39.07.01 · operator-algebras / cyclic-cohomology

Cyclic Cohomology and the Pairing with K-Theory

shipped3 tiersLean: none

Anchor (Master): Connes *Noncommutative Geometry* (1994) Ch. III–IV; Connes 1985 *IHÉS* 62; Connes-Moscovici 1995 (local index formula); Connes & Marcolli *Noncommutative Geometry, Quantum Fields and Motives* §1; Gracia-Bondía-Várilly-Figueroa Ch. 8, 10

Intuition Beginner

K-theory packages the projections inside an algebra — the idempotent elements that pick out direct summands — into a group 39.02.02. It is a coarse but robust invariant: it survives deformation and counts something integral, like a winding number. The natural next question is how to extract a number from a projection in a way that remembers more than its bare K-theory class. The answer is to pair it against a dual object.

A trace is the simplest such pairing. Feed it a projection and it returns the projection's size, a number that depends only on the K-theory class. Connes' insight was that a trace is the bottom rung of a whole ladder of dual objects, one in each even degree, and the higher rungs read off finer information. These dual objects are the cyclic cocycles, and the ladder they form is cyclic cohomology. It is built to be exactly the partner of K-theory: where K-theory has classes, cyclic cohomology has the functionals that evaluate on them.

The payoff is that an integer hiding inside an analytic problem — the index of an operator, the number of its solutions minus the number of its obstructions — can be computed by evaluating a cyclic cocycle on a projection. Geometry that used to require integrating a curvature form over a manifold becomes an algebraic pairing that still makes sense when there is no manifold at all.

Visual Beginner

Cyclic cohomology sits opposite K-theory and the two meet in a pairing that returns a number.

The dictionary reads: K-theory supplies the classes, cyclic cohomology supplies the functionals, and the pairing of an even cyclic class with a $K_{0}$ class is the cohomological way to compute an index. The smooth subalgebra box records a caveat: the pairing is rich only when the algebra is small enough to carry differential structure.

Worked example Beginner

Take the simplest pairing: a trace against a projection. Let $A$ be the algebra of $2 \times 2$ complex matrices, and let $τ$ be the usual matrix trace, the sum of the diagonal entries. Let $e$ be the projection onto the first coordinate axis,

e = (1000), e^{2} = e, e^{*} = e .

The trace of this projection is $τ (e) = 1 + 0 = 1$ . Now take any other projection of the same rank, say the projection onto the line spanned by $(1, 1)$ :

f = \frac{1}{2} (1111), τ (f) = \frac{1}{2} + \frac{1}{2} = 1.

Both projections give the trace value $1$ , because both have rank one. A projection of rank two (the identity matrix) would give $2$ , and the zero projection gives $0$ .

What this tells us: the trace does not see which line a rank-one projection picks out, only that it is rank one. It depends solely on the K-theory class — the rank — and is constant across all projections in that class. That stability is the whole point: the pairing of a cyclic cocycle (here the degree-zero one, a trace) with a K-theory class returns a number that is a deformation invariant.

Check your understanding Beginner

Formal definition Intermediate+

Let $A$ be a unital algebra over $C$ (in the analytic setting, a dense unital subalgebra of a C*-algebra $A_{0}$ closed under holomorphic functional calculus). For $n \geq 0$ write $C^{n} (A) = Hom_{C} (A^{\otimes (n + 1)}, C)$ for the space of $(n + 1)$ -linear functionals $φ (a_{0}, a_{1}, \dots, a_{n})$ . The Hochschild coboundary $b : C^{n} (A) \to C^{n + 1} (A)$ is $$ (b\varphi)(a_0, \dots, a_{n+1}) = \sum_{i=0}^{n} (-1)^i \varphi(a_0, \dots, a_i a_{i+1}, \dots, a_{n+1}) + (-1)^{n+1} \varphi(a_{n+1} a_0, a_1, \dots, a_n), $$ with $b^{2} = 0$ ; its cohomology is the Hochschild cohomology $H H^{*} (A)$ 04.03.20. The cyclicity operator $λ : C^{n} (A) \to C^{n} (A)$ is $$ (\lambda \varphi)(a_0, \dots, a_n) = (-1)^n \varphi(a_n, a_0, \dots, a_{n-1}). $$

Definition (cyclic cocycle). A cyclic $n$ -cochain is a $φ \in C^{n} (A)$ with $λ φ = φ$ , equivalently $$ \varphi(a_1, \dots, a_n, a_0) = (-1)^n \varphi(a_0, a_1, \dots, a_n). $$ A cyclic $n$ -cocycle is a cyclic $n$ -cochain with $b φ = 0$ . Write $C_{λ}^{n} (A) = ker (1 - λ)$ ; then $b$ restricts to $C_{λ}^{*} (A)$ , and the cohomology of $(C_{λ}^{*} (A), b)$ is the cyclic cohomology $H C^{n} (A)$ of Connes ^{[Connes 1985]}.

