The Chern Character in K-Homology

Intuition Beginner

A vector bundle over a curved space carries a number you can read off by integrating its curvature. Wrap the curvature into a closed form, integrate, and out comes the Chern character — an even-degree gadget that turns geometric twisting into ordinary cohomology. It is the bridge that lets you compute with a bundle as if it were a list of differential forms.

Noncommutative geometry asks the dual question. Instead of a bundle, you start with an abstract operator that knows the geometry: a reflection whose commutators with the algebra are small 39.06.03. There is no manifold to integrate over and no curvature form in the ordinary sense. The Chern character in K-homology is the recipe that, even so, produces a closed object — a cyclic cocycle — out of that operator. You feed it the operator and a few algebra elements, and it returns a number built from a trace.

The payoff is a dictionary. On one side sits an operator, hard to compute with directly. On the other sits a cyclic cocycle 39.07.01, a functional you can pair against projections to read off integers. The Chern character translates the first into the second, so that the index of the operator becomes an algebraic evaluation. What was a curvature integral on a manifold becomes a trace of commutators when the manifold disappears.

Visual Beginner

The Chern character is an arrow from operators to cyclic cocycles, mirroring the classical arrow from bundles to curvature forms.

The dictionary reads: an operator $F$ plays the role a vector bundle played classically; the Chern character sends it to a cyclic cocycle just as the classical character sent a bundle to a curvature form; and pairing that cocycle against a projection returns the same integer the curvature integral would. Two routes to one whole number.

Worked example Beginner

Take the smallest operator that does something. Work on two stacks of basis vectors and let $F$ be the swap that exchanges the stacks; doing the swap twice returns every vector to where it started, so $F$ applied twice is the identity. This is the toy reflection from 39.06.03.

Now compute the lowest piece of its Chern character. In degree zero the recipe asks for the trace of $F$ times a single algebra element, weighted by a constant. Take the algebra element to be the grading $g$ that reports $+1$ on the first stack and $-1$ on the second. The product $Fg$ swaps the stacks and then flips a sign; on a balanced pair of one vector per stack the diagonal entries of this product are both $0$, so the trace over that pair is

$$\text{trace}=0+0=0.$$

Now unbalance the pair by adding one extra vector to the first stack that the swap sends to itself. That extra vector contributes $+1$ to the trace of $Fg$, and the running total becomes

$$\text{trace}=0+1=1.$$

What this tells us: the Chern character did not count vectors; it counted the imbalance between the two stacks that the operator could not pair up. That leftover integer is exactly the kind of number an index reports — solutions minus obstructions — and the Chern character is the machine that surfaces it from the operator.

Check your understanding Beginner

Formal definition Intermediate+

Let $A$ be a unital $\ast$-algebra (a dense subalgebra of a C*-algebra, stable under holomorphic functional calculus, as for the Fredholm modules of 39.06.03). Let $(\mathcal{H},F,\pi )$ be a $p$-summable Fredholm module over $A$: a $\ast$-representation $\pi$ with $F=F^{\ast }$, $F^2=1$, and $[F,\pi (a)]\in {\mathcal{L}}^p(\mathcal{H})$ for all $a\in A$. In the even case $\mathcal{H}$ carries a grading $\gamma ={\gamma }^{\ast }$, ${\gamma }^2=1$, with $\gamma \pi (a)=\pi (a)\gamma$ and $\gamma F=-F\gamma$; in the odd case no grading is given. Write ${\mathrm{Tr}}_s(T)=\mathrm{Tr}(\gamma T)$ for the supertrace in the even case and $\mathrm{Tr}$ for the trace in the odd case. Throughout, $a_i$ is shorthand for $\pi (a_i)$.

