03.06.18 · modern-geometry / characteristic-classes

Chern character $ch (E)$ as a ring homomorphism

shipped3 tiersLean: partial

Anchor (Master): Atiyah-Hirzebruch *Riemann-Roch theorems for differentiable manifolds* (Bull. AMS 65, 1959, 276-281); Atiyah *K-Theory* §2-§3; Milnor-Stasheff *Characteristic Classes* §16; Bott-Tu §21

Intuition [Beginner]

The Chern character is a recipe that turns a complex vector bundle into a list of cohomology classes that behave like a polynomial in the bundle's twisting data. The recipe is short to state once you know Chern classes: write the bundle as a sum of Chern roots, take the formal exponential of each root, add the results, and read off the cohomology classes that fall out in each degree.

The point of having such a recipe is that it converts the structure of vector bundles into the structure of cohomology in a way that respects both addition and multiplication. Direct sum of bundles becomes addition of cohomology classes. Tensor product of bundles becomes multiplication. The Chern character is the precise translator between two different ways of recording the same global information about a space.

The slogan is: ordinary Chern classes record twisting one degree at a time, while the Chern character bundles them into a single exponential package whose algebra mirrors the algebra of bundles. Over the rational numbers, this translator becomes a perfect dictionary, an honest isomorphism between K-theory and even cohomology.

Visual [Beginner]

The picture to keep in mind is a complex vector bundle being split into line bundles, each line bundle contributing a single Chern root, and then each root being run through the formal exponential to produce a polynomial in cohomology. The whole bundle's Chern character is the sum of these single-line contributions.

This visual matches the algebra: direct sums of bundles correspond to taking a longer list of roots; tensor products correspond to taking pairwise sums of roots. The exponential converts these list operations into addition and multiplication of cohomology classes.

Worked example [Beginner]

Take the tautological line bundle $L$ over the projective line. Its first Chern class is the standard generator $x$ of $H^{2} (CP^{1}; Z)$ . Its only Chern root is $x$ itself, since $L$ has rank one.

Step 1. The Chern character of a line bundle is the formal exponential of its first Chern class: $ch (L) = e^{x} = 1 + x + x^{2} /2 + x^{3} /6 + \dots$ .

Step 2. The base $CP^{1}$ has $x^{2} = 0$ in its cohomology ring, so the higher terms vanish. The exponential collapses to $ch (L) = 1 + x$ .

Step 3. Form the tensor product of two copies of $L$ , written $L \cdot L$ , which has first Chern class $2 x$ . Then $ch (L \cdot L) = 1 + 2 x$ , which matches $ch (L) \cdot ch (L) = (1 + x) (1 + x) = 1 + 2 x + x^{2} = 1 + 2 x$ in this cohomology ring.

What this tells us: the Chern character of a tensor product is the product of the Chern characters, and on a low-dimensional space the formula reduces to a tidy polynomial identity.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $X$ be a compact Hausdorff space and let $E \to X$ be a complex vector bundle of rank $n$ . By the splitting principle 03.06.04, pulled back to a suitable flag bundle $f : F (E) \to X$ with injective $f^{*}$ on cohomology, the bundle $f^{*} E$ splits as a direct sum of line bundles $L_{1} \oplus L_{2} \oplus \dots \oplus L_{n}$ . Write $x_{j} = c_{1} (L_{j}) \in H^{2} (F (E); Z)$ for the Chern roots of $E$ . The total Chern class of $E$ is the symmetric polynomial $c (E) = \prod_{j} (1 + x_{j})$ in the roots.

Definition. The Chern character of $E$ is the inhomogeneous rational cohomology class

ch (E) := j = 1 \sum n e^{x_{j}} = j = 1 \sum n k \geq 0 \sum \frac{x _{j}^{k}}{k !} \in H^{even} (X; Q),

where the symmetric function $\sum_{j} e^{x_{j}}$ expands as a polynomial in the elementary symmetric functions of the $x_{j}$ , that is, in the Chern classes of $E$ . The result is independent of the choice of splitting and descends from $F (E)$ to $X$ because $f^{*}$ is injective and the expression is invariant under the Weyl-group action permuting the roots.

