dense: the algebraic tensors with first vector in \(\operatorname{Dom}D\) are dense, and every such vector is in that range. Apply the converse Cayley construction to \(V\), obtaining a self-adjoint regular \(S\). The formulas for \(Q_\pm\) show that \(S(x\otimes y)=Dx\otimes y\) on the algebraic domain. This domain is a graph core: its image under \(S+i\) contains all elementary tensors and is dense, while \(S+i\), with its bounded inverse and \(S Q_+=1-iQ_+\), identifies its graph norm with an equivalent complete norm on \(E\otimes_B Y\). Consequently \(S\) is precisely the closure of the algebraic tensor operator. Continuous circle calculus commutes with \(T\mapsto T\otimes1\). The bounded-function extension does too, since uniformly bounded vector convergence on \(E\) gives vector convergence on elementary tensors and hence on their dense span. This proves the claimed calculus identity. If \(D\) is odd, its grading sends \(U\) to \(U^*\), and the same formulas give \(f(D)\mapsto f(-D)\) and oddness of the tensor extension. \(\square\)

The free Kaad–Lesch author preprint, Section 2.3 and Proposition 4.1 supplies the regular-operator formulation used here. Its references are not imported as proof premises: the resolvent, Cayley, functional-calculus and tensor assertions needed below have just been proved.

An unbounded module \((E,\phi,D)\) means a countably generated graded Hilbert module, a graded representation, and an odd self-adjoint regular operator. We require \((1+D^2)^{-1}\phi(a)\) compact for every source element. A dense graded \(*\)-subalgebra preserves \(\operatorname{Dom}D\) and has bounded adjointable graded commutators with \(D\). The representation need not be essential.

Lemma 0.1 (bounded transformation with local compactness). For an unbounded module, \(F_D=D(1+D^2)^{-1/2}\) is a Kasparov-cycle operator.

Proof. Put \(q(t)=(1+t^2)^{-1}\). Cutoffs \(e_n\) on \((0,1]\), zero near zero and eventually one on every compact subinterval, make \(e_n(q)/q\) bounded. Hence \(e_n(q(D))\phi(a)\) is compact. For \(f\in C_0(\mathbb R)\), the functions \(fe_n(q)\) converge uniformly to \(f\); therefore \(f(D)\phi(a)\) is compact. Adjointing gives compactness on the other side.

For a homogeneous smooth source element write \(\epsilon=(-1)^{|a|}\), \(c=[D,\phi(a)]_{\rm gr}\), and \(R_z=(D-z)^{-1}\). Domain preservation, followed by multiplication by inverse resolvents, gives \[ R_z\phi(a)-\epsilon\phi(a)R_{\epsilon z} =-\epsilon R_zcR_{\epsilon z},\qquad z=\pm i\rho. \tag{0.1} \] For \(Q_\rho=D(D^2+\rho^2)^{-1}=(R_{i\rho}+R_{-i\rho})/2\), it follows that \(\|[Q_\rho,\phi(a)]_{\rm gr}\|\leq\|c\|\rho^{-2}\). The commutator becomes compact after source multiplication on either side, since the corresponding end resolvent is locally compact.

Functional calculus gives the vector integral \[ F_D\xi=\frac2\pi\int_0^\infty D(1+D^2+t^2)^{-1}\xi\,dt. \tag{0.2} \] Indeed its partial-integral scalar functions are bounded by one, converge uniformly on compact subsets of the real line, and converge pointwise to \(x/\sqrt{1+x^2}\). Apply them first to the dense set of vectors \(f(D)\eta\), \(f\in C_0(\mathbb R)\), and then use the uniform bound. The uncommuted integral need not converge in operator norm. Its commutator does: (0.1) bounds its integrand by \(\|c\|/(1+t^2)\). Thus both source-localized commutators of \(F_D\) are compact. For homogeneous smooth \(a,b\), the identity \[ [F_D,\phi(ab)]_{\rm gr} =[F_D,\phi(a)]_{\rm gr}\phi(b) +(-1)^{|a|}\phi(a)[F_D,\phi(b)]_{\rm gr} \] makes the unlocalized commutator for \(ab\) compact. Products from the dense smooth algebra linearly span a dense subset of the source: approximate an algebra approximate identity and the given element by smooth elements and multiply. Since the commutator is bounded by \(2\|a\|\) on each parity subspace, density proves compactness for every source element. Finally \(F_D\) is odd and self-adjoint, and \(F_D^2-1=-(1+D^2)^{-1}\). These are all cycle conditions. \(\square\)

Write \(E=E_1\widehat\otimes_B E_2\), \(\phi=\phi_1\widehat\otimes1\), and \(E_1^0=\overline{\phi_1(A)E_1}\). For a homogeneous vector \(x\), \(T_x\eta=x\widehat\otimes\eta\) is the creation operator.

Lemma 0.2 (the connection estimate). Suppose a homogeneous linear subspace \(\mathcal X\subset\operatorname{Dom}D_1\), dense in \(E_1^0\), satisfies the creation and adjoint domain inclusions and has bounded adjointable errors \[ DT_x-(-1)^{|x|}T_xD_2,\qquad T_x^*D-(-1)^{|x|}D_2T_x^*. \tag{0.3} \] Then \(F_D\) satisfies both \(F_{D_2}\)-connection conditions for every \(x\in E_1^0\).

Proof. On \(E\oplus E_2\), the two creations are the off-diagonal blocks of a homogeneous adjointable operator. Their domain inclusions and (0.3) give a bounded graded commutator with \(D\oplus D_2\). The inverse-resolvent identity (0.1) applies to this operator too. Multiply its bounded-transform commutator on the right by \(\operatorname{diag}(0,\phi_2(b))\), for a homogeneous smooth second-source element \(b\). The localized end resolvent is compact, and the bound \(C/(1+t^2)\) in (0.2) is integrable. The top-right block proves \[ \bigl(F_DT_x-(-1)^{|x|}T_xF_{D_2}\bigr)\phi_2(b) \] compact. Since \(T_{xb}=T_x\phi_2(b)\), moving \(F_{D_2}\) past \(\phi_2(b)\) adds only the compact commutator proved in Lemma 0.1. Thus the creation error for \(xb\) is compact; adjointing gives its second condition. The linear span of these vectors is dense in \(E_1^0\), by nondegeneracy of its right coefficient action, density of \(\mathcal X\), and density of the smooth coefficient algebra. The inequality \(\|T_x\|\leq\|x\|\) extends both assertions to every vector of \(E_1^0\). \(\square\)

Lemma 0.3 (bounded alignment, including essential connections). Let \(A\) be separable and the intermediate algebra be \(\sigma\)-unital. Suppose \(F_1\) is a normalized first-cycle operator and \(G\) is a self-adjoint contraction cycle on \(E\), with the \(F_2\)-connection conditions on \(E_1^0\). If, for a fixed \(0\leq\kappa<2\), \[ \phi(a)[F_1\widehat\otimes1,G]_{\rm gr}\phi(a)^* \geq-\kappa\phi(aa^*)\pmod{\mathcal K(E)} \qquad(a\in A), \tag{0.4} \] then \(G\) represents the Kasparov product.

Proof. First assume a full connection. The product-existence proof, using this connection and its technical partition \(M+N=1\), gives a product operator \(H=M^{1/4}SM^{1/4}+N^{1/4}GN^{1/4}\), \(S=F_1\widehat\otimes1\). The same root commutation and defect annihilation used there give, after a source sandwich, \[ [G,H]_{\rm gr} \equiv M^{1/2}[G,S]_{\rm gr}+2N^{1/2} \pmod{\text{source-localized compacts}}. \] Consequently (0.4), \(0\leq M^{1/2}\leq1\), and positivity of the second term bound this sandwich below by \(-\kappa\phi(aa^*)\).

Here are the quotient and homotopy details of that conclusion. Let \(\mathcal D_\phi\) be the graded algebra of bounded operators whose graded source commutators are compact, and let \(\mathcal C_\phi\) be its ideal of two-sided source-locally compact operators. Closure under products and adjoints follows from the graded Leibniz identity. For an even self-adjoint \(Z\in\mathcal D_\phi\), positivity of every source sandwich modulo compacts implies \(Z_-\in\mathcal C_\phi\). To see this, work modulo \(\mathcal K(E)\), where \(Z\) commutes with the represented source. The negative part of \(\phi(aa^*)Z\) is \(\phi(aa^*)Z_-\); it is zero. Hence \(Z_-\phi(a)\) has zero product with its adjoint in this quotient and is compact. Adjointing gives the other side. Continuous functional calculus is legitimate because it follows by polynomial approximation in the unital algebra generated by \(Z\).

Apply this to \(Z=[G,H]_{\rm gr}+\kappa\). In \(\mathcal D_\phi/\mathcal C_\phi\), the cycle operators \(G,H\) are self-adjoint unitaries and their anticommutator is at least \(-\kappa\). For \(L_t=(1-t)G+tH\) this gives \[ L_t^2\geq (1-t)^2+t^2-\kappa t(1-t) \geq(2-\kappa)/4. \] Choose \(0<\delta<(2-\kappa)/4\) and normalize by \(L_t\bigl(\max(L_t^2,\delta)\bigr)^{-1/2}\). This is a norm-continuous path of cycles: it is an odd self-adjoint unitary in the defect quotient, and its source commutators are compact by functional calculus. At its endpoints normalization changes the original cycle only by a two-sided locally compact operator. The straight interpolation for such a change is a cycle, as follows by expanding its square and source commutator. Thus \(G\) and \(H\) have the same class.

We now remove the full-connection assumption. Put \(E^0=\overline{\phi(A)E}\); the balancing rule identifies it with \(E_1^0\widehat\otimes_B E_2\). Proposition 5.1 of the product lesson supplies essential replacement cycles \(F_1^0,G^0\) on these modules in the original classes. Its connection relations for multiplication maps \(A_a^1:E_1\to E_1^0\), \(A_a:E\to E^0\) are \[ F_1^0A_a^1-(-1)^{|a|}A_a^1F_1\in\mathcal K(E_1,E_1^0),\qquad G^0A_a-(-1)^{|a|}A_aG\in\mathcal K(E,E^0). \tag{0.5} \] These maps are adjointable: their adjoints are multiplication by \(a^*\) followed by inclusion. Normalizing the replacement cycles preserves (0.5).