Definition (the Connes character cocycle). For $n\ge p-1$ of the parity matching the module ($n$ even in the even case, $n$ odd in the odd case), the character of $(\mathcal{H},F,\pi )$ in degree $n$ is the $(n+1)$-linear functional $$ \tau_n(a_0, a_1, \dots, a_n) = \lambda_n, \mathrm{Tr}_s!\big(F,[F, a_0],[F, a_1]\cdots[F, a_n]\big), $$ with normalisation ${\lambda }_n=(-1{)}^{n(n-1)/2}\Gamma (\frac{n}{2}+1)$ in the even case and ${\lambda }_n=\sqrt{2i}(-1{)}^{n(n-1)/2}\Gamma (\frac{n}{2}+1)$ in the odd case (conventions follow ^{[Connes 1994]}). The product $[F,a_0]\cdots [F,a_n]$ lies in ${\mathcal{L}}^{p/(n+1)}\subseteq {\mathcal{L}}^1$ once $n+1\ge p$, so the trace converges; the summability degree is exactly the lower bound on $n$. Using $F^2=1$ one has $F[F,a]=-[F,a]F$, so $F$ may be moved through to the right, an identity used repeatedly below.

Definition (the Chern character in K-homology). The class $[{\tau }_n]\in H{C}^n(A)$ is the Chern character of the module in cyclic degree $n$. The collection $\{[{\tau }_n]{\}}_{n\equiv \epsilon }$ (with $\epsilon =0$ even, $\epsilon =1$ odd) is stable under the periodicity operator $S$ of 39.07.01: $S[{\tau }_n]=[{\tau }_{n+2}]$. The induced class in the direct limit is the Chern character $$ \mathrm{ch}^[F] \in HP^\varepsilon(A) = \varinjlim_S HC^{n}(A), \qquad \varepsilon \in \mathbb{Z}/2, $$ and the assignment $[F] \mapsto \mathrm{ch}^[F]$definesahomomorphism$\mathrm{ch}^* : K^\varepsilon(A) \to HP^\varepsilon(A)$ from the K-homology of 39.06.03 into periodic cyclic cohomology, additive and homotopy-invariant.

Definition (the commutative model). When $A=C^{\infty }(M)$ for a closed spin manifold and $F=D|D{|}^{-1}$ is the phase of the Dirac operator 39.06.03, the cocycle ${\tau }_n$ represents, under the isomorphism $H{P}^{\mathrm{ev}}(C^{\infty }(M))\cong {⨁}_k{H}_{\mathrm{dR}}^{2k}(M;\mathbb{C})$ of 04.03.22, the de Rham class $$ \mathrm{ch}^[D] \ \longleftrightarrow\ \widehat{A}(M) \in \bigoplus_k H^{2k}_{\mathrm{dR}}(M;\mathbb{C}), $$ the $\hat{A}$-genus. This is the noncommutative continuation of the Chern-Weil construction: a curvature polynomial becomes a cyclic cocycle, and "${\int }_M\hat{A}\mathrm{ch}(E)$" becomes "$\langle \mathrm{ch}^[D], \mathrm{ch}_*[E]\rangle$".

Counterexamples to common slips

• The character ${\tau }_n$ is defined only for $n+1\ge p$; in lower degree the trace diverges. Writing ${\tau }_0(a)={\lambda }_0{\mathrm{Tr}}_s(Fa)$ for a module that is merely $2$-summable is meaningless — $Fa$ is not trace-class. Summability fixes the lowest cyclic degree in which the Chern character can be written, and only the $S$-stable class in $HP$ is independent of that degree.
• ${\tau }_n$ is the character of the bounded datum $F$, not of the unbounded $D$. Two spectral triples with the same phase $F$ have the same ${\mathrm{ch}}^{\ast }[F]$ even when their metrics differ; the Connes distance 39.06.03 is invisible to the Chern character. The metric re-enters only through the local representative of Connes-Moscovici, not through the class.
• The cocycle is cyclic and $(b,B)$-closed but is generally not a Hochschild cocycle: $b{\tau }_n\ne 0$ in isolation. What vanishes is $b{\tau }_n+B{\tau }_{n+2}$ in the periodic $(b,B)$-bicomplex of 39.07.01. Treating ${\tau }_n$ as a single-complex Hochschild cocycle, ignoring the $B$-coupling to the next degree, breaks the periodicity and the index theorem with it.
• The normalisation ${\lambda }_n$ is not cosmetic: it is the unique scaling making $S[{\tau }_n]=[{\tau }_{n+2}]$ on the nose, so that the index value is the same in every degree. Dropping it leaves a degree-dependent multiple of the index.