Definition (the $(b, B)$ -bicomplex). Let $b^{'} : C^{n} \to C^{n + 1}$ be the truncated coboundary (the sum above without the last wrap-around term), and let $A^{'} = \sum_{i = 0}^{n} λ^{i}$ be the cyclic averaging operator on $C^{n}$ . The Connes coboundary $B : C^{n + 1} (A) \to C^{n} (A)$ is $B = A^{'} B_{0}$ , where $(B_{0} φ) (a_{0}, \dots, a_{n}) = φ (1, a_{0}, \dots, a_{n}) - (- 1)^{n + 1} φ (a_{0}, \dots, a_{n}, 1)$ . One has $b^{2} = 0$ , $B^{2} = 0$ , $b B + B b = 0$ . The $(b, B)$ -bicomplex $B^{p, q} (A) = C^{q - p} (A)$ (for $q \geq p \geq 0$ ) carries vertical differential $b$ and horizontal differential $B$ ; its total cohomology is again $H C^{n} (A)$ . This is the cohomological mirror of the homological $(b, B)$ -bicomplex of 04.03.22, obtained by dualizing every chain group and reversing every arrow; the equivalence of the cyclic-subcomplex and bicomplex descriptions, and the construction of $B$ from $1 - λ$ , $s$ and the norm, are imported from there and not reproved here.

Definition (periodicity operator and SBI). The periodicity operator $S : H C^{n} (A) \to H C^{n + 2} (A)$ raises degree by two; cup product with the generator of $H C^{2} (C) = C$ realizes it. It assembles the SBI periodicity exact sequence $$ \cdots \to HH^n(A) \xrightarrow{B} HC^{n-1}(A) \xrightarrow{S} HC^{n+1}(A) \xrightarrow{I} HH^{n+1}(A) \xrightarrow{B} HC^{n}(A) \to \cdots, $$ where $I$ forgets cyclicity (the inclusion of cyclic into Hochschild cochains at the cohomology level). The sequence is the cohomological dual of the homological SBI sequence of 04.03.22; its existence and exactness are quoted from there.

Definition (periodic cyclic cohomology). The periodic cyclic cohomology is the direct limit under $S$ , $$ HP^i(A) = \varinjlim \big( HC^i(A) \xrightarrow{S} HC^{i+2}(A) \xrightarrow{S} \cdots \big), \qquad i \in \mathbb{Z}/2, $$ $Z /2$ -graded into $H P^{ev}$ and $H P^{odd}$ . It is the receptacle on which the K-theory pairing is well defined, because $S$ -translation does not change the pairing value.

The motivating degree-zero example is a trace: a linear $τ : A \to C$ with $τ (ab) = τ (ba)$ is exactly a cyclic $0$ -cocycle, since $b τ (a_{0}, a_{1}) = τ (a_{0} a_{1}) - τ (a_{1} a_{0}) = 0$ and cyclicity in degree $0$ is vacuous. Thus $H C^{0} (A)$ is the space of traces on $A$ .

Counterexamples to common slips

A bounded trace need not exist on a C*-algebra at all, and even when it does, the C*-algebra typically has $H P^{*}$ that is too coarse to see the geometry. The Cuntz algebra $O_{2}$ has no trace; $O_{\infty}$ has $K_{0} = Z$ yet its cyclic cohomology pairs to zero. Cyclic cohomology is computed on a dense smooth subalgebra (e.g. $C^{\infty} (M) \subset C (M)$ , or the smooth noncommutative torus), not on the C*-completion.
The sign in $λ$ matters. Dropping the $(- 1)^{n}$ gives a genuine $Z / (n + 1)$ -action but breaks compatibility of $1 - λ$ with $b$ , so the bicomplex collapses; this is the same failure recorded at the chain level in 04.03.22.
The periodicity operator $S$ raises degree by $+ 2$ in cohomology (it lowers degree by $2$ in the homology of 04.03.22). Mixing the two conventions reverses the SBI sequence.
The pairing is defined on $H P$ , not on a single $H C^{n}$ . Two cocycles differing by $S$ of a lower cocycle pair identically with K-theory, so only the $S$ -stable class is an invariant of the K-group.