The operator \(G^0\) is a full connection on \(E_1^0\). Indeed \(T_{ax}=A_aT_x\). Expanding its error with (0.5) leaves a compact term and the source-localized creation error of \(G\). That latter term is compact after inclusion in \(E\), because the connection error for \(ax\in E_1^0\) is compact and the source commutator of \(G\) is compact. It is compact with target \(E^0\) as well: for an adjointable map \(R\), compactness of \(R^*R\) implies compactness of \(R\). In fact, with \(q_\varepsilon(t)=t/(t+\varepsilon)\), the maps \(Rq_\varepsilon(R^*R)\) are compact and converge in norm to \(R\), since \(\sup_{t\geq0}\sqrt t\,\varepsilon/(t+\varepsilon)\to0\). Taking adjoints and using density of the vectors \(ax\) proves both full connection conditions.

For \(k\in\mathcal K(E_1)\), the weak connection also gives \[ \phi(a)[G,k\widehat\otimes1]_{\rm gr}\phi(b)\in\mathcal K(E). \tag{0.6} \] Indeed \(\phi_1(a)k\phi_1(b)\) is a norm limit of rank-one operators with both vectors in \(E_1^0\). Their tensor images are \(T_xT_y^*\); the connection identities cancel the two \(F_2\) terms in their commutator. Expanding the commutator with the two outer source factors adds only compact source commutators of \(G\). This proves (0.6).

Take even \(c\in A\). By (0.5), \(R_c=(A_c^1)^*F_1^0A_c^1-\phi_1(c)^*F_1\phi_1(c)\) is compact. Put \(S^0=F_1^0\widehat\otimes1\). Moving \(G^0\) past \(A_c\), then sandwiching by \(\phi(d)\), gives \[ \phi(d)^*A_c^*[S^0,G^0]_{\rm gr}A_c\phi(d) \equiv\phi(cd)^*[S,G]_{\rm gr}\phi(cd) \pmod{\mathcal K(E)}. \tag{0.7} \] The extra bracket of \(R_c\widehat\otimes1\) is compact by (0.6); the remaining errors are (0.5) and compact source commutators. Equation (0.4), applied with \(a=(cd)^*\), transfers its lower bound through (0.7).

This transfer holds on \(E^0\), not only after inclusion into \(E\). A compact operator on \(E\) whose range and adjoint range lie in \(E^0\) belongs to \(\mathcal K(E^0)\). For a proof, first regard it as an adjointable map \(E\to E^0\); the compactness test just proved makes that map compact. An approximate identity \(u_\lambda\) of \(\mathcal K(E^0)\) then has \(u_\lambda T\to T\) in norm, by finite rank approximation with target vectors in \(E^0\). Apply the same argument to \(T^*\) to obtain \(Tu_\lambda\to T\). The sandwiches \(u_\lambda T u_\lambda\) lie in \(\mathcal K(E^0)\) and converge to \(T\). For the source sandwich in (0.7), its negative part has the same range property, since its continuous defining function vanishes at zero. Thus its compact negative part restricts compactly to \(E^0\). Positive even approximate identities \(c\) give \(cd\to d\), proving (0.4) for all sources on the essential module. The full-connection case applies there. Essential replacement and homotopy invariance of products return the assertion to the original modules. \(\square\)

Lemma 0.4 (integrated positivity). Let \(D,S\) be odd self-adjoint regular operators with \(\operatorname{Dom}D\subset\operatorname{Dom}S\). If \[ Q(\xi):=2\operatorname{Re}\langle D\xi,S\xi\rangle \geq-c\langle\xi,\xi\rangle\quad(\xi\in\operatorname{Dom}D), \tag{0.8} \] for \(c\geq0\), then \([F_D,F_S]_{\rm gr}\geq-c\). Replacing \(S\) by \(\alpha S\), \(\alpha>0\), replaces this bound by \(-\alpha c\).

Proof. For \(\lambda=1+u^2\), \(\mu=1+v^2\), set \[ \begin{gathered} r_D=(D^2+\lambda)^{-1},\quad h_D=Dr_D,\quad k_D=\sqrt\lambda r_D,\\ r_S=(S^2+\mu)^{-1},\quad h_S=Sr_S,\quad k_S=\sqrt\mu r_S. \end{gathered} \] For \(\eta\in\operatorname{Dom}S\), expanding \(Dh_D=1-\lambda r_D\), \(Dk_D=\sqrt\lambda h_D\) gives \[ Q(h_D\eta)+Q(k_D\eta)=2\operatorname{Re}\langle S\eta,h_D\eta\rangle. \] The cancelling difference is \(2\lambda\operatorname{Re}(\langle h_D\eta,Sr_D\eta\rangle-\langle r_D\eta,Sh_D\eta\rangle)\), zero by symmetry of \(S\). All these vectors are in the indicated domains by the domain inclusion. Apply the identity to \(h_S\psi,k_S\psi\). Using \(Sh_S=1-\mu r_S\), \(Sk_S=\sqrt\mu h_S\), the remaining two terms cancel by self-adjointness of \(h_D\). Hence \[ \sum_{w\in\{h_Dh_S,k_Dh_S,h_Dk_S,k_Dk_S\}}Q(w\psi) =2\operatorname{Re}\langle\psi,h_Dh_S\psi\rangle. \tag{0.9} \] Integrate first over finite rectangles \(0\leq u,v\leq R\), with factor \(4/\pi^2\). Formula (0.2) makes the right side converge to the quadratic form of the bounded anticommutator. The lower bound (0.8) bounds the left side below by minus \(c\) times the sum of the four positive forms \(\langle w\psi,w\psi\rangle\). Their total integral is at most \(\langle\psi,\psi\rangle\): first use \[ h_D^2+k_D^2=r_D,\qquad \frac2\pi\int_0^\infty r_D\,du=(1+D^2)^{-1/2}\leq1, \] then bound the two remaining sandwiches by \(h_S^2+k_S^2=r_S\) and integrate once more. These increasing positive integrals also justify passage to the rectangle limit without asserting absolute integrability of the unsplit anticommutator integrand. The inequality holds for every \(\psi\), hence as an operator inequality. Scaling \(S\) scales (0.8) by \(\alpha\), proving the last assertion. \(\square\)

Theorem 0.5 (the unbounded product criterion). Let \(A\) be separable and \(B\) be \(\sigma\)-unital. Suppose \((E_1,\phi_1,D_1)\), \((E_2,\phi_2,D_2)\), and \((E_1\widehat\otimes_B E_2,\phi_1\widehat\otimes1,D)\) are unbounded modules. Suppose (0.3) holds on a homogeneous subspace of \(\operatorname{Dom}D_1\) dense in \(\overline{\phi_1(A)E_1}\), with both creation domain inclusions. Suppose \(\operatorname{Dom}D\subset\operatorname{Dom}(D_1\widehat\otimes1)\), and (0.8) holds with \(S=D_1\widehat\otimes1\). Then the bounded-transform class of \(D\) is the Kasparov product of the classes of \(D_1,D_2\). The first representation may be nonessential.

Proof. Lemma 0.1 supplies the cycles and Lemma 0.2 their essential-source connection. Choose \(\alpha>0\) with \(\alpha c<2\). Lemma 0.4 gives (0.4) for \(F_D\) and \(F_{\alpha D_1}\widehat\otimes1\); functional calculus commutes with the tensor representation. Lemma 0.3 identifies their product. Finally positive rescaling preserves the class of the first operator: on a compact positive interval of parameters, \(tx/\sqrt{1+t^2x^2}\) varies uniformly in \(x\), continuously in \(t\). Its transforms form a norm-continuous path of cycles by Lemma 0.1. Domains and bounded commutators are unchanged by this positive scaling, and local compactness follows from the vanishing-functions argument in its proof. Thus \([\alpha D_1]=[D_1]\), proving the assertion. \(\square\)

The freely readable van den Dungen preprint, Sections 2.4 and 3.1, printed pp. 11–16, provides the comparison account for connection estimates and integrated positivity. The proof used here is the complete local argument above. In particular its nonessential representation step does not rely on the preprint's reference to an external original proof.

1. Central coefficients and compact bases

Let \(X\) be compact Hausdorff and put \(R=C(X)\), trivially graded. An \(R\)-algebra is a graded C*-algebra with a specified even central action of \(R\). An \(R\)-linear cycle has the property that the left and right actions of \(R\) on its module agree. Write \(KK_R(A,B)\) for its homotopy classes. Forgetting this agreement gives a usual KK class.

The tensor products of modules below are balanced over their displayed coefficient algebra. External products over \(R\) use the balanced algebra tensor product over \(R\). Thus both factors refer to the same point of \(X\).

Lemma 1.1 (the central coefficient argument). Suppose the closed \(R\)-linear *-algebra generated by countably many elements is \(A\), and all coefficient and intermediate algebras under consideration are \(\sigma\)-unital. For \(R\)-linear countably generated cycles, the product-existence, uniqueness, associativity and unbounded product-recognition arguments used earlier apply with this hypothesis in place of separability of \(A\). These operations preserve \(R\)-linearity.

Proof. We specify the modification, since \(C(X)\) need not be separable. Every adjointable right-module operator commutes with the right action of the central algebra \(R\), hence with its left action in an \(R\)-linear cycle. This includes all compact operators, connection operators, technical partitions and their functional calculi.

In the technical theorem, take as the derivating space the separable space generated by the countable noncentral source generators, the required cycle and connection operators, and their adjoints. Include countably many words in the source generators. All the ideals used in the product proof remain \(\sigma\)-unital: the compact ideal of a countably generated module is \(\sigma\)-unital, its represented images are \(\sigma\)-unital, and the proof uses only sums of these images and finitely or countably many added operators. The technical theorem supplies the same partition. Its commutators with the omitted elements of \(R\) are exactly zero.

The source conditions extend from the selected words to products with elements of \(R\), since the latter commute with every operator in the argument. Finite sums of these products are dense in \(A\). Norm continuity then gives the cycle, connection and localized positivity conditions for every source element. This proves existence and the uniqueness comparison by the same convex-path and normalization argument as before.

For associativity, use precisely the two compact images and the localized negative-part ideal in Theorem 4.1 of Homotopy, associativity, the index pairing and KK-equivalence. The extra derivating operators there are again countable; central coefficients commute with them and with the partition. Both positivity terms in that proof are retained. This gives associativity whenever the displayed binary products exist. The argument for balanced external products is the identical argument on the canonically associated module tensor products.