Key theorem with proof Intermediate+

Theorem (the Chern character is a periodic cyclic cocycle representing the index). Let $(\mathcal{H},F,\pi )$ be an even $p$-summable Fredholm module over $A$ and $n$ an even integer with $n+1\ge p$. Then:

(i) ${\tau }_n$ is cyclic: ${\tau }_n(a_1,\dots ,a_n,a_0)=(-1{)}^n{\tau }_n(a_0,\dots ,a_n)={\tau }_n(a_0,\dots ,a_n)$.

(ii) ${\tau }_n$ is a cyclic cocycle, $b{\tau }_n=0$ on the cyclic complex, and its class is $S$-stable: $S[{\tau }_n]=[{\tau }_{n+2}]$ in $H{C}^{n+2}(A)$, so $\{[{\tau }_n]\}$ defines $\mathrm{ch}^[F] \in HP^{\mathrm{ev}}(A)$.*

(iii) For a projection $e\in M_q(A)$ representing $[e]\in K_0(A)$, $$ \big\langle \mathrm{ch}^*[F], [e] \big\rangle = \mathrm{index}\big(eF^+e : e\mathcal{H}^+_q \to e\mathcal{H}^-_q\big) \in \mathbb{Z}, $$ the integer index pairing of 39.06.03. ^{[Connes 1985; Connes 1994]}

Proof. Throughout write $\delta a=[F,a]$ and recall $F\delta a=-\delta aF$ from $F^2=1$, and $\delta (ab)=(\delta a)b+a(\delta b)$ since $\delta$ is a derivation. Each $\delta a_i\in {\mathcal{L}}^p$, so any product of $n+1\ge p$ of them lies in ${\mathcal{L}}^1$ and the (super)trace is finite and cyclic under permutation.

(i) Cyclicity. Move the leading $F$ through the product using $F\delta a_0=-\delta a_{0}F$ repeatedly: $$ F,\delta a_0,\delta a_1\cdots\delta a_n = (-1)^{n+1},\delta a_0\cdots\delta a_n,F. $$ Apply the trace property ${\mathrm{Tr}}_s(XF)={\mathrm{Tr}}_s(FX)$ (valid since $\gamma$ commutes with $F$ up to the sign already accounted for, and the supertrace is cyclic on trace-class operators) to bring $F$ back to the front, and separately cycle $\delta a_0$ to the end. Combining the sign $(-1{)}^{n+1}$ from commuting $F$ past $n+1$ factors with the parity of the cyclic permutation of the $n+1$ commutators yields ${\tau }_n(a_1,\dots ,a_n,a_0)=(-1{)}^n{\tau }_n(a_0,\dots ,a_n)$; for $n$ even this is ${\tau }_n$ itself, so ${\tau }_n\in C_{\lambda }^n(A)$.

(ii) Cocycle and periodicity. Compute $b{\tau }_n(a_0,\dots ,a_{n+1})={\sum }_{i=0}^n(-1{)}^i{\tau }_n(\dots ,a_i{a}_{i+1},\dots )+(-1{)}^{n+1}{\tau }_n(a_{n+1}{a}_0,a_1,\dots ,a_n)$. Expand each merged term with the Leibniz rule $\delta (a_i{a}_{i+1})=(\delta a_i)a_{i+1}+a_i(\delta a_{i+1})$. The first piece of term $i$ cancels against the second piece of term $i-1$ in the telescoping sum because adjacent commutators meet, and the algebra element $a_i$ slides through under the trace; the boundary terms cancel against the wrap-around using cyclicity from (i). Hence $b{\tau }_n=0$ on the cyclic complex. The periodicity claim is the computation that $S{\tau }_n$ — cup product with the generator of $H{C}^2(\mathbb{C})$ — equals ${\tau }_{n+2}$ up to a coboundary; the normalisation ${\lambda }_n$ is fixed by matching ${\lambda }_{n+2}/{\lambda }_n$ to the combinatorial constant $(\frac{n}{2}+1{)}^{-1}$ produced by $S$, so $S[{\tau }_n]=[{\tau }_{n+2}]$ and $\{[{\tau }_n]\}$ assembles into ${\mathrm{ch}}^{\ast }[F]\in H{P}^{\mathrm{ev}}(A)$.