Key theorem with proof Intermediate+

Theorem (the index pairing of cyclic cohomology with K-theory). Let $A$ be a unital $C$ -algebra and $φ$ a cyclic $2 m$ -cocycle on $A$ . For an idempotent $e \in M_{q} (A)$ representing a class $[e] \in K_{0} (A)$ 39.02.02, set $$ \langle [\varphi], [e] \rangle = \frac{(-1)^m}{m!}, (\varphi ,#, \mathrm{tr})(e, e, \dots, e), $$ where $φ # tr$ extends $φ$ to $M_{q} (A) = A \otimes M_{q} (C)$ by $(φ # tr) (a_{0} \otimes μ_{0}, \dots, a_{2 m} \otimes μ_{2 m}) = φ (a_{0}, \dots, a_{2 m}) tr (μ_{0} \dots μ_{2 m})$ and there are $2 m + 1$ entries $e$ . Then the number depends only on the cyclic-cohomology class $[φ] \in H C^{2 m} (A)$ and the K-theory class $[e] \in K_{0} (A)$ , and it is unchanged under the periodicity operator: $⟨ S [φ], [e]⟩ = ⟨[φ], [e]⟩$ . Consequently the pairing descends to a bilinear map $$ \langle,\cdot,,,\cdot,\rangle : HP^{\mathrm{ev}}(A) \times K_0(A) \to \mathbb{C}. $$ ^{[Connes 1985; Connes 1994]}

Proof. Write $N = 2 m$ and reduce to $q = 1$ by replacing $φ$ with $φ # tr$ , which is again a cyclic $N$ -cocycle on $M_{q} (A)$ because $tr$ is a trace and the Hochschild coboundary commutes with the cup product by a trace. So assume $e = e^{2} \in A$ .

Independence of the cocycle within its class. Suppose $φ = b ψ$ for a cyclic $(N - 1)$ -cochain $ψ$ . Evaluate $b ψ$ on $(e, \dots, e)$ . Each face term $ψ (\dots, e \cdot e, \dots) = ψ (\dots, e, \dots)$ collapses since $e^{2} = e$ , so all $N + 1$ telescoping terms become the same functional $ψ (e, \dots, e)$ up to sign $(- 1)^{i}$ , and the last wrap term contributes $(- 1)^{N} ψ (e, \dots, e)$ . The alternating sum $\sum_{i = 0}^{N} (- 1)^{i} = 1$ (since $N$ is even there are $N + 1$ odd-many terms), but cyclicity of $ψ$ forces $ψ (e, \dots, e) = (- 1)^{N - 1} ψ (e, \dots, e) = - ψ (e, \dots, e)$ , hence $ψ (e, \dots, e) = 0$ and the whole expression vanishes. Thus coboundaries pair to zero, and the value depends only on $[φ] \in H C^{N} (A)$ .

Independence of the idempotent within its K-class. Two idempotents represent the same $K_{0}$ -class iff they are connected by a smooth path of idempotents $e_{t}$ (after stabilization). Differentiate $t \mapsto (φ # tr) (e_{t}, \dots, e_{t})$ . Using $\overset{e}{˙} = \overset{e}{˙} e + e \overset{e}{˙}$ (from $e_{t}^{2} = e_{t}$ ) and the cyclic invariance of $φ$ , the derivative reorganizes into $b$ applied to a transgression cochain evaluated on the $e_{t}$ , which vanishes by the cocycle condition $b φ = 0$ exactly as in the previous paragraph. So the pairing is constant along the path: it depends only on $[e] \in K_{0} (A)$ .