The first few homogeneous components are explicit polynomials in the Chern classes:

ch_{0} (E) ch_{1} (E) ch_{2} (E) ch_{3} (E) ch_{4} (E) = n, = c_{1} (E), = \frac{1}{2} (c_{1} (E)^{2} - 2 c_{2} (E)), = \frac{1}{6} (c_{1} (E)^{3} - 3 c_{1} (E) c_{2} (E) + 3 c_{3} (E)), = \frac{1}{24} (c_{1} (E)^{4} - 4 c_{1} (E)^{2} c_{2} (E) + 4 c_{1} (E) c_{3} (E) + 2 c_{2} (E)^{2} - 4 c_{4} (E)) .

The denominators $1, 1, 2, 6, 24, \dots$ are the factorials from the exponential series; their presence is the basic reason the Chern character is defined over $Q$ rather than $Z$ . The Chern character of a line bundle is the bare exponential $ch (L) = e^{c_{1} (L)}$ , and the Chern character of a rank- $n$ product bundle is the integer $n$ .

Counterexamples to common slips

The Chern character is not the total Chern class. The total class $c (E) = 1 + c_{1} + c_{2} + \dots$ is multiplicative on direct sums via $c (E \oplus F) = c (E) c (F)$ but is not additive. The Chern character is additive on direct sums and multiplicative on tensor products — the opposite pairing.
The Chern character is not integral. The denominators $k!$ in $ch_{k}$ make $ch$ a class in $H^{even} (X; Q)$ , not in $H^{even} (X; Z)$ . The image of $ch$ inside rational cohomology is generally not the integral lattice; the failure is precisely what makes $ch$ a rational isomorphism, not an integral one.
The grading is by degree, not by rank. The Chern character lives in $⨁_{k} H^{2 k} (X; Q)$ with the $k$ -th piece being a polynomial in Chern classes of total degree $k$ (each $c_{i}$ counted as degree $i$ ). The rank of the bundle is the degree- $0$ part, not a separate piece of grading.

Key theorem with proof [Intermediate+]

Theorem (Chern character is a ring homomorphism). Let $X$ be a compact Hausdorff space. The Chern character induces a ring homomorphism

ch : K (X) ⟶ H^{even} (X; Q),

where $K (X)$ is the Grothendieck group of complex vector bundles under direct sum with multiplication induced by tensor product, and $H^{even} (X; Q) = ⨁_{k \geq 0} H^{2 k} (X; Q)$ is the even rational cohomology ring under cup product.

Proof. The proof has four steps: (i) additivity on direct sums, (ii) multiplicativity on tensor products, (iii) extension to the Grothendieck group, (iv) compatibility with the ring structure.

(i) Additivity. Let $E$ and $F$ be complex vector bundles on $X$ . By the splitting principle applied jointly to $E$ and $F$ , there is a base $f : Y \to X$ with $f^{*}$ injective on cohomology such that $f^{*} E$ and $f^{*} F$ split as sums of line bundles. Write $f^{*} E = ⨁_{i} L_{i}$ with Chern roots $x_{i}$ , and $f^{*} F = ⨁_{j} M_{j}$ with Chern roots $y_{j}$ . Then $f^{*} (E \oplus F) = ⨁_{i} L_{i} \oplus ⨁_{j} M_{j}$ , with Chern roots given by the concatenated multiset ${x_{i}} \cup {y_{j}}$ . By definition,

ch (f^{*} (E \oplus F)) = i \sum e^{x_{i}} + j \sum e^{y_{j}} = ch (f^{*} E) + ch (f^{*} F) .

Injectivity of $f^{*}$ transports the identity back to $X$ : $ch (E \oplus F) = ch (E) + ch (F)$ .

(ii) Multiplicativity. With the same splitting cover, the Chern roots of $f^{*} (E \otimes F)$ are the pairwise sums ${x_{i} + y_{j}}$ . The tensor product of line bundles satisfies $c_{1} (L_{i} \otimes M_{j}) = c_{1} (L_{i}) + c_{1} (M_{j}) = x_{i} + y_{j}$ ; this is the additivity of the first Chern class for line-bundle tensor products. So

ch (f^{*} (E \otimes F)) = i, j \sum e^{x_{i} + y_{j}} = i, j \sum e^{x_{i}} e^{y_{j}} = (i \sum e^{x_{i}}) (j \sum e^{y_{j}}) = ch (f^{*} E) \cdot ch (f^{*} F),

using that the exponential of a sum is the product of exponentials. Injectivity of $f^{*}$ on cohomology gives $ch (E \otimes F) = ch (E) \cdot ch (F)$ on $X$ .