(iii) Index. By the Calderón-Fedosov formula 39.06.03, for $m$ with $2m\ge p$, $$ \mathrm{index}(eF^+e) = \mathrm{Tr}s\big(e,(F,\delta e)^{2m}\big)\cdot c_m, $$ a supertrace of $2m$ commutators squeezed by $e$, where $c_m$ is the Calderón constant. Substituting $a_0 = \cdots = a{2m} = e$into$\tau_{2m}$andusing$e^2 = e$tocollapsetherepeatedfactorsreproducesexactlythissupertracewith$c_m = \lambda_{2m}^{-1}\cdot(\text{pairing normalisation})$,so$\langle[\tau_{2m}],[e]\rangle = \mathrm{index}(eF^+e)$.TherightsideisanintegerbecauseitisaFredholmindex,and$S$-stabilityfrom(ii)makesthevalueindependentof$m$.$\square$

Bridge. The Chern character builds toward the whole local index theory of noncommutative geometry, and it appears again in the local index formula, where the same class ${\mathrm{ch}}^{\ast }[D]$ acquires a residue representative computable from the spectral data of a triple 39.06.01. The foundational reason the construction closes up is exactly the identity $F^2=1$: it makes $F\delta a=-\delta aF$, which both forces cyclicity and turns $b{\tau }_n$ into a telescoping cancellation, and this is exactly the bounded-side shadow of the index pairing of 39.06.03. The central insight is that K-homology and periodic cyclic cohomology are matched by a single homomorphism ${\mathrm{ch}}^{\ast }$, dual to the K-theory Chern character ${\mathrm{ch}}_{\ast }:K_{\ast }(A)\to H{P}_{\ast }(A)$ of 39.07.01; putting these together, the analytic index $\mathrm{index}(e{F}^{+}e)$ becomes the cohomological pairing $⟨{\mathrm{ch}}^{\ast }[F],{\mathrm{ch}}_{\ast }[e]⟩$, and the bridge is that periodicity makes this number the same in every cyclic degree, so the Chern character generalises the curvature integral that computes the classical index.

Exercises Intermediate+

Lean formalization Intermediate+

lean_status: none — Mathlib has Fredholm operators, Schatten ideals, and the operator trace, but it has neither Fredholm modules nor cyclic cohomology, so the Chern character cannot be stated. The intended statement is schematic:

-- Schematic. Mathlib lacks: FredholmModule, the cyclic cochain complex,
-- the (b,B)-bicomplex, periodic cyclic cohomology HP, and K-homology K^*.

/-- The Connes character cocycle of an even p-summable Fredholm module in
    cyclic degree n:  τ_n(a₀,…,a_n) = λ_n · Tr_s(F [F,a₀] ⋯ [F,a_n]). -/
noncomputable def chernCocycle {A : Type*} [Ring A] [Algebra ℂ A]
    (m : FredholmModule A) (n : ℕ) : CyclicCochain A n :=
  sorry  -- needs the supertrace of a product of Schatten-class commutators

/-- The Chern character lands in HP and is S-stable; pairing it with a
    K₀-class returns the Fredholm index of the compression e F⁺ e. -/
theorem chern_pairs_to_index {A : Type*} [Ring A] [Algebra ℂ A]
    (m : FredholmModule A) (e : A) (he : IsIdempotentElem e) :
    True := trivial  -- placeholder: ⟨ch* m, [e]⟩ = index (compress m.F e)

The honest gap is large: the cyclic and Hochschild cochain complexes, the operators $b$, $B$, $S$, the colimit $HP$, the Calderón-Fedosov index formula, and the Schatten-power bookkeeping that makes the supertrace converge all need building on top of a Fredholm-module structure Mathlib does not carry.

Advanced results Master

The Chern character is the universal vehicle carrying analytic index data into cyclic cohomology, and three refinements sharpen it ^{[Connes 1994; Gracia-Bondía-Várilly-Figueroa]}.

The $(b,B)$ description and entire growth. In the periodic $(b,B)$-bicomplex of 39.07.01 the character is more naturally a cochain system $({\tau }_n{)}_n$ than a single cocycle: the defining closedness is $(b+B)\tau =0$, i.e. $b{\tau }_n+B{\tau }_{n+2}=0$ for each $n$, expressing that the family is closed in the total complex. The single-degree cocycle ${\tau }_n$ recovered above is the leading component of this system. Writing the character as a $(b,B)$-cochain is what makes the passage to the entire theory uniform: one keeps the whole sequence $({\tau }_n)$ rather than truncating at the summability degree.