Finally the creation-operator resolvent identities and the integrated positivity estimate in the unbounded product criterion require no separable source. Separability enters its product comparison only through the technical partition just described. Thus its domain, connection and lower-bound hypotheses give product recognition here as well. All operators constructed are right-module operators, so remain \(R\)-linear. The same construction on interval modules proves compatibility with homotopies. \(\square\)

This lemma concerns central module actions. It makes no assertion about an arbitrary representation of a nonseparable source algebra.

Let \(V\to X\) be a rank-\(n\) Euclidean real vector bundle. Write \[ \begin{aligned} C&=\Gamma(\operatorname{Cl}(V)),\\ B&=C_0(V),\\ A&=B\widehat\otimes_R C. \end{aligned} \tag{1.1} \] The complex Clifford convention is \(vw+wv=2(v,w)\). The Clifford algebra is graded; \(B\) is not.

The algebra \(C\) is unital. The radial functions of \(\|v\|\), equal to one on expanding balls, give a countable approximate identity of \(B\) and \(A\). All three are generated over \(R\) by countably many elements. For \(C\), take finitely many bundle sections spanning each fibre, obtained from a finite trivializing cover and a partition of unity. For \(B\), embed \(V\) isometrically in a finite trivial bundle and use its coordinate functions multiplied by countably many radial cutoffs. Polynomials in these coordinates approximate continuous functions on bounded fibre balls; a finite bundle cover and a partition of unity make the approximation uniform over \(X\). Cut off at larger radii to approximate \(C_0(V)\). The same argument with the finite Clifford sections proves the assertion for \(A\). Thus Lemma 1.1 applies without a metrizability hypothesis on \(X\).

Rank may be locally constant rather than constant. A finite-rank bundle on a compact base has only finitely many ranks; their inverse images are clopen. Apply the constructions separately and take the finite direct sum. We write constant rank to simplify the formulas.

2. The fibrewise position and Dirac cycles

On the standard right \(A\)-module let \[ \begin{gathered} X_V(v)=v\in\operatorname{Cl}(V_{\pi(v)}),\\ F_V(v)=\frac{v}{\sqrt{1+\|v\|^2}}. \end{gathered} \tag{2.1} \] The source \(R\) acts by pullback along \(\pi:V\to X\).

Proposition 2.1 (the position class). Formula (2.1) defines \[ b_V\in KK_R(R,A). \tag{2.2} \]

Proof. It is odd and self-adjoint, its source commutators vanish, and \[ F_V^2-1=-\frac1{1+\|v\|^2}\in A. \tag{2.3} \] This is a function vanishing at infinity because the base is compact and the norm-bounded ball bundles are compact. Multiplication by it is compact on the standard module. The module is countably generated by the radial approximate identity. These verify every cycle condition. The unbounded multiplier also has the explicit inverse resolvents \[ (X_V\pm i)^{-1}=\frac{X_V\mp i}{1+\|v\|^2}; \] their dense ranges contain the compactly supported continuous sections. Thus it is self-adjoint and regular. \(\square\)

Over \(x\in X\) put \[ \mathcal H_x=L^2(V_x)\otimes\Lambda^*(V_x\otimes_{\mathbb R}\mathbb C). \tag{2.4} \] Orthogonal frame changes act by their unitary change of variable on \(L^2\) and their exterior action on forms. These give a continuous Hilbert bundle: its transition operators are strongly continuous, which suffices for continuity of its sections. Let \(H_V\) be the Hilbert \(R\)-module of continuous sections. Its inner product is the fibre integral.

In an orthonormal frame set \[ \begin{gathered} c_j=\varepsilon_j+\iota_j,\\ \widehat c_j=\varepsilon_j-\iota_j,\\ D_V=\sum_j\widehat c_j\partial_j. \end{gathered} \tag{2.5} \] Both Clifford families transform with the frame, so the operator and the representation of \(A\) by multiplication and \(c_j\) are globally defined. Grade \(H_V\) by form parity.

Proposition 2.2 (the vertical Dirac class). The fibre operator (2.5) defines an odd self-adjoint regular operator on \(H_V\). Its domain is the continuous field of first Sobolev domains with the graph norm. Its bounded transform gives \[ a_V\in KK_R(A,R). \tag{2.6} \]

Proof. In a bundle chart this is the fixed Euclidean operator from Lemma 3.1 of the Bott lesson. Its inverse resolvents are uniformly bounded, and are carried into one another by the frame-transition unitaries. They therefore give adjointable global operators, with adjoints the opposite resolvents. Their ranges contain sections locally given by compactly supported smooth fibre vectors, times continuous base functions; finite partitions of unity show that such sections are dense. The resolvent identities, symmetry and dense range prove self-adjointness and regularity on the stated graph domain.

Countable generation also follows from a finite cover. Choose partitions with supports in bundle charts and a countable dense family of smooth fibre vectors in each chart. Multiplying these by the partition functions gives countably many global sections; their \(R\)-linear span is dense. Strong continuity of the transition unitaries is enough for this assertion.

For local compactness take a section of \(A\) supported in one chart and compactly supported in its fibre variable. Approximate it uniformly by finite sums of a continuous base coefficient times a compactly supported fibre function and a Clifford matrix. The localized Euclidean inverse resolvent is compact by the frequency-cutoff and square-integrable-kernel proof in the Bott lesson. Multiplying its finite rank approximations on each side by chart partition functions gives finite rank operators on \(H_V\). Summing finitely many chart pieces and taking norm limits proves local compactness for every element of \(A\).

There is a dense graded *-algebra of sections continuous in the base and smooth with compact support in the fibre in each chart. On it the graded commutator is Clifford multiplication by the fibre derivative, a bounded multiplier. Domain preservation is the fibre Sobolev product rule. The bounded-transform theorem now applies. All operators commute with \(R\), so the cycle is \(R\)-linear. \(\square\)

3. The two inverse identities

Theorem 3.1 (the parametrized Bott theorem). The classes above satisfy \[ \begin{aligned} b_V\widehat\otimes_A a_V&=1_R,\\ a_V\widehat\otimes_R b_V&=1_A. \end{aligned} \tag{3.1} \]

Proof of the first identity. Identify the tensor module \(A\widehat\otimes_A H_V\) with \(H_V\). The candidate product is the bounded transform of \[ Q_V=D_V+\sum_jv_jc_j. \tag{3.2} \] Its graph domain is the field of weighted Sobolev spaces \[ \{u:\partial_j u,\ v_j u\in L^2\text{ for every }j\}. \tag{3.3} \] The earlier local oscillator proof in Bott periodicity in KK: the Bott and Dirac elements, Lemma 4.0, proves the Euclidean self-adjoint closure, compact graph inclusion, even Gaussian kernel and zero odd kernel on this domain. Its Lemma 0.0a proves the Euclidean Fourier unitarity used in Proposition 2.2. These are frame-invariant statements. The chart resolvents glue as in Proposition 2.2.

Here the inverse resolvent is globally compact on the Hilbert \(R\)-module. Indeed the Euclidean oscillator inverse resolvent is compact, so its conjugation by the strongly continuous frame unitaries is norm continuous. This follows first for rank-one operators, then for compact operators by uniform finite rank approximation. In finitely many charts approximate this field in norm by finite rank operators, multiply by partition functions on both sides, and sum. The resulting operators are finite sums of global rank-one operators.

The creation operator associated to a smooth compactly supported Clifford section \(d\) is multiplication by \(d\). Its graded error relative to \(D_V\) is \[ \begin{gathered} Q_V M_d-(-1)^{|d|}M_dD_V\\ =[D_V,M_d]_{\rm gr}+X_VM_d, \end{gathered} \tag{3.4} \] which is bounded. Its adjoint satisfies the corresponding domain and error conditions. The domain (3.3) is contained in that of \(X_V\), and on its smooth core \[ \begin{gathered} \{Q_V,X_V\}\\ =2\|v\|^2+2N-n\geq -n. \end{gathered} \tag{3.5} \] Here \(N\) is form degree. The inequality extends to the graph domain. The unbounded product criterion, with the central coefficient argument of Lemma 1.1, identifies (3.2) as the product.

The normalized Gaussian \[ g_x(v)=\pi^{-n/4}e^{-\|v\|^2/2} \tag{3.6} \] is a global even unit section of the kernel, so that kernel is the module \(R\). The complementary operator is invertible, with a uniform spectral gap: every fibre is unitarily the same Euclidean oscillator with its kernel removed. Deform its bounded transform to its sign, leaving the kernel at zero. This is a norm-continuous functional-calculus homotopy; its defects are compact and its source commutators are zero. The complementary summand becomes degenerate. The kernel summand is \((R,1,0)\), proving the first identity.

Proof of the reverse identity. Work on \(V\oplus V\), with vectors \(v,w\) in the same fibre and Clifford families \(e_j,f_j\). For \(0\leq\theta\leq\pi/2\) put \[ \begin{aligned} u&=\cos\theta\,v+\sin\theta\,w,\\ t&=-\sin\theta\,v+\cos\theta\,w,\\ e_j^\theta&=\cos\theta\,e_j+\sin\theta\,f_j,\\ f_j^\theta&=-\sin\theta\,e_j+\cos\theta\,f_j . \end{aligned} \tag{3.7} \] These formulas commute with every orthogonal change of frame, so define global data. Use the interval standard module over \(A\widehat\otimes_R A\), source \(A\) represented by its functions in \(t\) and its Clifford generators \(f_j^\theta\), and operator \[ \frac{\sum_ju_je_j^\theta}{\sqrt{1+\|u\|^2}}. \tag{3.8} \] The graded source commutators vanish. Its localized square defect vanishes uniformly at infinity: \(\|u\|^2+\|t\|^2=\|v\|^2+\|w\|^2\), so either the inverse quadratic factor is small or the source function is small. Joint continuity gives strict multiplier continuity and an actual interval cycle. Operator-norm continuity of the multiplier is unnecessary.

The endpoints prove \[ b_V\boxtimes_R1_A=[\rho]\boxtimes_R b_V, \tag{3.9} \] where \(\rho\) reflects both the fibre coordinate and the Clifford vector. Compose with \(1_A\boxtimes_R a_V\). The left side is \(a_Vb_V\); the right side is \([\rho]\), by the first identity. Thus \(p=a_Vb_V\) is invertible, since \(\rho\) is an automorphism. It is also idempotent: \[ p^2=a_V(b_Va_V)b_V=p. \tag{3.10} \] An invertible idempotent is the identity, proving the second equation of (3.1). Every product and homotopy is \(R\)-linear; Lemma 1.1 justifies the displayed associativity even for nonmetrizable \(X\). \(\square\)

4. Removing the doubled Clifford algebra

There is a useful version with the Clifford algebra in the source rather than the coefficient: \[ \begin{gathered} x_V\in KK_R(C,B),\\ y_V\in KK_R(B,C). \end{gathered} \tag{4.1} \] We describe its Morita correspondence, including its grading.