(iii) Extension to $K (X)$ . The Grothendieck completion universal property says that any abelian-group-valued function on the monoid of isomorphism classes of bundles under direct sum extends uniquely to a homomorphism from $K (X)$ . By step (i), $ch$ is such a function from bundles to $H^{even} (X; Q)$ , so it extends to a homomorphism of abelian groups $ch : K (X) \to H^{even} (X; Q)$ . On a formal difference $[E] - [F]$ , the extension reads $ch ([E] - [F]) = ch (E) - ch (F)$ .

(iv) Ring compatibility. The multiplication on $K (X)$ is induced by tensor product on bundle classes. Step (ii) shows that the bundle-level map respects this multiplication, and step (iii) extends additively to formal differences. The product compatibility on the extension follows from the bilinearity of tensor product over direct sum: $(E - F) \otimes (E^{'} - F^{'}) = E \otimes E^{'} - F \otimes E^{'} - E \otimes F^{'} + F \otimes F^{'}$ , and applying $ch$ term by term gives $(ch (E) - ch (F)) (ch (E^{'}) - ch (F^{'})) = ch ((E - F) (E^{'} - F^{'}))$ . The unit relation $ch (1) = 1$ holds because the rank-one product bundle has vanishing first Chern class, so $ch (\underline{C}) = e^{0} = 1$ . $□$

Bridge. The Chern character builds toward 03.09.10 Atiyah-Singer by way of the index formula's topological side: the integer-valued analytical index of an elliptic operator is identified with $\int_{X} ch (σ (D)) \cdot Td (T X)$ , and the Chern character is exactly the natural transformation that converts the K-theory class of the symbol into a degree-graded cohomology class integrable against the fundamental class. The same identifying-K-with-cohomology map appears again in 03.08.07 Bott periodicity, where it identifies the K-theoretic Bott element of $K (S^{2 n})$ with the cohomological fundamental class of $H^{2 n} (S^{2 n}; Q)$ up to a factorial scaling. The central insight is that the Chern character carries the multiplicative structure of K-theory faithfully into cohomology, with the splitting principle reducing every identity to a symmetric-function statement about formal Chern roots, and putting these together identifies the rational K-theory ring with the rational even cohomology ring — the foundational reason all K-theoretic computations descend to ordinary cohomology over $Q$ .

Exercises [Intermediate+]

Exercise 7 (hard, short-answer).

Use the Chern character to prove that two complex vector bundles $E, F$ on a compact CW complex $X$ with $ch (E) = ch (F)$ in $H^{even} (X; Q)$ are rationally K-equivalent, but need not be stably isomorphic over $Z$ .

Hint

The Atiyah-Hirzebruch rational isomorphism makes $ch \otimes Q$ injective on $K (X) \otimes Q$ . Torsion in $K (X)$ is annihilated by tensoring with $Q$ .

Answer

By Atiyah-Hirzebruch, $ch \otimes Q : K (X) \otimes Q \to H^{even} (X; Q)$ is an isomorphism. If $ch (E) = ch (F)$ , then $[E] \otimes 1 = [F] \otimes 1$ in $K (X) \otimes Q$ , hence $[E] - [F]$ is a torsion element of $K (X)$ : there exists a positive integer $N$ with $N \cdot ([E] - [F]) = 0$ , equivalently $E^{\oplus N} \oplus G ≅ F^{\oplus N} \oplus G$ for some bundle $G$ . This is rational stable equivalence. Integral stable equivalence $[E] = [F]$ in $K (X)$ requires the torsion class $[E] - [F]$ to itself vanish, which $ch$ cannot detect — $ch$ sees only the rationalisation. Example: $X = RP^{2}$ with $K (RP^{2}) ≅ Z /2$ generated by a torsion class; the Chern character vanishes on this generator. Rubric: full credit for the rationalisation argument and the torsion-kernel structural point.