Compatibility with periodicity. By construction $S φ = φ \cup σ$ where $σ$ generates $H C^{2} (C)$ , normalized so that $⟨ σ, [1]⟩ = 1$ in the universal computation. The combinatorial factor $\frac{( - 1 ) ^{m}}{m !}$ is fixed precisely so that the cup product with $σ$ reproduces the same number in degree $N + 2$ : evaluating $(S φ) # tr$ on $(e, \dots, e)$ with $2 m + 3$ entries and using $e^{2 m + 3} = e$ collapses to the degree- $N$ value. Hence $⟨ S [φ], [e]⟩ = ⟨[φ], [e]⟩$ , and the pairing factors through $H P^{ev} (A) = lim_{S} H C^{2 *} (A)$ . $□$

Bridge. The index pairing builds toward the entire index theory of noncommutative geometry, and it appears again in the local index formula, where the cyclic cocycle that does the pairing is computed from the spectral data of a triple 39.06.01. The foundational reason the formula works is exactly the collapse $e^{k} = e$ : every higher tensor power of an idempotent folds back to the idempotent, so an $S$ -tower of cocycles evaluates to one stable number, and this is dual to the construction in 04.03.22, where the same $(b, B)$ machinery built the homological SBI sequence. The central insight is that K-theory and periodic cyclic cohomology are paired vector spaces over $C$ , with the trace as the degree-zero pairing and the Chern character of a Fredholm module supplying the higher-degree cocycle that computes a Fredholm index; putting these together, the analytic index of an elliptic problem becomes the algebraic value $⟨[φ], [e]⟩$ , and the bridge is that periodicity makes this value independent of the degree in which it is computed.

Exercises Intermediate+

Exercise 4 (medium, short-answer).

State why the pairing $⟨[φ], [e]⟩$ must be defined on $H P^{ev}$ rather than on an individual $H C^{2 m}$ , and what goes wrong if one ignores periodicity.

Hint

Consider two cocycles $φ$ and $S φ$ in different degrees that pair to the same number.

Answer

A given $K_{0}$ -class can be paired with cocycles of any even degree, and $φ$ and $S φ$ (living in degrees $2 m$ and $2 m + 2$ ) give the same value by the theorem. So a single $H C^{2 m}$ over-counts: it distinguishes $φ$ from $S φ$ though they are indistinguishable to K-theory. Passing to the direct limit $H P^{ev} = lim_{S} H C^{2 *}$ identifies exactly the cocycles that pair identically, making the pairing well defined and the invariant intrinsic. Ignoring periodicity yields a degree-dependent number that is not an invariant of the K-class. Rubric: full credit for the $S$ -invariance argument and the direct-limit conclusion.

Exercise 6 (hard, short-answer).

On the smooth noncommutative torus $A_{θ}^{\infty}$ generated by unitaries $u, v$ with $v u = e^{2 π i θ} uv$ , exhibit a cyclic $2$ -cocycle and explain why it is needed in addition to the canonical trace to detect all of $K_{0}$ .

Hint

There are two canonical derivations $δ_{1}, δ_{2}$ with $δ_{1} (u) = u$ , $δ_{2} (v) = v$ ; antisymmetrize them against the trace.

Answer

Let $τ$ be the canonical trace ( $τ (u^{m} v^{n}) = δ_{m, 0} δ_{n, 0}$ ) and let $δ_{1}, δ_{2}$ be the two commuting derivations dual to the generators. The functional $$ \varphi(a_0, a_1, a_2) = \tfrac{1}{2\pi i},\tau\big(a_0(\delta_1(a_1)\delta_2(a_2) - \delta_2(a_1)\delta_1(a_2))\big) $$ is a cyclic $2$ -cocycle (the "fundamental class" of the noncommutative torus). The trace $τ$ pairs with $K_{0} (A_{θ}) = Z^{2}$ to give $τ_{*} (K_{0}) = Z + θ Z$ , but it cannot separate the two $Z$ summands; the $2$ -cocycle $φ$ supplies the second independent linear functional, pairing the Rieffel projection to $1$ and the unit to $0$ . Together ${τ, φ}$ span $H P^{ev} (A_{θ}^{\infty}) = C^{2}$ and pair nondegenerately with $K_{0}$ . Rubric: full credit for the explicit $2$ -cocycle and the explanation that one functional cannot resolve a rank-two K-group.

Lean formalization Intermediate+

lean_status: none — Mathlib has neither cyclic cohomology nor a usable operator-K-theory pairing. The intended statement is schematic:

-- Schematic. Mathlib lacks: the cyclic cochain complex, the (b,B)-bicomplex,
-- periodic cyclic cohomology HP, operator K-theory K_0, and the pairing.
-- A cyclic n-cocycle is a φ : (A^⊗(n+1) →ₗ[ℂ] ℂ) with λ φ = φ and b φ = 0.