Let \(M=\Gamma(\Lambda^*V_{\mathbb C})\) as a finite Hilbert \(R\)-module. Represent the first copy of \(C\) by \(c_j\), and the second by \(i\widehat c_j\). These are odd self-adjoint anticommuting Clifford families. Give \(M\) the grading \[ \Gamma_M=(-1)^{n(n-1)/2}(-1)^N. \tag{4.2} \] It is the chirality for the ordered doubled frame, first all \(e_j\), then all \(f_j\). To check the sign, the chirality for the interleaved frame is \(\prod_j(-i c_j\,i\widehat c_j)=(-1)^N\); sorting that frame takes \(n(n-1)/2\) exchanges of odd generators.

Lemma 4.1 (double Clifford Morita correspondence). The representation gives a graded isomorphism \[ C\widehat\otimes_R C\cong\operatorname{End}_R(M). \tag{4.3} \] Thus \(M\) is a graded Morita correspondence to \(R\).

Proof. In a fibre, the two Clifford families recover exterior multiplication and contraction by \[ \varepsilon_j=(c_j-i f_j)/2,\qquad \iota_j=(c_j+i f_j)/2. \] The product \(\prod_j\iota_j\varepsilon_j\) is the vacuum projection. Multiplication on the left and right by exterior creations and contractions gives every matrix unit in the exterior basis. The representation is onto. Both sides have dimension \(2^{2n}\), so it is injective. The construction respects frame changes and (4.2), proving the bundle and section assertion.

The usual left inner product is the rank-one operator \(\theta_{\xi,\eta}\), identified through (4.3), and the right inner product is the Hermitian fibre inner product. Their compatibility is the rank-one formula. Finite partitions of unity and local frames make the module finitely generated projective. Its inner products span both coefficient algebras; the right span contains the constant one through the vacuum section. Its dual is the inverse Morita correspondence by evaluation of these inner products. All left actions are compact, so the zero operators give the two Morita cycles and their explicit inverse products. \(\square\)

Extend \(M\) to \(B\), denoting that correspondence by \(M_B\). Define \[ \begin{aligned} x_V&=(b_V\boxtimes_R1_C) \widehat\otimes_{B\widehat\otimes_R C\widehat\otimes_R C}M_B,\\ y_V&=M_B^* \widehat\otimes_{B\widehat\otimes_R C\widehat\otimes_R C} (a_V\boxtimes_R1_C). \end{aligned} \tag{4.4} \] The first Clifford copy is the position copy in \(b_V\); the source \(C\) is the second copy. This order matters.

Theorem 4.2 (Clifford Thom equivalence). For every compact Hausdorff base, \[ \begin{aligned} x_V\widehat\otimes_B y_V&=1_C,\\ y_V\widehat\otimes_C x_V&=1_B. \end{aligned} \tag{4.5} \]

Proof. In the first product cancel \(M_B\widehat\otimes_BM_B^*\) to the identity correspondence, and then use \((b_Va_V)\boxtimes_R1_C=1_C\). In the second use \((a_Vb_V)\boxtimes_R1_C=1_{B\widehat\otimes_R C\widehat\otimes_R C}\), and cancel the dual Morita pair. These are the explicit evaluation tensor unitaries from Lemma 4.1 and the two equations of Theorem 3.1. Associativity is supplied by Lemma 1.1. \(\square\)

In particular, \(x_V\) has the concrete module \(C_0(V,\pi^*\Lambda^*V_{\mathbb C})\), source Clifford action \(i\widehat c\), grading (4.2), and position operator \(c(v)/\sqrt{1+\|v\|^2}\). Formula (4.4) supplies its Dirac inverse.

5. Spinors and the positive Thom class

For an oriented Euclidean bundle, a spin\(^{c}\) structure is a lift of its oriented orthonormal frame bundle to \[ \begin{gathered} \operatorname{Spin}^{c}(n)\\ =(\operatorname{Spin}(n)\times U(1))/\{\pm(1,1)\}. \end{gathered} \tag{5.1} \] The spin group acts on its usual complex spin representation and \(U(1)\) acts by scalars. The determinant line \(L\) is the associated line for the character \([s,\lambda]\mapsto\lambda^2\).

In even rank \(n=2r\), the associated spinor bundle \(S=S^+\oplus S^-\) has rank \(2^r\), and Clifford multiplication is odd and self-adjoint. Its grading is the chirality \[ \Gamma_S=(-i)^r c(e_1)\cdots c(e_{2r}) \tag{5.2} \] in a positively oriented orthonormal frame. This is the convention established by the Pauli-matrix model in the Clifford lesson.

Proposition 5.1 (spinor Morita equivalence). In even rank the spinor action identifies \[ \Gamma(\operatorname{Cl}(V)) \cong\operatorname{End}_{C(X)}\Gamma(S). \tag{5.3} \] Thus \(\Gamma(S)\), and also \(\Gamma(S\otimes L^{-1})\), are graded Morita correspondences from \(C\) to \(R\). In odd rank, a spin\(^{c}\) structure gives a graded Morita correspondence from \(C\) to \(R\widehat\otimes C_1\).

Proof. The even-rank assertion is the fibre matrix-algebra isomorphism from the Clifford lesson, conjugated by the spinor transition maps. It gives an isomorphism of algebra bundles and hence of section algebras. The spinor section module is finitely generated projective, by finite local frames and a partition of unity. Its compact operators are all its endomorphisms. The inner-product evaluation unitaries prove the two Morita identities as in Lemma 4.1. Twisting by a line bundle does not change the endomorphism algebra or fullness.

For rank \(2r+1\), let \(\Delta\) be the ungraded irreducible spinor bundle of rank \(2^r\), in the Clifford branch fixed by the orientation. On \(\Delta\otimes C_1\), regarded as a right \(R\otimes C_1\)-module, represent a real vector \(v\) by \(c_\Delta(v)\otimes\epsilon\), where \(\epsilon\) is the odd generator of \(C_1\); grade only the \(C_1\) factor. These operators are odd self-adjoint Clifford operators. Even Clifford words give all endomorphisms of \(\Delta\) times \(1\), and odd words give all endomorphisms times \(\epsilon\). The fibre map onto \(\operatorname{End}(\Delta)\widehat\otimes C_1\) is therefore an isomorphism by the equal dimensions. Spin\(^{c}\) transition maps intertwine it. Finite generation, fullness and the evaluation identities follow exactly as in even rank. \(\square\)

A choice of spinor convention is needed to turn (4.5) into a particular Thom class. We choose the outward spinor symbol. In even rank it is \[ \begin{gathered} \mathcal E_V=C_0(V,\pi^*S),\\ F_S(v)=\frac{c_S(v)}{\sqrt{1+\|v\|^2}},\\ \tau_V=[(\mathcal E_V,\pi^*,F_S)]\\ \in KK_R(R,B). \end{gathered} \tag{5.4} \] Its cycle axioms follow directly from \(c_S(v)^2=\|v\|^2\), finite rank, and compactness of the base. The following sign and line-bundle check identifies the correct Morita reduction.

Lemma 5.2 (spinor reduction of the Clifford Thom cycle). Let \(N=\Gamma(S\otimes L^{-1})\), with its \(C\)-\(R\) Morita structure. Then \[ \tau_V=[N^*]\widehat\otimes_C x_V. \tag{5.5} \]

Proof. Work first in an oriented orthonormal fibre. The doubled module of Lemma 4.1 decomposes under the second Clifford action as a spinor for that action tensored with a multiplicity space. Its graded decomposition has \(\Gamma_M=\Gamma_c\Gamma_f\), by the sorted chirality in (4.2). After removing the source spinor, the remaining position Clifford action therefore has grading \(\Gamma_c\). It is the positive irreducible spinor representation, rather than its grading reverse. The matrix-algebra classification from the Clifford lesson supplies an even unitary identifying it with \(S\).

We check transition maps, which is necessary for a bundle statement. A spin lift acts on the doubled module by the product of its actions in the two Clifford copies. Indeed it conjugates both vector families by the same orthogonal map; on exterior forms this is precisely the exterior action. One can check this on a plane rotation: the infinitesimal generator is \((c_jc_k-\widehat c_j\widehat c_k)/2\), the generator of the exterior rotation. Plane rotations generate the spin group, and the central \(-1\) acts twice, hence trivially, on the doubled module.

The doubled module is associated to the orthogonal frame bundle, so the scalar \(\lambda\) in (5.1) acts trivially on it. The source module \(N=S\otimes L^{-1}\) has scalar action \(\lambda^{-1}\). Removing it leaves scalar action \(\lambda\), exactly the action on \(S\). The remaining spin action is the position spin representation already identified above. Thus the fibre unitary intertwines the full spin\(^{c}\) transition maps and is a global even bundle unitary. Under the graded tensor evaluation it sends the position operator to \(c_S(v)\) and the source action to pullback of \(R\). It therefore identifies the cycles in (5.5). \(\square\)

The determinant twist in (5.5) fixes which spinor symbol is used. Using \(S\) instead of \(S\otimes L^{-1}\) for the source Morita correspondence would give the outward symbol twisted by \(L^{-1}\).

Theorem 5.3 (the spin\(^{c}\) Thom isomorphism). A spin\(^{c}\) rank-\(n\) bundle over a compact Hausdorff base has an invertible class \[ \tau_V\in KK_R^n(R,B). \tag{5.6} \] Consequently, in both degrees modulo two, \[ \begin{gathered} K_i(C(X))\xrightarrow{\ \cong\ }K_{i+n}(C_0(V)),\\ a\longmapsto a\widehat\otimes_R\tau_V. \end{gathered} \tag{5.7} \] On an oriented trivial fibre this is \((-1)^{n(n-1)/2}\) times the ordered product of the positive one-dimensional Bott classes; in a complex line its even symbol is \(z\) from even to odd spinors.

Proof. For even \(n\), combine (5.5), Proposition 5.1 and Theorem 4.2. An explicit inverse is \[ \eta_V=y_V\widehat\otimes_C[N]. \tag{5.8} \] The two Morita evaluation identities and (4.5) give \(\tau_V\eta_V=1_R\) and \(\eta_V\tau_V=1_B\).