Exercise 8 (hard, symbolic).

Compute the compatibility identity between Adams operations and the Chern character: $ch_{k} (ψ^{j} (E)) = j^{k} ch_{k} (E)$ in $H^{2 k} (X; Q)$ , for any complex vector bundle $E$ and any positive integers $j, k$ .

Hint

Reduce to a line bundle by the splitting principle. On a line bundle $L$ , $ψ^{j} (L) = L^{\otimes j}$ , so $c_{1} (ψ^{j} (L)) = j c_{1} (L)$ .

Answer

By the splitting principle, pull $E$ back to a flag bundle where it splits as $L_{1} \oplus \dots \oplus L_{n}$ with Chern roots $x_{j}$ . On each line bundle $L_{j}$ , $ψ^{j} (L_{i}) = L_{i}^{\otimes j}$ has Chern root $j x_{i}$ , so $ch (ψ^{j} (L_{i})) = e^{j x_{i}}$ . Additivity gives $ch (ψ^{j} (E)) = \sum_{i} e^{j x_{i}}$ , and expanding the exponential, the degree- $2 k$ part is $\sum_{i} (j x_{i})^{k} / k! = j^{k} \sum_{i} x_{i}^{k} / k! = j^{k} ch_{k} (E)$ . Injectivity transports back. Rubric: full credit for the splitting reduction, the line-bundle identity $ψ^{j} (L) = L^{\otimes j}$ , and the explicit power extraction.

Lean formalization [Intermediate+]

The companion Lean module Codex.Modern.CharClasses.ChernCharacter records the four anchor theorems with placeholder bundle and ring carriers and sorry-proofs pending the upstream Mathlib infrastructure described in the frontmatter lean_mathlib_gap field. The module statements include:

-- Additivity on direct sums.
theorem chernCharacter_directSum
    (X : CompactBase) (E F : ComplexVectorBundle X) (H : HEvenQ X) :
    chernCharacter (directSum E F) H
      = chernCharacter E H + chernCharacter F H

-- Multiplicativity on tensor products.
theorem chernCharacter_tensorProd
    (X : CompactBase) (E F : ComplexVectorBundle X) (H : HEvenQ X) :
    chernCharacter (tensorProd E F) H
      = chernCharacter E H * chernCharacter F H

-- Atiyah-Hirzebruch rational isomorphism (statement).
theorem atiyah_hirzebruch_rational
    (X : FiniteCWBase) (K : KRing X.toCompactBase) (H : HEvenQ X.toCompactBase) :
    True

Low-degree explicit formulas $ch_{0} = n$ , $ch_{1} = c_{1}$ , $ch_{2} = (c_{1}^{2} - 2 c_{2}) /2$ , $ch_{3} = (c_{1}^{3} - 3 c_{1} c_{2} + 3 c_{3}) /6$ are recorded as concrete ℚ-valued functions of integer Chern classes, with line-bundle reduction lemmas (ch2_lineBundle, ch3_lineBundle) proved by ring. The Hirzebruch-Riemann-Roch and Atiyah-Singer index statements appear as True-bodied placeholders pending bundled cohomology, the symbol-class K-theory, and the integration-along-the-fibre machinery; promotion to lean_status: full requires the bundle-level characteristic-class library plus the rational cohomology and Atiyah-Hirzebruch spectral sequence work catalogued in the lean_mathlib_gap field.

Advanced results [Master]

Definition via Chern roots and explicit formulas through degree $4$

For a complex vector bundle $E$ of rank $n$ , the splitting principle 03.06.04 yields a flag-bundle pullback on which $E$ splits as $L_{1} \oplus \dots \oplus L_{n}$ with line-bundle Chern roots $x_{j} = c_{1} (L_{j})$ . The total Chern class becomes $c (E) = \prod_{j} (1 + x_{j}) = 1 + e_{1} + e_{2} + \dots + e_{n}$ where $e_{k} = e_{k} (x_{1}, \dots, x_{n})$ is the $k$ -th elementary symmetric polynomial, identified with $c_{k} (E)$ . The Chern character

ch (E) = j = 1 \sum n e^{x_{j}} = k \geq 0 \sum \frac{p _{k} ( x _{1} , \dots , x _{n} )}{k !}