/-- The index pairing of an even cyclic cocycle with a K₀-class,
    φ ↦ e ↦ (-1)^m / m! · (φ # tr)(e, …, e). -/
noncomputable def cyclicPairing {A : Type*} [Ring A] [Algebra ℂ A]
    (m : ℕ) (φ : CyclicCocycle A (2 * m)) (e : Idempotent A) : ℂ :=
  sorry  -- needs CyclicCocycle, Idempotent → K₀, and the # cup product

/-- Well-definedness on HP^ev × K₀: stable under S and under the K-class. -/
theorem cyclicPairing_factors :
    True := trivial  -- placeholder for the S-invariance + homotopy-invariance theorem

The honest gap is large: the Hochschild and cyclic cochain complexes, the Connes coboundary $B$ , periodicity $S$ , the colimit defining $H P$ , and the homotopy invariance of the pairing all need building, on top of an operator $K_{0}$ that Mathlib does not yet package.

Advanced results Master

The pairing acquires its index-theoretic meaning through Fredholm modules and their Chern characters ^{[Connes 1994; Gracia-Bondía-Várilly-Figueroa]}.

Fredholm modules and the Chern character. An even Fredholm module over $A$ is a representation on a $Z /2$ -graded Hilbert space together with an odd self-adjoint $F$ , $F^{2} = 1$ , such that $[F, a]$ is compact for $a \in A$ ; it is $p$ -summable if $[F, a] \in L^{p}$ . Connes' Chern character sends such a module to a periodic cyclic cocycle $$ \tau_n(a_0, \dots, a_n) = \lambda_n, \mathrm{Tr}\big(\gamma, a_0 [F, a_1] \cdots [F, a_n]\big), \qquad n \ge p,\ n \text{ even}, $$ with normalization $λ_{n}$ , and the classes for different $n$ are $S$ -equivalent, defining $ch (F) \in H P^{ev} (A)$ . The fundamental computation is that the pairing reproduces a Fredholm index: for a projection $e$ , $$ \langle \mathrm{ch}(F), [e] \rangle = \mathrm{index}\big(e F^+ e : e\mathcal{H}^+ \to e\mathcal{H}^-\big), $$ an integer. This is the cohomological computation of the index pairing of K-theory with K-homology, and it is where the analytic content of 39.06.01 enters: a spectral triple $(A, H, D)$ yields the Fredholm module with $F = D ∣ D ∣^{- 1}$ , whose Chern character is the transverse fundamental class of the geometry.

The local index formula (Connes-Moscovici). For a regular spectral triple with discrete dimension spectrum, the Chern character has a representative built from residues of zeta functions $ζ (s) = Tr (\dots ∣ D ∣^{- s})$ , the local cocycle $$ \phi_n(a_0, \dots, a_n) = \sum_{k} c_{n,k}, \mathrm{Res}_{s}\ \mathrm{Tr}\big(a_0 [D, a_1]^{(k_1)} \cdots [D, a_n]^{(k_n)} |D|^{-2|k|-n-2s}\big), $$ where the superscripts denote iterated commutators with $D^{2}$ . This expresses the index pairing through local, computable spectral data, generalizing the Atiyah-Singer index theorem 03.09.10 to the noncommutative setting; in the commutative case it recovers the $\hat{A}$ -genus integral ^{[Connes-Moscovici 1995]}.

Smooth subalgebras. A C*-algebra has poor cyclic cohomology: $H P^{*}$ of $C (M)$ is the topological K-homology with complex coefficients, but the cyclic groups $H C^{n} (C (M))$ for the C*-algebra are not the de Rham groups, because continuous multilinear functionals cannot encode differentiation. One passes to a dense smooth subalgebra $A \subset A$ stable under holomorphic functional calculus, so that the inclusion induces an isomorphism on K-theory ( $K_{*} (A) = K_{*} (A)$ , the density theorem) while $A$ carries enough differentiable structure for $H P^{*} (A)$ to be the de Rham-type receptacle. For $C^{\infty} (M) \subset C (M)$ this recovers $H P^{ev} (C^{\infty} (M)) = ⨁_{k} H_{dR}^{2 k} (M; C)$ , matching the homological computation of 04.03.22.