For odd rank take the ungraded spinor bundle \(\Delta\) in the positive-volume branch, so that \((-i)^{(n-1)/2}\prod_jc_\Delta(e_j)=1\). The right odd cycle has module \(C_0(V,\pi^*\Delta)\widehat\otimes C_1\), graded by the last factor, and operator \(c_\Delta(v)\otimes\epsilon/\sqrt{1+\|v\|^2}\). Its compact defect and source commutator are exactly those of the even position cycle. This defines the outward class \(\tau_V\).

Let \(\ell\) be a positive trivial real line. Proposition 7.2 below proves directly from the position operators, without using invertibility, that \(\tau_V\boxtimes_R\tau_\ell=-\tau_{V\oplus\ell}\). Put \(u=1_B\boxtimes\tau_{\mathbb R}\) and \(v=1_B\boxtimes\eta_{\mathbb R}\), with \(uv=1_B\) and \(vu=1_{C_0(V\oplus\ell)}\). The even case gives the inverse \(\eta_{V\oplus\ell}\). Consequently \(\tau_V=-\tau_{V\oplus\ell}v\) and \(\eta_V=-u\eta_{V\oplus\ell}\) are inverse classes. Associating these two displayed products proves both inverse identities. This is the ordered addition and removal of a last line, including its required sign.

Here is the orientation check in those maps. For one real line the doubled model has source \(i\widehat c=\sigma_2\), operator \(h(v)\sigma_1\), and grading \(\sigma_3\), where \(h(v)=v/\sqrt{1+v^2}\). The even unitary \(\exp(i\pi\sigma_3/4)\) sends it to source \(\sigma_1\) and operator \(-h(v)\sigma_2\). Proposition 7.1 of the Bott lesson identifies precisely this left-Clifford cycle with the positive cone class. The right odd transfer therefore gives the positive one-dimensional class. When several odd classes are multiplied, the course's right-Clifford product orders the second cycle's original generator before the first cycle's auxiliary generator. Proposition 7.2 below computes the resulting sign: on a positively oriented \(\mathbb R^n\), the outward spinor class is \((-1)^{n(n-1)/2}\) times the ordered exterior product of the positive one-dimensional classes. Adding and removing a last line uses this same sign. The graded Clifford equivalence before scalar transfer has no such scalar-degree conversion.

In a complex line the spinor model is \(S^+=\mathbb C\), \(S^-=\mathbb C\), with \[ c_S(x+iy)= \begin{pmatrix}0&x-iy\\x+iy&0\end{pmatrix}. \tag{5.9} \] Its lower-left symbol is \(z=x+iy\). The explicit self-adjoint disc lift in Theorem RK.6 therefore gives exactly the positive planar class \([q]-[\operatorname{diag}(0,1)]\) in that comparison convention. Graded tensor products of these models prove the complex higher-rank assertion.

Finally multiply K-theory classes by the inverse pair. The associativity proof applies by Lemma 1.1; for scalar-source cycles one can adjoin the central right \(R\)-action on the essential scalar summand. Clifford dilation gives the same argument in odd degree. This proves bijectivity in (5.7). For a vector bundle \(E\) representing \(a\in K^0(X)\), the tensor module is \(\pi^*E\otimes\pi^*S\) and its operator is \(1_E\otimes c_S(v)/\sqrt{1+\|v\|^2}\). Thus the formula is the usual \(\pi^*a\) times the Thom symbol, including bundle differences. \(\square\)

6. Complex bundles and relative symbols

Proposition 6.1 (the complex spinor bundle). A Hermitian complex rank-\(r\) bundle \(E\), viewed as a real rank-\(2r\) bundle, has a spin\(^{c}\) structure with \[ \begin{gathered} S=\Lambda^*E,\quad L=\det E,\\ c(v)=\varepsilon_v+\iota_v. \end{gathered} \tag{6.1} \] Its Thom class is the relative triple \[ \begin{gathered} \left(\pi^*\Lambda^{\rm even}E,\pi^*\Lambda^{\rm odd}E,\sigma\right),\\ \sigma=c(v)^+\big|_{S(E)}. \end{gathered} \tag{6.2} \] The comparison on the sphere is unitary. At the zero section \(s\), \[ \begin{gathered} s^*\tau_E=\lambda_{-1}(E)\\ =\sum_{j=0}^r(-1)^j[\Lambda^jE]. \end{gathered} \tag{6.3} \]

Proof. Exterior multiplication and its Hermitian adjoint give \(\{c(v),c(w)\}=2\operatorname{Re}\langle v,w\rangle\). In a unitary frame with ordered real basis \(e_1,ie_1,\ldots,e_r,ie_r\), these are the tensor Pauli generators. They generate every endomorphism of \(\Lambda^*E\), and their chirality is its form parity. Hence this is an irreducible graded Clifford module.

For completeness, this supplies an actual spin\(^{c}\) lift. The group of pairs \((g,u)\), with \(g\in SO(2r)\) and \(u\) an even unitary implementing \(c(gv)\), is \(\operatorname{Spin}^{c}(2r)\). A spin lift supplies one such unitary for every \(g\); two implementers differ by a scalar because the Clifford matrices generate the full endomorphism algebra. The map from \(\operatorname{Spin}(2r)\times U(1)\) consequently has precisely the kernel \(\{(1,1),(-1,-1)\}\). The unitary group \(U(r)\) maps continuously to this implementer group by \(g\mapsto(g,\Lambda^*g)\). Applying it to the unitary frame transitions gives the required lift.

Its determinant character is \(\det g\). On a diagonal unitary rotation of angles \(\theta_j\), the exterior representation differs from the spin representation by the scalar \(\exp(i\sum_j\theta_j/2)\), whose square is \(\det g\). Every unitary is unitarily diagonalizable; conjugation preserves both characters, proving the equality for all \(g\). This proves \(L=\det E\).

The compact Hausdorff ball/sphere identification and both inverse relative-bundle maps in Theorems RK.3–RK.4 turn the outward symbol (5.4) into (6.2). Corollary RK.4a proves the symbol and zero-section rules for these actual comparisons. On the unit sphere \(c(v)^2=1\), so its even-to-odd block is unitary. Restriction to the zero section forgets that boundary comparison and gives the difference of the two bundles, namely (6.3). \(\square\)

For a line bundle the triple is simply \[ (\mathbf1,\pi^*E,\ \lambda\mapsto\lambda v). \tag{6.4} \] This formula keeps track of the actual bundle and comparison, rather than just the fibre winding.

7. Pullback and direct sums

Theorem 7.1 (naturality). Let \(f:Y\to X\) be a continuous map of compact Hausdorff spaces. Give \(f^*V\) the pulled-back metric and spin\(^{c}\) structure. Its total-space map \(\widetilde f:f^*V\to V\) is proper, and \[ \widetilde f^*(\tau_V)=\tau_{f^*V} \tag{7.1} \] as Thom symbols. In K-theory, \[ \widetilde f^*\bigl(\pi^*a\,\tau_V\bigr) =\pi_Y^*f^*a\,\tau_{f^*V}. \tag{7.2} \] The Clifford equivalence and its inverse pull back as well.

Proof. A compact subset of \(V\) is norm bounded. Its inverse image under \(\widetilde f\) is closed in a norm-bounded ball bundle over the compact base \(Y\), hence compact. This proves properness and gives the pullback homomorphism on \(C_0\)-algebras.

Pulling back the position module gives the module of sections of the pulled-back spinor bundle, and its operator is still \(c(v)/\sqrt{1+\|v\|^2}\). The fibre isometry in the definition of \(f^*V\) preserves \(v\), its norm, and both Clifford actions. Thus the cycle identification proves (7.1), including in the odd model with the additional last line.

More explicitly, the position module scalar-extension map sends \(\xi\otimes b\) to \((\xi\circ\widetilde f)b\). Its inner-product formula is that for scalar extension. Its range is dense by local frames and finite partitions, so it is the claimed module unitary. The same argument for the finite Morita modules preserves (4.2).

For the inverse, scalar extension of the vertical \(L^2\) module has the pulled-back Hilbert field. Local smooth fibre sections are dense in both modules; on them the fibre-integral inner products, Dirac operators and inverse resolvents agree. This proves the inverse-cycle identification and its graph-domain compatibility. Every homotopy in Theorem 3.1 has these same frame-invariant formulas, so the inverse identities also pull back.

For a bundle representative of \(a\), tensor its pullback with the pulled-back spinor symbol. This gives (7.2) directly; differences and Clifford suspension give both degrees. Equivalently, the Clifford statement is the KK equality \[ [C_f]\widehat\otimes_{C_{f^*V}}x_{f^*V} =x_V\widehat\otimes_B[B_f], \tag{7.3} \] where \(C_f\) and \(B_f\) are the Clifford and total-space pullback homomorphisms. Both sides have the same position cycle just identified. \(\square\)

Metric choices cause no ambiguity. Join two metrics through their convex interpolation, and transport Clifford vectors by the positive fibrewise isometry between them. The same formulas give a cycle on the compact base cylinder. Endpoint evaluation proves metric independence with the given spin\(^{c}\) structure.

Proposition 7.2 (ordered direct sums). For spin\(^{c}\) bundles \(V,W\) of ranks \(r,s\) over \(X\), using the ordinary oriented direct-sum spinor convention with \(V\) first and the course's right-Clifford degree products, \[ \tau_{V\oplus W}=(-1)^{rs}\tau_V\boxtimes_R\tau_W. \tag{7.4} \] For a trivial bundle this is \(1_X\boxtimes\tau_{\mathbb R^n}\).

Proof. In even rank the spinor module is the graded tensor product, and the unbounded symbol is \[ c_V(v)\widehat\otimes1+1\widehat\otimes c_W(w). \tag{7.5} \] Its square is \(\|v\|^2+\|w\|^2\), since the two odd summands anticommute. Its explicit inverse resolvents prove regularity. The domain is contained in each summand's domain. Creations from compactly supported smooth first-factor spinors have bounded error, namely creation by \(c_V(v)\xi\), and the positivity form for the first summand is \(2\|v\|^2\geq0\). Its inverse resolvent is locally compact on the finite-rank symbol module. The product criterion identifies its bounded transform with the external product. It is exactly the direct-sum symbol. The central argument applies over any compact \(X\).