The JLO cocycle for $\theta$-summable modules. When only $\theta$-summability holds — $e^{-t{D}^2}$ trace-class for all $t>0$, with no finite Schatten bound — the finite-degree ${\tau }_n$ diverges and one needs entire cyclic cohomology $H{E}^{\ast }(A)$, defined on cochain systems with the entire growth condition ${\sum }_n\frac{\Vert {\phi }_n\Vert }{\lfloor n/2\rfloor !}{z}^n<\infty$. The Chern character is then the Jaffe-Lesniewski-Osterwalder cocycle $$ \mathrm{ch}^n(a_0, \dots, a_n) = \int{\Delta_n}\mathrm{Tr}_s!\big(a_0,e^{-s_0 D^2}[D, a_1],e^{-s_1 D^2}\cdots[D, a_n],e^{-s_n D^2}\big),ds, $$ integrated over the simplex ${\sum }_i{s}_i=1$, $s_i\ge 0$. The JLO family $(\mathrm{ch}^n)$is$(b + B)$-closedandentire,definesaclassin$HE^*(A)$,andpairswith$K*(A)$tocomputetheindexevenintheinfinite-dimensionalregimeofquantumfieldtheory[ref:TOD{O}_{R}EFConnes1988].As$t \to 0$itinterpolatestothefinite-degreecharacter;as$t \to \infty$ it localises on the index, exhibiting the McKean-Singer heat-kernel mechanism in cyclic form.

The local index formula. For a regular spectral triple with discrete dimension spectrum, the Chern character has a residue representative (Connes-Moscovici): the local cocycle $$ \phi_n(a_0, \dots, a_n) = \sum_{k}c_{n,k},\mathrm{Res}_{s=0},\mathrm{Tr}!\big(a_0[D, a_1]^{(k_1)}\cdots[D, a_n]^{(k_n)},|D|^{-2|k|-n-2s}\big), $$ where $T^{(j)}$ denotes the $j$-fold commutator with $D^2$. This represents ${\mathrm{ch}}^{\ast }[D]\in H{P}^{\mathrm{ev}}$ by local, computable spectral data, replacing the divergent commutator traces by residues of zeta functions; in the commutative case it recovers the $\hat{A}\mathrm{ch}$ integrand of the heat-kernel proof of Atiyah-Singer 03.09.10 ^{[Connes-Moscovici 1995]}. The same class ${\mathrm{ch}}^{\ast }[D]$ thus has two faces: the global supertrace cocycle ${\tau }_n$ and the local residue cocycle ${\varphi }_n$, $S$-equivalent in $H{P}^{\mathrm{ev}}$.

Compatibility of the two Chern characters. The K-theory Chern character ${\mathrm{ch}}_{\ast }:K_{\ast }(A)\to H{P}_{\ast }(A)$ of 39.07.01 and the K-homology Chern character ${\mathrm{ch}}^{\ast }:K^{\ast }(A)\to H{P}^{\ast }(A)$ of this unit are adjoint with respect to the index pairing: for $[e]\in K_0(A)$ and $[F]\in K^0(A)$, $$ \langle [F], [e]\rangle_{K^* \times K_*} = \big\langle \mathrm{ch}^[F],, \mathrm{ch}_[e]\big\rangle_{HP^* \times HP_*}, $$ both sides the same integer 39.06.03. This square — index pairing on top, cyclic pairing on the bottom, the two Chern characters as vertical arrows — commutes, and its commutativity is the cohomological form of the index theorem.

Synthesis. These results are one map seen at increasing resolution, and putting these together is the analytic engine of noncommutative index theory: the foundational reason cyclic cohomology receives the index is that ${\mathrm{ch}}^{\ast }$ is dual to the K-theory character ${\mathrm{ch}}_{\ast }$, so the index pairing of 39.06.03 factors through cohomology and becomes the algebraic value $⟨{\mathrm{ch}}^{\ast }[F],{\mathrm{ch}}_{\ast }[e]⟩$. The single-degree cocycle generalises the curvature polynomial of the classical Chern-Weil character, the JLO cocycle is the receptacle that survives the loss of finite summability, and the local formula is exactly the residue refinement that recovers Atiyah-Singer in the commutative case. The central insight is that K-homology and periodic cyclic cohomology are matched functors over $\mathbb{C}$, with the supertrace cocycle the global representative and the residue cocycle the local one; this is dual to the homological picture of 04.03.22, where the same $(b,B)$ machinery computes de Rham cohomology, so the bridge is that ${\mathrm{ch}}^{\ast }$ generalises the curvature integral exactly as the index pairing generalises the count of solutions minus obstructions, and periodicity makes the resulting integer independent of the degree in which it is read.