Entire cyclic cohomology. When no finite summability holds — only $θ$ -summability, $Tr (e^{- t D^{2}}) < \infty$ for $t > 0$ — finite-degree cocycles fail and one needs entire cyclic cohomology $H E^{*} (A)$ , defined on cochain systems $(φ_{n})_{n \geq 0}$ with the entire growth condition $\sum_{n} \frac{∥ φ _{n} ∥}{n !} z^{n} < \infty$ for all $z$ . The Chern character of a $θ$ -summable module is then the JLO cocycle $$ \varphi_n(a_0, \dots, a_n) = \int_{\Delta_n} \mathrm{Tr}\big(\gamma, a_0 e^{-s_0 D^2}[D,a_1]e^{-s_1 D^2}\cdots [D,a_n]e^{-s_n D^2}\big), ds, $$ integrated over the simplex $\sum s_{i} = 1$ , which pairs with $K_{0}$ to give the index even in the infinite-dimensional regime relevant to quantum field theory ^{[Connes 1988]}.

Synthesis. These results are one theorem seen at increasing resolution, and putting these together is the content of noncommutative index theory: the foundational reason cyclic cohomology is the right home is that it is dual to K-theory, with the Chern character $ch (F)$ the universal even cocycle whose pairing computes a Fredholm index, and this is exactly the cohomological face of the analytic index pairing of 39.06.01. The local index formula generalises Atiyah-Singer, the smooth-subalgebra passage is what makes the pairing nondegenerate, and entire cyclic cohomology is the receptacle that survives the loss of finite dimension. The central insight is that K-theory and periodic cyclic cohomology are dual functors over $C$ ; the trace is the degree-zero pairing, the transverse fundamental class the top-degree pairing, and the bridge is the Chern character, which appears again wherever an index must be read off algebraically. This is dual to the homological picture of 04.03.22, where the same $(b, B)$ machinery computes de Rham cohomology in the commutative case, so periodic cyclic cohomology generalises de Rham theory exactly as K-theory generalises the topological count of vector bundles.

Full proof set Master

Proposition (cyclic cohomology in degree zero is the space of traces). For any unital algebra $A$ , $H C^{0} (A) = {τ : A \to C linear : τ (ab) = τ (ba)}$ . Indeed the cyclic condition in degree $0$ is vacuous, and the cocycle condition is $b τ (a_{0}, a_{1}) = τ (a_{0} a_{1}) - τ (a_{1} a_{0}) = 0$ , the tracial identity; there are no coboundaries into degree $0$ , so every trace is a nonzero class. $□$

Proposition (the pairing is well defined on cohomology and on K-theory). Let $φ$ be a cyclic $2 m$ -cocycle and $e = e^{2} \in M_{q} (A)$ . (i) If $φ = b ψ$ with $ψ$ cyclic, then $(φ # tr) (e, \dots, e) = 0$ . (ii) If $e_{0}, e_{1}$ are smoothly homotopic idempotents, the pairing agrees. For (i), the collapse $e^{2} = e$ turns $(b ψ) (e, \dots, e)$ into $(\sum_{i} (- 1)^{i}) ψ (e, \dots, e)$ , and cyclicity gives $ψ (e, \dots, e) = (- 1)^{2 m - 1} ψ (e, \dots, e) = - ψ (e, \dots, e)$ , so it vanishes. For (ii), set $f (t) = (φ # tr) (e_{t}, \dots, e_{t})$ ; differentiating and using $\overset{e}{˙} = \overset{e}{˙} e + e \overset{e}{˙}$ rewrites $f^{'} (t)$ as $(b φ)$ applied to a transgression cochain in the $e_{t}, \overset{e}{˙}_{t}$ , which is zero. Hence $f$ is constant. $□$

Proposition (the Chern-character pairing is integral). For an even $p$ -summable Fredholm module $(H, F, γ)$ over $A$ and a projection $e \in A$ with $[F, e] \in L^{p}$ , the operator $T = e F^{+} e : e H^{+} \to e H^{-}$ is Fredholm and $$ \langle \mathrm{ch}(F), [e]\rangle = \mathrm{index}(T) \in \mathbb{Z}. $$ The compactness of $[F, a]$ makes $T$ Fredholm; the McKean-Singer–type identity $index (T) = Tr (γ (1 - F^{2}) \dots)$ evaluated on $e$ matches the normalized supertrace formula defining $τ_{n}$ , and the integer values of an index force integrality. $□$