We spell out the odd transfer. If just one rank is odd, the scalar odd operator is the usual direct-sum spinor position operator: for an even first factor it is \(c_V(v)\otimes1+\Gamma_V\otimes c_W(w)\); for an even second factor it is \(c_V(v)\otimes\Gamma_W+1\otimes c_W(w)\). Ordered volume multiplication gives the positive odd spinor convention. Their squares, creation errors and first-factor positivity are the same ones checked above, in the right odd Clifford module. There is no sign in these two cases.

If both ranks are odd, use the ungraded positive-volume spinor bundles \(S_V,S_W\). The ordinary oriented direct sum has bundle \(S_V\otimes S_W\otimes\mathbb C^2\), grading \(\sigma_3\), and position operator \[ Q_+=c_V(v)\otimes\sigma_1+c_W(w)\otimes\sigma_2. \tag{7.6} \] The omitted tensor identities act on the other spinor factor. Write \(r=2a+1,s=2b+1\). The positive odd volumes satisfy \(\prod c_V=i^a\), \(\prod c_W=i^b\); multiplying the ordered total volume gives \((-i)^{a+b+1}i^{a+b}\sigma_1\sigma_2=\sigma_3\). This verifies the stated direct-sum grading globally. Spin transitions act on their own two spinor factors, so this calculation is a bundle identification.

In the product, however, the original second Clifford generator comes first and the auxiliary first generator second. Their matrices are \(-\sigma_2,\sigma_1\), exactly as in Proposition 8.1 of Connections and the existence of the Kasparov product. Consequently its position operator is \[ \begin{gathered} Q_-=c_V(v)\otimes\sigma_1-c_W(w)\otimes\sigma_2,\\ \Gamma=\sigma_3. \end{gathered} \tag{7.7} \] Its square is again \(\|v\|^2+\|w\|^2\). Dense compactly supported creating sections have the fixed first position error; the Clifford anticommutation gives their creation sign and the adjoint error. The first-operator domain inclusion and positivity form \(2\|v\|^2\) hold as before. The product criterion therefore gives this actual cycle. Conjugation by \(1\otimes\sigma_1\) sends \(Q_-\) to \(Q_+\) and sends its grading to \(-\sigma_3\). Thus its class is the additive inverse of the ordinary direct-sum Thom class. This proves the minus sign when both ranks are odd and hence (7.4) in every parity.

For two positive lines this is the elementary distinction between the symbols \(x-iy\) and \(x+iy\). Iteration gives the factor \((-1)^{n(n-1)/2}\) stated in Theorem 5.3. For a trivial bundle all modules are the corresponding exterior scalar extensions of the actual fibre module, proving the last assertion. \(\square\)

Corollary 7.3. Translating an ordered-coordinate normalization

Set \(\epsilon_n=(-1)^{n(n-1)/2}\). In the right-Clifford product convention of this course, the rank-normalized symbols \[ \widehat\tau_V=\epsilon_r\tau_V, \qquad r=\operatorname{rank}V, \] satisfy \[ \widehat\tau_{V\oplus W} =\widehat\tau_V\boxtimes_R\widehat\tau_W. \tag{7.8} \] For an oriented trivial bundle, \(\widehat\tau_{\mathbb R^n}\) is the ordered right-Clifford external product of its positive one-dimensional symbols. Equivalently, one may retain the outward symbols \(\tau_V\) and use the degree product \[ \begin{gathered} x\boxtimes_{\mathrm{out}}y =(-1)^{rs}x\boxtimes_R y,\\ |x|=r,\qquad |y|=s. \end{gathered} \tag{7.9} \] Then the outward direct-sum symbols multiply without an additional sign.

Proof. The integer identity \[ \begin{aligned} \frac{(r+s)(r+s-1)}2 &=\frac{r(r-1)}2\\ &\quad+\frac{s(s-1)}2+rs. \end{aligned} \] gives \(\epsilon_{r+s}=\epsilon_r\epsilon_s(-1)^{rs}\). Substitute Proposition 7.2 into \(\epsilon_{r+s}\tau_{V\oplus W}\); the two factors \((-1)^{rs}\) cancel. This proves (7.8). Starting with \(\epsilon_1=1\) and applying it successively to ordered trivial lines proves the coordinate assertion. Formula (7.9) and Proposition 7.2 give the second formulation directly. The product in (7.9) is associative: the signs for three degrees on its two sides are \((-1)^{rs+(r+s)t}\) and \((-1)^{st+r(s+t)}\), which agree. It has the same naturality and bilinearity as the original product.

The degreewise conversion is explicit: multiplication by \(\epsilon_n\) takes the right-Clifford product to (7.9), since \[ \epsilon_{r+s}(x\boxtimes_R y) =(\epsilon_r x)\boxtimes_{\mathrm{out}}(\epsilon_s y). \] One must retain the integer degree while making this conversion. Although KK is two-periodic, \(\epsilon_{n+2}=-\epsilon_n\); changing the convention therefore changes the chosen degree-two periodicity identification as well. A sign cannot be discarded merely by replacing a rank with its parity. \(\square\)

In rank two the distinction is visible before any Chern-character calculation. Two positive lines give the right-product operator \(x\sigma_1-y\sigma_2\), while the outward plane has \(x\sigma_1+y\sigma_2\), with grading \(\sigma_3\). Proposition 7.2 proves their classes are opposite by an explicit grading-reversing conjugation. Thus \(\epsilon_2=-1\). The next values are \(\epsilon_3=-1\), \(\epsilon_4=1\). These comparisons specify the actual product and periodicity conventions, rather than using the word “positive” alone to identify them.

8. The Chern character and the Todd factor

The following comparison uses rational cohomology on a finite CW base, or on a compact base of finite CW homotopy type. The KK Thom theorem above requires neither assumption. Let \(U_E\) be the ordinary cohomological Thom class for the complex orientation, normalized to integrate to \(+1\) on each real \(2r\)-dimensional fibre. We identify its group \(H^*(E,E\setminus s(X))\) with \(H^*(D(E),S(E))\): the disc bundle is homotopy equivalent to \(E\), and the sphere bundle is homotopy equivalent to the punctured bundle, so their pair sequences give the identification.

We use the complete local proofs in Cohomological foundations for the complex Thom comparison: CF.1b–CF.1c prove the ordinary Thom isomorphism, its naturality and its ordered product; CF.4a proves the splitting pullback injection; and CF.5a–CF.5b construct the even relative Chern character and prove its products and bundle action. Its normalization is \(\operatorname{ch}(L)=e^{c_1(L)}\); CF.3a and CF.5b prove that the positive planar Bott projection has Chern value \(-1\). Only this natural even character is needed; no rational comparison isomorphism on arbitrary compact Hausdorff spaces is used.

Define the Todd class by its line factors \[ \operatorname{Td}(E)=\prod_j\frac{x_j}{1-e^{-x_j}}, \tag{8.1} \] where \(x_j\) are the Chern roots after the splitting pullback. The factors are power series with constant one; positive-degree classes are nilpotent, so only finitely many terms contribute.

Proposition 8.1 (the factor in the fixed convention). For the complex spinor symbol (6.2), \[ \begin{gathered} \operatorname{ch}_c(\tau_E)\\ =(-1)^r\,\pi^*g_E\smile U_E,\\ g_E=e^{c_1(E)}\operatorname{Td}(E)^{-1}. \end{gathered} \tag{8.2} \]

Proof. First take a line \(L\) with \(x=c_1(L)\). The cohomological Thom isomorphism writes its compactly supported character uniquely as \(\pi^*g(x)U_L\), with an even-degree coefficient on the base. By (6.3), pulling back to the zero section gives \[ x\,g(x)=1-e^x. \tag{8.3} \] We justify determination of the coefficient instead of dividing a possibly zero-divisor in a particular base. On the tautological line over \(\mathbb{CP}^N\), the ring calculation in CF.3a makes all coefficients polynomials in \(x=-h\). Equation (8.3) determines their terms below the top degree. Pull back the identical construction over \(\mathbb{CP}^{N+1}\); its next relation determines that last term as well. Naturality of the relative Chern character in CF.5b and of the ordinary Thom class in CF.1c makes these coefficients compatible. A line bundle over a finite CW base is pulled back from a finite projective space by the finite isometric embedding proved in Lemma RK.7a. Thus all line bundles have the coefficient \[ g(x)=\frac{1-e^x}{x} =-e^x\frac{1-e^{-x}}{x}. \tag{8.4} \] The expression at \(x=0\) means its power-series value \(-1\). This also agrees with the actual planar projection's Chern number.

The relative Chern character preserves these actual symbol products by CF.5b. Its proof constructs the graded tensor comparison, verifies the neighborhood deformation needed at the quotient wedge, and proves injectivity of the smash-quotient pullback from the pair sequence. Ordinary bundle-character multiplicativity then gives the relative product, and base-diagonal pullback gives the balanced symbol product. These are the disc/sphere pairs covered by that proof.

Now apply the splitting pullback proved in CF.4a. It is injective on base cohomology; the two ordinary Thom isomorphisms imply that it is injective on the corresponding relative bundle cohomology too. On this base \(E\) is a sum of lines. Proposition 7.2 multiplies their K-symbols, and the ordinary Thom classes multiply in the same ordered complex orientation. Multiplying (8.4) gives \[ \prod_j\frac{1-e^{x_j}}{x_j} =(-1)^r e^{\sum_jx_j}\operatorname{Td}(E)^{-1}. \] Injectivity brings this equality back to the original base and proves (8.2). \(\square\)

If one instead uses the dual Koszul symbol with zero-section class \(\lambda_{-1}(E^*)\), the same line calculation gives \((1-e^{-x})/x\). Multiplication after the splitting pullback gives its factor \(\operatorname{Td}(E)^{-1}\), with fibre Chern value \(+1\). Formula (8.2) records both changes for our outward symbol: its zero-section class is \(\lambda_{-1}(E)\), and its ordered positive fibre class has Chern value \((-1)^r\).

9. Compact group actions

We supply the compact equivariant operations needed here before using them. Thus the inverse identities in this section depend only on preceding programme proofs and the following averaging argument.