Full proof set Master

Proposition (anticommutation and degree-zero collapse). For a Fredholm module with $F^2=1$ and any $a$, $F[F,a]=-[F,a]F$, and consequently $[F,a^2]=[F,a]a+a[F,a]$ while $F[F,a{]}^2=[F,a{]}^{2}F$. The first is the computation $F[F,a]=a-FaF=-([F,a]F)$ of Exercise 3; the second follows by applying it twice, so an even number of commutators commutes with $F$ and an odd number anticommutes. This parity is what makes the supertrace ${\mathrm{Tr}}_s(F[F,a_0]\cdots [F,a_n])$ nonzero only in the correct degree parity. $\square$

Proposition (cyclicity of ${\tau }_n$). For $n$ even, ${\tau }_n(a_1,\dots ,a_n,a_0)={\tau }_n(a_0,\dots ,a_n)$. Moving $F$ past the $n+1$ commutators gives $F\delta a_0\cdots \delta a_n=(-1{)}^{n+1}\delta a_0\cdots \delta a_{n}F$; trace cyclicity returns $F$ to the front and rotates $\delta a_0$ to the end, contributing the cyclic-permutation sign. The two signs combine to $(-1{)}^n=+1$ for $n$ even, so ${\tau }_n$ is invariant under the cyclic rotation $\lambda$, hence ${\tau }_n\in C_{\lambda }^n(A)$. $\square$

Proposition ($b{\tau }_n=0$). On the cyclic complex, $b{\tau }_n=0$. Expanding $b{\tau }_n(a_0,\dots ,a_{n+1})$ and applying the Leibniz rule $\delta (a_i{a}_{i+1})=(\delta a_i)a_{i+1}+a_i(\delta a_{i+1})$ to each merged slot, the "right" half of slot $i$ and the "left" half of slot $i+1$ produce equal-and-opposite contributions under the trace (the intervening algebra element slides through), telescoping to zero; the wrap-around term $(-1{)}^{n+1}{\tau }_n(a_{n+1}{a}_0,\dots )$ cancels the surviving boundary term by the cyclicity of the previous proposition. Hence ${\tau }_n$ is a cyclic cocycle and $[{\tau }_n]\in H{C}^n(A)$. $\square$

Proposition ($S$-stability and the homomorphism). $S[{\tau }_n]=[{\tau }_{n+2}]$ in $H{C}^{n+2}(A)$, and $[F]\mapsto \{[{\tau }_n]\}$ is a well-defined additive homomorphism $K^0(A)\to H{P}^{\mathrm{ev}}(A)$. The operator $S$ is cup product with the generator $\sigma \in H{C}^2(\mathbb{C})$ 39.07.01; a direct expansion of ${\tau }_n\cup \sigma$ against ${\tau }_{n+2}$ shows they differ by a coboundary once ${\lambda }_{n+2}/{\lambda }_n=(\frac{n}{2}+1{)}^{-1}$, which is the value of the chosen normalisation. Degenerate modules give zero (all commutators vanish) and operator homotopies give norm-continuous, hence cohomologically constant, families, so the map factors through K-homology and is additive over direct sums. $\square$

Proposition (the index theorem in cohomological form). For an even $p$-summable module and a projection $e\in M_q(A)$, $$ \langle\mathrm{ch}^[F], [e]\rangle = \mathrm{index}(eF^+e) \in \mathbb{Z}. $$ By the Calderón-Fedosov formula 39.06.03, $\mathrm{index}(e{F}^{+}e)=c_m{\mathrm{Tr}}_s(e(F\delta e{)}^{2m})$ for $2m\ge p$. Putting $a_0=\cdots =a_{2m}=e$ in ${\tau }_{2m}$ and collapsing the repeated factors with $e^2=e$ matches this supertrace, and the normalisation makes the constants agree, so $⟨[{\tau }_{2m}],[e]⟩=\mathrm{index}(e{F}^{+}e)$. The value is an integer because it is a Fredholm index, and $S$-stability makes it independent of $m$, so it is a well-defined pairing of $\mathrm{ch}^[F] \in HP^{\mathrm{ev}}$with$[e] \in K_0$.$\square$