Proposition (periodicity invariance of the pairing). With $S : H C^{2 m} \to H C^{2 m + 2}$ , one has $⟨ S [φ], [e]⟩ = ⟨[φ], [e]⟩$ . The operator $S$ is cup product with the generator $σ \in H C^{2} (C)$ normalized by $⟨ σ, [1]⟩ = 1$ ; the combinatorial constant $\frac{( - 1 ) ^{m}}{m !}$ is the value that makes $(S φ) (e, \dots, e)$ with $2 m + 3$ slots collapse under $e^{k} = e$ to $φ (e, \dots, e)$ with $2 m + 1$ slots. Hence the value is stable along the $S$ -tower and factors through $H P^{ev} (A) = lim_{S} H C^{2 *} (A)$ . $□$

Proposition (density / smooth-subalgebra invariance of the K-side). If $A \subset A$ is a dense subalgebra stable under holomorphic functional calculus in a C*-algebra $A$ , the inclusion induces an isomorphism $K_{*} (A) \sim K_{*} (A)$ . Stability under holomorphic functional calculus makes idempotents and invertibles of $A$ deformable into $A$ (spectral projections lie in $A$ ), giving surjectivity, and a relative version gives injectivity. Thus the pairing $H P^{*} (A) \times K_{*} (A) \to C$ is well posed even though $H P^{*} (A)$ for the C*-algebra is too coarse: the smooth subalgebra supplies the cocycles, the C*-algebra supplies the same K-theory. $□$

Connections Master

Operator K-theory $K_{0}$ , $K_{1}$ 39.02.02 — this unit supplies the dual theory: the index pairing $H P^{ev} \times K_{0} \to C$ (and the odd pairing with $K_{1}$ ) evaluates cyclic cocycles on the very projection and unitary classes that K-theory organizes, so the K-groups are exactly the domain of the pairing constructed here.
Spectral triples and the reconstruction theorem 39.06.01 — a spectral triple yields the Fredholm module $F = D ∣ D ∣^{- 1}$ whose Chern character is the transverse fundamental class living in this cohomology; the index pairing of that unit is computed cohomologically here, and the orientation Hochschild cycle of the triple is the top-degree cyclic cocycle.
Algebraic cyclic homology and Connes' long exact sequence 04.03.22 — the $(b, B)$ -bicomplex, the SBI periodicity sequence, and the periodicity operator $S$ are the homological mirror of the cohomological versions used here; this unit dualizes that construction and is the analytic, K-theory-paired side of the same machine.
Hochschild cohomology 04.03.20 — Hochschild cohomology $H H^{*}$ is the input the cyclic theory refines; the SBI sequence interleaves $H H^{*}$ and $H C^{*}$ , and the forgetful map $I : H C^{*} \to H H^{*}$ records how much cyclic structure is lost.
Atiyah-Singer index theorem 03.09.10 — the local index formula of Connes-Moscovici is the noncommutative generalization, expressing the cyclic-cohomology index pairing through residues of zeta functions; the commutative case reduces to the $\hat{A}$ -genus integral of the classical theorem.

Historical & philosophical context Master

Cyclic cohomology was introduced by Connes in a 1983 Comptes Rendus announcement and developed fully in Noncommutative differential geometry (Publ. Math. IHÉS 62, 1985) ^{[Connes 1985]}, where the cyclic cocycles, the $(b, B)$ -bicomplex, the periodicity operator $S$ , and the pairing with K-theory all appear together; the motivation was an index problem for foliations where the relevant invariant was a cyclic cocycle rather than a measure. Tsygan introduced the homological version independently in 1983 from the Lie-algebra cohomology of matrices, the parallel recorded in 04.03.22. The pairing with K-theory and the Chern character of a Fredholm module are systematized in Connes' 1994 monograph Noncommutative Geometry ^{[Connes 1994]}.