For a second-countable compact group \(G\), a graded equivariant Hilbert \(B\)-module has an even strongly continuous action \(U_g\) with \(U_g(\xi b)=U_g\xi\,\beta_g(b)\) and \(\langle U_g\xi,U_g\eta\rangle=\beta_g\langle\xi,\eta\rangle\). An equivariant cycle is an ordinary graded Kasparov cycle with equivariant representation \(\phi\), norm-continuous orbit \(g\mapsto U_gFU_g^{-1}\), and extra localized defect \((U_gFU_g^{-1}-F)\phi(a)\in\mathcal K(E)\). Equivariant homotopies are cycles over the interval coefficient algebra, with trivial action on the interval. Their homotopy classes, with direct sum, define \(KK^G\). The opposite-grading and rotation proofs of the preceding cycle lesson give additive inverses: their scalar matrices commute with the group action, and a degenerate cycle contracts by the C_0-path construction of Proposition 2.3 of Lesson 06, with diagonal action. Auxiliary Clifford factors have trivial action.

Lemma 9.0 (compact equivariant products). For separable graded G-algebras and countably generated equivariant modules, the connection and localized positivity criterion of Connections and the existence of the Kasparov product defines a bilinear equivariant product. It is associative and has equivariant homomorphism cycles as identities. The exterior extensions and signed flips of Homotopy, associativity, the index pairing and KK-equivalence are equivariant. An invariant unbounded candidate satisfying the local Theorem 0.5 represents this product.

Proof. Write \(g\cdot T=U_gTU_g^{-1}\). Rank-one operators satisfy \(g\cdot\theta_{\xi,\eta}=\theta_{U_g\xi,U_g\eta}\), so compact operators have norm-continuous orbits. The orbit of an adjointable operator is continuous on every vector, as is its adjoint orbit: this follows by inserting \(U_g^{-1}\xi\) and using strong continuity and the uniform operator bound. A bounded such field has a strict integral, obtained by integrating its action and its adjoint action on vectors; these actions respect the right module and the adjoint identity. Haar translation makes its average invariant.

First make an input operator self-adjoint and contractive by the preceding cycle normalization, whose scalar functional calculus preserves the equivariance defect. Its Haar average \(\bar F\) is invariant, self-adjoint and contractive. For each \(a\), \[ (\bar F-F)\phi(a)=\int_G(g\cdot F-F)\phi(a)\,dg \] is compact: the integrand is a norm-continuous compact field. The adjoint formula supplies \(\phi(a)(\bar F-F)\in\mathcal K(E)\). Expanding the commutator and square shows that \((1-t)F+t\bar F\) is an equivariant cycle for every \(t\). Its localized equivariance error is \((1-t)(g\cdot F-F)\). Its orbit is norm continuous. This also works on interval modules, so classes and homotopies can be represented invariantly.

For two invariant inputs, choose an ordinary self-adjoint odd connection \(G_0\) by Theorem 1.3 of the product lesson and average it strictly. For homogeneous \(\xi\), conjugation sends the creation operator \(T_\xi\) to \(T_{U_g\xi}\). The creation error of \(g\cdot G_0\) is therefore the conjugate of the error of \(G_0\) for \(U_g^{-1}\xi\). It varies norm continuously with \(g\): creation operators are norm continuous in their vectors, and conjugation is norm continuous on compact operators. Both creation errors have compact integrals. The averaged \(G\) is consequently an invariant self-adjoint odd connection.

The only further change to the ordinary existence proof is an invariant technical partition. Suppose its separation data \(J,A_1,A_2,\Delta\) are invariant and consist of norm-continuous orbit elements. Apply Theorem 3.1 of Kasparov's technical theorem, then strictly average its even positive contractions \(M_0,N_0\). For \(a\in A_1\) and \(d\in\Delta\), the identities \[ (g\cdot M_0)a=g\cdot\bigl(M_0(g^{-1}\cdot a)\bigr),\qquad [g\cdot M_0,d]=g\cdot[M_0,g^{-1}\cdot d] \] give norm-continuous \(J\)-valued fields. Their integrals prove \(MA_1\subset J\), \([M,\Delta]\subset J\); the identical calculation proves \(NA_2\subset J\). Positivity, parity, \(M+N=1\), and invariance are preserved. Corollary 3.2 of that lesson supplies the required root properties by functional calculus in the quotient.

Use exactly the separation data in Theorem 3.1 of the product lesson: with \(S=F_1\widehat\otimes1\), they are \[ \begin{aligned} J&=\mathcal K(E_1\widehat\otimes_D E_2),\qquad I=\mathcal K(E_1)\widehat\otimes1,\\ A_1&=C^*(J,I),\\ A_2&=C^*(J,G^2-1,[S,G]_{\mathrm{gr}}, [G,\phi(A)]_{\mathrm{gr}}),\\ \Delta&=\overline{\operatorname{span}} \{S,G,\phi(A),S^*,G^*,\phi(A)^*\}. \end{aligned} \] Invariance of \(S,G\), equivariance of \(\phi\) and invariance of the compact images make these data invariant. Their orbit continuity follows from their displayed generators. The countability, separation and derivation conditions are proved in that theorem before its partition step; none uses a group action. The averaged partition therefore gives the invariant candidate \(F=M^{1/4}SM^{1/4}+N^{1/4}GN^{1/4}\). Equations (3.5)–(3.7) of that proof verify its cycle defects, connection errors and localized positivity. Its additional equivariance defect is zero.

For invariant product candidates \(F,F'\), use the comparison algebra \(C^*(J,[S,F]_{\mathrm{gr}},[S,F']_{\mathrm{gr}},F-F')\) and include \(S,F,F',\phi(A)\) and their adjoints in the derivating space. These are invariant separation data by the zero-connection calculation in Theorem 4.1 of the product lesson. The averaged partition gives the invariant comparison operator of its equation (4.3); its positive-anticommutator normalization paths are invariant functional calculus. This proves invariant uniqueness. A possibly noninvariant equivariant product can first be averaged: its connection errors integrate as above, its localized positivity integrates to a positive element of the compact-operator quotient, and the two-sided locally compact straight homotopy just proved identifies its class with the average. Thus uniqueness holds among all equivariant products.

Construct a product over an interval module to prove homotopy invariance in either variable; evaluation gives the endpoint products by their same creation and positivity equations. Block sums prove bilinearity. For an equivariant homomorphism, the essential-replacement interval ideal of Proposition 5.1 of the product lesson is invariant. Choose its connection by averaging. Its endpoint tensor maps are equivariant, so Theorem 5.2 of that lesson gives both identity laws even for degenerate representations.

For three inputs use the proof of Theorem 4.1 of the homotopy-and-associativity lesson, with invariant intermediate product operators. The compact images, bracket connections and negative parts in that proof are invariant. Replace its technical partition by its Haar average. The two terms of its positivity equation (4.3) remain positive: one uses the first intermediate product's positive lift and annihilation of its compact error, and the other uses annihilation of the final bracket's negative part. The simultaneous candidate is now invariant, so uniqueness proves associativity. The exterior tensor and flip unitaries are equivariant for diagonal actions; the same uniqueness comparison proves the signed exterior laws. Finally the resolvent connection and integrated positivity estimates of the unbounded criterion are equations on these same modules. For an invariant candidate its resolvents, bounded transform and normalization homotopies are invariant; the estimates verify the two product conditions just proved sufficient. This proves the last assertion. \(\square\)

Theorem 9.1 (equivariant linear Thom theorem). Let a compact group \(G\) act continuously and linearly on a finite-dimensional real vector space \(W\). Then there are inverse classes \[ \begin{gathered} x_W\in KK^G(\operatorname{Cl}(W),C_0(W)),\\ y_W\in KK^G(C_0(W),\operatorname{Cl}(W)). \end{gathered} \tag{9.1} \] If the action has a spin\(^{c}\) lift, it gives a degree-\(\dim W\) equivariant Thom equivalence with the scalar algebra. In particular \(C_0(W)\) with this action is equivariantly KK-equivalent to \(C_0(W)\) with the trivial action. A complex linear action has the required lift.

Proof. Average an inner product over normalized Haar measure, making the action orthogonal. Give \(L^2(W)\otimes\Lambda W_{\mathbb C}\) its change-of-variable and exterior representation. Position Clifford multiplication and the vertical Dirac operator intertwine this action exactly. Their inverse resolvents and bounded transforms are invariant. Thus the cycles of Sections 2–4 are equivariant, with zero equivariance defect.

The oscillator product is equivariant. The Gaussian kernel is the trivial one-dimensional representation, because its norm is radial and its form degree is zero. The complementary sign homotopy is invariant functional calculus. Hence the first product is the equivariant identity. The simultaneous coordinate and Clifford rotation commutes with the diagonal orthogonal action on \(W\oplus W\), at every parameter. Its uniform compactness estimate is unchanged. The invertible-idempotent argument consequently proves the reverse equivariant product. The doubled Clifford Morita correspondence has the exterior action and the invariant grading (4.2), so its two evaluation tensor maps are equivariant. This proves (9.1).

A spin\(^{c}\) lift gives the equivariant spinor and determinant-line correspondences of Section 5; every transition-map computation there is an intertwining computation. They supply the degree-shifted scalar equivalence. The trivial linear action has the same degree-shifted scalar equivalence, so composing one with the inverse of the other gives the asserted equivalence between the two function algebras. For a complex linear action average a Hermitian metric and use the homomorphism \(U(r)\to\operatorname{Spin}^{c}(2r)\) constructed in Proposition 6.1.

There is no hidden second-countability restriction on the compact group in this linear assertion. Its action factors through a closed subgroup of the finite-dimensional orthogonal group, hence through a compact metrizable group. In the spinor case include its finite-dimensional spinor representation in that quotient. The equivariant products just used can be taken in this second-countable quotient, where Lemma 9.0 applies, and pulled back along \(G\)'s quotient homomorphism. Pullback leaves each creation equation, positivity sandwich and homotopy unchanged and supplies a continuous action by composition. Their explicit cycles and homotopies are still equivariant for \(G\), and their inverse identities remain the same identities. \(\square\)

10. Examples

For \(X\times\mathbb C^r\), take \(S=X\times\Lambda^*\mathbb C^r\). The symbol is the graded sum of the coordinate symbols (5.9), so Theorem 5.3 is external product with the fixed positive Bott class. In degree zero its inverse sends that class back to the trivial line on \(X\).

Let \(E_k\to S^2\) be a complex line with \(c_1(E_k)=k h\), where \(h\) evaluates to one on the complex-oriented sphere. Its Thom triple is \((\mathbf1,\pi^*E_k,v)\). Its zero-section class is \(1-[E_k]\); the comparison map contains the tautological fibre vector, so is invertible off the zero section. Formula (8.2) gives \[ \begin{gathered} \operatorname{ch}_c(\tau_{E_k})\\ =-\left(1+\frac{k}{2}\pi^*h\right)U_{E_k}. \end{gathered} \tag{10.1} \] Here \(h^2=0\). The Thom group is free of rank two, because \(K^0(S^2)\cong\mathbb Z^2\), and its odd group is zero. The symbol generates it as a \(K^0(S^2)\)-module, which is stronger than saying its restriction generates each fibre group.