Proposition (the JLO cochain is entire and $(b+B)$-closed). For a $\theta$-summable spectral triple, the JLO family $({\mathrm{ch}}_n^{\ast })$ satisfies $b{\mathrm{ch}}_n^{\ast }+B{\mathrm{ch}}_{n+2}^{\ast }=0$ and the entire growth bound ${\sum }_n\frac{\Vert {\mathrm{ch}}_n^{\ast }\Vert }{\lfloor n/2\rfloor !}{z}^n<\infty$, hence defines a class in $H{E}^{\ast }(A)$. Closedness is the Duhamel expansion of $\frac{d}{dt}{e}^{-t{D}^2}$ together with the simplex integration by parts, which converts a $b$-term in degree $n$ into a $B$-term in degree $n+2$; the growth bound follows from the heat-kernel estimate $\Vert e^{-s{D}^2}{\Vert }_1\le C{s}^{-d/2}$ on the simplex, integrated to a factorial decay matching the entire condition. Pairing $({\mathrm{ch}}_n^{\ast })$ with a K-theory class computes the index by the $t$-independence of the pairing and the $t\to \infty$ localisation. $\square$

Connections Master

• Cyclic cohomology and the pairing with K-theory 39.07.01 — this unit constructs the dual arrow: where that unit built the K-theory Chern character ${\mathrm{ch}}_{\ast }:K_{\ast }(A)\to H{P}_{\ast }(A)$ and the pairing $H{P}^{\ast }\times K_{\ast }\to \mathbb{C}$, here ${\mathrm{ch}}^{\ast }:K^{\ast }(A)\to H{P}^{\ast }(A)$ sends a Fredholm module to the cyclic cocycle ${\tau }_n$, and the two characters are adjoint, so the cyclic pairing $⟨{\mathrm{ch}}^{\ast }[F],{\mathrm{ch}}_{\ast }[e]⟩$ computes the same number defined there.

• Fredholm modules and the index pairing 39.06.03 — the integer $\mathrm{index}(e{F}^{+}e)$ defined there is reproduced cohomologically here: ${\mathrm{ch}}^{\ast }$ converts the analytic K-homology class $[F]$ into a cyclic class, and the Calderón-Fedosov supertrace formula of that unit is exactly the evaluation $⟨[{\tau }_{2m}],[e]⟩$, making the index pairing factor through cyclic cohomology.

• Spectral triples and the reconstruction theorem 39.06.01 — a spectral triple $(\mathcal{A},\mathcal{H},D)$ supplies the unbounded data whose phase $F=D|D{|}^{-1}$ feeds this character; the local index formula represents ${\mathrm{ch}}^{\ast }[D]$ by residues of the zeta functions of $|D|$ over the dimension spectrum of that unit, the local counterpart of the global supertrace cocycle here. The Connes-Moscovici local index formula, the residue refinement of this Chern character, is the next unit in this chapter (39.06.06).

• The Dixmier trace and the noncommutative integral 39.06.05 — the leading residue producing the local representative of ${\mathrm{ch}}^{\ast }[D]$ is computed by the Dixmier trace of that unit, the noncommutative integral that turns the top-degree component of the Chern character into the noncommutative volume integrand.

• Algebraic cyclic homology and Connes' long exact sequence 04.03.22 — the $(b,B)$-bicomplex in which the character $({\tau }_n)$ is $(b+B)$-closed is the cohomological mirror of the homological machine there; in the commutative case ${\mathrm{ch}}^{\ast }[D]$ becomes the $\hat{A}$-genus in de Rham cohomology, the same target that unit's $(b,B)$ computation produces.

• Atiyah-Singer index theorem 03.09.10 — the commutative model of this Chern character is the classical Chern-Weil character, and the index pairing $⟨{\mathrm{ch}}^{\ast }[D],{\mathrm{ch}}_{\ast }[E]⟩={\int }_M\hat{A}(M)\mathrm{ch}(E)$ is exactly the twisted-Dirac index of that theorem; the Connes-Moscovici residue cocycle generalises its heat-kernel proof to the noncommutative setting.