The entire theory and the JLO cocycle for $θ$ -summable modules were given by Connes in Entire cyclic cohomology of Banach algebras and characters of $θ$ -summable Fredholm modules (K-Theory 1, 1988) ^{[Connes 1988]}, building on the heat-kernel formula of Jaffe, Lesniewski and Osterwalder. The local index formula, which makes the pairing computable from residues of zeta functions, was proved by Connes and Moscovici (Geom. Funct. Anal. 5, 1995) ^{[Connes-Moscovici 1995]}. The textbook account of the Chern character and the index pairing is in Gracia-Bondía, Várilly and Figueroa ^{[Gracia-Bondía-Várilly-Figueroa]}. The lineage runs from the Atiyah-Singer index theorem, through Connes' foliation index theory, to the recognition that cyclic cohomology is the de Rham theory dual to K-theory in the noncommutative world.

Bibliography Master

Connes, A., "Cohomologie cyclique et foncteurs $Ext^{n}$ ", C. R. Acad. Sci. Paris 296 (1983), 953–958.
Connes, A., "Noncommutative differential geometry", Publ. Math. IHÉS 62 (1985), 257–360.
Connes, A., "Entire cyclic cohomology of Banach algebras and characters of $θ$ -summable Fredholm modules", K-Theory 1 (1988), 519–548.
Connes, A., Noncommutative Geometry, Academic Press, 1994. Ch. III–IV.
Connes, A. & Moscovici, H., "The local index formula in noncommutative geometry", Geom. Funct. Anal. 5 (1995), 174–243.
Connes, A. & Marcolli, M., Noncommutative Geometry, Quantum Fields and Motives, AMS Colloquium Publications 55, 2008. §1.
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.

Operator-algebras spine, cyclic-cohomology chapter. The cohomology theory dual to K-theory in noncommutative geometry: cyclic cocycles, the $(b, B)$ -bicomplex and SBI periodicity, periodic cyclic cohomology $HP^ $, an d t h e in d e x p ai r in g$ HP^{\mathrm{ev}} \times K_0 \to \mathbb{C}$ computed by the Chern character of a Fredholm module. Builds on operator K-theory (39.02.02) and spectral triples (39.06.01); cross-refs the algebraic version (04.03.22).*

Prerequisites

39.02.02
39.06.01

Tier anchors

beginner: Connes' slogan 'cyclic cohomology is the de Rham theory dual to K-theory'; the trace as a degree-zero current, recast from Connes *Noncommutative Geometry* (1994) Ch. III §1 and the index-pairing picture
intermediate: Connes 1985 *Publ. Math. IHÉS* 62 §I (cyclic cocycles, the $(b,B)$-bicomplex, the periodicity sequence) and Connes *Noncommutative Geometry* (1994) Ch. III §1–3; Gracia-Bondía, Várilly & Figueroa *Elements of Noncommutative Geometry* Ch. 8 and 10
master: Connes *Noncommutative Geometry* (1994) Ch. III–IV; Connes 1985 *IHÉS* 62; Connes-Moscovici 1995 (local index formula); Connes & Marcolli *Noncommutative Geometry, Quantum Fields and Motives* §1; Gracia-Bondía-Várilly-Figueroa Ch. 8, 10

References

Connes, A. — Noncommutative differential geometry · *Publ. Math. IHÉS* 62 (1985), 257–360 — cyclic cocycles, the $(b,B)$-bicomplex, the SBI periodicity exact sequence relating Hochschild and cyclic cohomology, the periodicity operator $S$, the character of a cycle, and the pairing of cyclic cohomology with K-theory
Connes, A. — Noncommutative Geometry · Academic Press 1994, Ch. III (cyclic cohomology, the bicomplex, periodicity, the pairing with K-theory) and Ch. IV (the index formula, Fredholm modules, the Chern character in cyclic cohomology)
Connes, A. — Entire cyclic cohomology · *K-Theory* 1 (1988), 519–548 — entire cyclic cohomology, the $\theta$-summable case, the JLO cocycle (with the Jaffe-Lesniewski-Osterwalder construction)
Connes, A. & Moscovici, H. — The local index formula in noncommutative geometry · *Geom. Funct. Anal.* 5 (1995), 174–243 — the local (residue) cocycle representing the Chern character of a regular spectral triple, computing the index pairing through the dimension spectrum
Gracia-Bondía, J. M., Várilly, J. C. & Figueroa, H. — Elements of Noncommutative Geometry · Birkhäuser 2001, Ch. 8 (Hochschild and cyclic cohomology, the bicomplexes, periodicity) and Ch. 10 (the Chern character of a Fredholm module and the index pairing)

Estimated time

beginner: 18m
intermediate: 50m
master: 85m