For a smooth embedding \(i:M\hookrightarrow N\) with compact \(M\), a spin\(^{c}\) normal bundle \(\nu\) gives the first part of the wrong-way construction: \[ K^*(M)\xrightarrow{\ \tau_\nu\ } K_c^{*+\operatorname{rank}\nu}(\nu). \tag{10.2} \] A tubular neighborhood identifies this compact-support group with that of the neighborhood in \(N\), and extension by zero gives a class in \(K^{*+\operatorname{rank}\nu}(N)\). The composition and independence questions for general wrong-way maps are proved in the lesson Wrong-way maps for K-oriented maps. The Thom step (10.2), including its inverse and orientation, is established here.

11. Exercises

11.1. A trivial bundle. Write the Clifford Thom cycle and the spinor Thom cycle for \(X\times\mathbb R^n\). Prove that they are the external scalar extensions of the corresponding fibre cycles.

11.2. Complex spinors. For a Hermitian rank-\(r\) complex bundle \(E\), construct its Clifford spinor module, prove the Morita equivalence, and compute its determinant line and zero-section Thom class.

11.3. Pullback. Prove the naturality formula (7.2), including properness of the total-space pullback map. Explain why the same proof identifies the inverse Dirac cycle.

11.4. The tautological line. Let \(L\to\mathbb{CP}^1\) be the tautological line and put \(u=1-[L]\). Give its Thom class as an actual relative difference-bundle triple, compute the two compactly supported K-groups of its total space with generators, and compute zero-section restriction on those generators. Compare the fibre sign with the relative-bundle convention.

12. Solutions

Solution to 11.1. All frame transitions are identity. The Clifford cycle has module \(C_0(X\times\mathbb R^n,\Lambda^*\mathbb C^n)\), source \(C(X)\widehat\otimes C_n\) acting by the scalar base function and \(i\widehat c\), operator \(c(v)/\sqrt{1+\|v\|^2}\), and grading (4.2). These are precisely the fibre cycle's scalar extensions. Its inverse is the module \(C(X,L^2(\mathbb R^n)\otimes\Lambda^*\mathbb C^n)\), followed by the finite Clifford Morita map in (4.4); its inverse resolvents are the constant fibre resolvents. With the trivial positive spinor bundle, the spinor cycle is \(C_0(X\times\mathbb R^n,S_n)\) with outward Clifford symbol. The scalar-extension tensor maps preserve inner products and have dense range by finite sums of base functions times fibre sections. Hence they are module unitaries identifying the cycles. Proposition 7.2 and its ordered one-dimensional check give \(1_X\boxtimes\tau_{\mathbb R^n}\).

Solution to 11.2. Set \(S=\Lambda^*E\), grade by degree, and use \(c(v)=\varepsilon_v+\varepsilon_v^*\). The exterior relations give its self-adjoint Clifford relations for the underlying real Euclidean metric. In a unitary frame, the real pair \(e_j,ie_j\) gives \(\varepsilon_j+\iota_j\) and \(i(\varepsilon_j-\iota_j)\). Recovering \(\varepsilon_j,\iota_j\) and using the vacuum projection gives every matrix unit, so \(\operatorname{Cl}(E_{\mathbb R})\cong\operatorname{End}(S)\). Finite frames and partitions make \(\Gamma(S)\) finitely generated projective and full; its dual and inner-product evaluation give the inverse Morita correspondence. The implementer lift is \(g\mapsto(g,\Lambda^*g)\). Its scalar-square character is \(\det g\), as checked on diagonal unitary rotations in Proposition 6.1, so the determinant line is \(\det E\). The sphere comparison is \(c(v)^+\), and the zero-section difference is \(\sum_j(-1)^j[\Lambda^jE]\).

Solution to 11.3. The pulled-back metric has \(\|v\|=\|\widetilde f(v)\|\). The inverse image of a compact set is closed and bounded in a ball bundle over compact \(Y\), so is compact. Pullback therefore acts on the function algebras. The pulled-back spinor module and position symbol are exactly those of \(f^*V\); local frames prove the scalar-extension map is an onto isometry. Pulling back a bundle representative of \(a\) identifies the tensor symbol with \(\pi_Y^*f^*a\) times this same symbol, proving (7.2). For the Dirac inverse, the fibre integral, smooth fibre core, Clifford multiplication and inverse resolvents agree on pulled-back local sections. Those sections are dense, so the module unitary also intertwines the closures and bounded transforms. This proves the inverse identification, rather than inferring it only from a fibrewise K-group calculation.

Solution to 11.4. On the closed ball and sphere bundle the triple is \[ \xi=(\mathbf1,\pi^*L,\ \sigma_{(\ell,v)}:\lambda\mapsto\lambda v). \tag{12.1} \] For \(\|v\|=1\) it is a unitary isomorphism from the trivial line to the pulled-back tautological line. This is the specified comparison in the difference-bundle theorem; the bundles alone would not specify the compact-support class.

The integral clutching calculation in Theorem RK.7 gives \[ \begin{gathered} K^0(\mathbb{CP}^1)=\mathbb Z[1]\oplus\mathbb Z u,\\ K^1(\mathbb{CP}^1)=0,\quad u^2=0. \end{gathered} \tag{12.2} \] For the last relation, the rotation path in Theorem RK.7 joins the clutching matrices \(\operatorname{diag}(z,z)\) and \(\operatorname{diag}(z^2,1)\). It gives the actual bundle isomorphism \(L\oplus L\cong L^{\otimes2}\oplus\mathbf1\). Hence \(u^2=(1-[L])^2=0\) integrally. The Thom isomorphism now gives \[ \begin{aligned} K_c^0(L)&=\mathbb Z\xi\oplus\mathbb Z(\pi^*u\,\xi),\\ K_c^1(L)&=0. \end{aligned} \tag{12.3} \] Zero-section restriction sends \(\xi\) to \(u\) and \(\pi^*u\,\xi\) to \(u^2=0\). In a unitary fibre frame, (12.1) has comparison \(z\); Theorem RK.6 and Lemma RK.6a prove its positive planar class and first Chern value \(-1\) on that compactified fibre. The zero-section comparison rules are Corollary RK.4a. Globally \[ \operatorname{ch}_c(\xi) =-\left(1-\frac12\pi^*h\right)U_L \tag{12.4} \] by Proposition 8.1. Thus the clutching sign, the zero-section difference, and the Todd factor agree.

What this lesson uses

The Euclidean Fourier theorem and oscillator domain, compact graph inclusion and kernel calculation are the exact earlier local results in Bott periodicity in KK: the Bott and Dirac elements, Lemmas 0.0a and 4.0. That lesson proves its criterion locally and does not use the present lesson, so this dependency has no cycle. The analytic bounded-transform and product-recognition arguments are proved locally in Lemmas 0.1–0.4 and Theorem 0.5, including local compactness and a nonessential first representation. The central extension of the product argument is proved in Lemma 1.1 above.

The bundle/projection and relative-triple correspondence, radial identification, actual symbol rules and integral sphere groups are proved in the earlier Relative difference bundles, radial pairs and the planar normalization, RK.1–RK.7. The exact cohomological inputs used in Section 8 are proved in Cohomological foundations for the complex Thom comparison: CF.0 supplies chains, products and coefficients; CF.1b–CF.1c prove the ordinary Thom theorem on every Hausdorff base and its orientation and product rules; CF.2 proves the quotient and radial comparisons; CF.3–CF.4 prove the projective ring, first-Chern normalization, projective-bundle formula and splitting injection; CF.5a–CF.5b prove the natural even relative character, its bundle action and its actual symbol products; and CF.6 gives the complete outward-symbol Todd calculation. No full odd-character or rational-isomorphism theorem is needed in Section 8, and no singular-cohomology comparison isomorphism on all compact Hausdorff spaces is asserted.

Lemma 9.0 proves the compact equivariant product and associativity needed in Theorem 9.1 from the preceding non-equivariant connection, partition and associativity proofs. The later lesson Equivariant KK-theory and the Green–Julg theorem develops stabilization, crossed-product comparison and induction; none of those results is needed here. The later lesson Wrong-way maps for K-oriented maps treats the further composition and independence assertions in the embedding example.

The inward-to-outward vector comparison in A fundamental class for an action on a manifold, in Cyclic cohomology, connections and transverse geometry, Lemma 8.12, uses the same geometric Clifford symbol. Its antipodal sign \((-1)^N\) is the parity sign of reversing all fibre coordinates. It is separate from the product conversion \((-1)^{rs}\) in Proposition 7.2. When an outward coordinate class is normalized as an ordered product of positive line classes, translate that statement into this course using Corollary 7.3: with \(\boxtimes_R\) the class is \(\epsilon_N\tau\), while retaining \(\tau\) uses \(\boxtimes_{\mathrm{out}}\). The planar projection in that lesson is constant-unitarily conjugate to the planar projection used here; that agreement of ordinary projection classes does not identify the odd-odd product or its degree-two Morita convention. These formulas provide the conversion needed before combining the two constructions in an index argument.

References

• B. Blackadar, K-Theory for Operator Algebras, author-posted corrected second edition. The author identifies this freely downloadable version as a slightly corrected second edition. The source retains its author copyright and the usage terms stated on that page. Sections 19.9.3–19.9.4 and 20.3.2, printed pp. 201–202 and 208, describe the Clifford, parametrized Thom and compact-equivariant constructions. Those passages include exercises and statements rather than complete proofs of all inverse identities. Sections 2–9 above prove the needed domain, product, sign, bundle and inverse assertions, using the exact earlier oscillator proof and the local analytic capsule in Section 0. Lemma 9.0 supplies the compact equivariant foundation before Theorem 9.1 uses it.

• K. van den Dungen, Localisations of half-closed modules and the unbounded Kasparov product, arXiv version 2, 28 March 2022. The freely readable Sections 2.4 and 3.1, printed pp. 11–16, supply the source comparison for creation estimates and integrated positivity. Section 0 above proves the complete criterion needed in this lesson, including its nonessential representation case. No open adaptation licence is inferred from free access.