Historical & philosophical context Master

The Chern character of a Fredholm module was introduced by Connes in Noncommutative differential geometry (Publ. Math. IHÉS 62, 1985) ^{[Connes 1985]}, in the same paper that introduced cyclic cohomology, the $(b,B)$-bicomplex, and the K-theory pairing; the character ${\tau }_n$ and the proof that the index pairing is computed by it appear there as §II, motivated by Connes' index theory for foliations, where the relevant invariant of a transversally elliptic operator was a cyclic cocycle. The construction generalises the Chern-Weil homomorphism, by which Chern (1946) and Weil expressed characteristic classes as curvature polynomials; Connes' insight was that the curvature form is the commutator $[F,a]$ and the integral is the operator (super)trace, so the whole Chern-Weil package survives the disappearance of the manifold.

The systematic account, including the even and odd cases, summability, and the bounded/unbounded dictionary, is in Connes' 1994 monograph Noncommutative Geometry ^{[Connes 1994]}. The entire theory and the JLO cocycle for $\theta$-summable modules were given by Connes in Entire cyclic cohomology of Banach algebras and characters of $\theta$-summable Fredholm modules (K-Theory 1, 1988) ^{[Connes 1988]}, building on the heat-kernel formula of Jaffe, Lesniewski and Osterwalder. The residue (local) representative making the Chern character computable from zeta-function residues was proved by Connes and Moscovici (Geom. Funct. Anal. 5, 1995) ^{[Connes-Moscovici 1995]}. The operator-theoretic treatment of the Chern character and the index pairing in K-homology is in Higson and Roe's Analytic K-Homology (2000) ^{[Higson-Roe Ch. 8]} and the textbook account in Gracia-Bondía, Várilly and Figueroa ^{[Gracia-Bondía-Várilly-Figueroa]}. The lineage runs from Chern-Weil theory, through Atiyah's analytic K-homology and the Atiyah-Singer index theorem, to Connes' recognition that the Chern character is a map from K-homology to cyclic cohomology valid for any algebra.

Bibliography Master

• Connes, A., "Noncommutative differential geometry", Publ. Math. IHÉS 62 (1985), 257–360.
• Connes, A., "Entire cyclic cohomology of Banach algebras and characters of $\theta$-summable Fredholm modules", K-Theory 1 (1988), 519–548.
• Connes, A., Noncommutative Geometry, Academic Press, 1994. Ch. IV.
• Connes, A. & Moscovici, H., "The local index formula in noncommutative geometry", Geom. Funct. Anal. 5 (1995), 174–243.
• Jaffe, A., Lesniewski, A. & Osterwalder, K., "Quantum $K$-theory I. The Chern character", Communications in Mathematical Physics 118 (1988), 1–14.
• Chern, S.-S., "Characteristic classes of Hermitian manifolds", Annals of Mathematics 47 (1946), 85–121.
• Higson, N. & Roe, J., Analytic K-Homology, Oxford University Press, 2000. Ch. 8–9.
• Gracia-Bondía, J. M., Várilly, J. C. & Figueroa, H., Elements of Noncommutative Geometry, Birkhäuser, 2001. Ch. 8, 10.
• Loday, J.-L., Cyclic Homology, Springer Grundlehren 301, 2nd ed. 1998.

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Operator-algebras spine, cyclic-cohomology chapter. The Chern character in K-homology $\mathrm{ch}^ : K^(A) \to HP^(A)$:theConnescharactercocycle$\tau_n(a_0,\dots,a_n) = \lambda_n,\mathrm{Tr}s(F[F,a_0]\cdots[F,a_n])$ofa$p$-summableFredholmmodule,itscyclicityand$(b,B)$-closedness,$S$-periodicity$S[\tau_n] = [\tau{n+2}]$,theevenandoddcases,thecohomologicalindextheorem$\langle\mathrm{ch}^[F],[e]\rangle = \mathrm{index}(eF^+e)$,theJLOcocyclefor$\theta$-summable modules, and the commutative Chern-Weil model. Builds on cyclic cohomology (39.07.01) and Fredholm modules (39.06.03).