is expressed in the power sums $p_{k} = \sum_{j} x_{j}^{k}$ , which the Newton-Girard identities convert to polynomials in the elementary symmetric functions: $p_{1} = e_{1}$ , $p_{2} = e_{1}^{2} - 2 e_{2}$ , $p_{3} = e_{1}^{3} - 3 e_{1} e_{2} + 3 e_{3}$ , $p_{4} = e_{1}^{4} - 4 e_{1}^{2} e_{2} + 4 e_{1} e_{3} + 2 e_{2}^{2} - 4 e_{4}$ . Dividing by the factorials $k!$ gives the Chern character components $ch_{k}$ as rational polynomials in the integer Chern classes $c_{i}$ .

The denominator factorials are central: $ch_{k}$ has denominator dividing $k!$ , and the existence of a class in $H^{2 k} (X; Q)$ rather than $H^{2 k} (X; Z)$ is exactly what makes the Chern character a rational comparison map rather than an integral one. The integer Chern classes recover from the Chern character by the inverse Newton-Girard transformation, but only after passing to rational coefficients. Theorem (Hirzebruch). The Chern classes are integer-valued cohomology classes whose rational images are computable from the rational Chern character by inverse Newton-Girard:

c_{1} = ch_{1}, c_{2} = \frac{ch _{1}^{2} - 2 ch _{2}}{2}, c_{3} = \frac{ch _{1}^{3} - 6 ch _{1} ch _{2} + 6 ch _{3}}{6}, \dots

The dual presentation makes precise that, rationally, the Chern character and the Chern classes carry the same information; integrally, the Chern classes carry strictly more (their denominators are pinned to $1$ ).

Multiplicativity via the splitting principle

The most important property of the Chern character is its multiplicativity on tensor products: $ch (E \otimes F) = ch (E) \cdot ch (F)$ . Combined with the additivity on direct sums, this makes $ch$ a ring homomorphism. The proof reduces to a calculation on line bundles via the splitting principle.

Lemma (line-bundle tensor formula). For complex line bundles $L, M$ on $X$ , $c_{1} (L \otimes M) = c_{1} (L) + c_{1} (M)$ in $H^{2} (X; Z)$ .

This is the basic fact that $\Pic (X) ≅ H^{1} (X; O^{*}) ≅ H^{2} (X; Z)$ is a group homomorphism from the multiplicative group of isomorphism classes of line bundles to the additive group of integer cohomology, via the first Chern class. It is the underlying reason the splitting argument works: line-bundle tensor products linearise to addition on Chern roots.

Theorem (multiplicativity). For complex vector bundles $E, F$ on $X$ , $ch (E \otimes F) = ch (E) \cdot ch (F)$ in $H^{even} (X; Q)$ .

The proof, given in the Full proof set, runs through the splitting principle applied jointly to $E$ and $F$ , the line-bundle tensor formula, and the exponential identity $e^{a + b} = e^{a} e^{b}$ . The Chern roots of $E \otimes F$ are the pairwise sums ${x_{i} + y_{j}}$ of the Chern roots of $E$ and $F$ , and the exponential converts pairwise sums into pairwise products.

Atiyah-Hirzebruch theorem: rational $K (X)$ is rational $H^{even} (X)$

Theorem (Atiyah-Hirzebruch 1959). For any finite CW complex $X$ , the Chern character induces a ring isomorphism

ch \otimes Q : K (X) \otimes_{Z} Q \sim H^{even} (X; Q) .

This was proved in the originator paper Riemann-Roch theorems for differentiable manifolds (Bull. Amer. Math. Soc. 65, 1959, 276-281) ^{[Atiyah-Hirzebruch]}. The proof runs through the Atiyah-Hirzebruch spectral sequence converging from $E_{2}^{p, q} = H^{p} (X; K^{q} (pt))$ to $K^{p + q} (X)$ . Rationally, the differentials of the AHSS are torsion: the lowest non-zero differential is $d_{3} = Sq^{3}$ acting on integer cohomology and is identically zero on $Q$ -coefficient classes. After collapsing at $E_{2} \otimes Q$ , the associated graded of the resulting filtration of $K (X) \otimes Q$ is identified with $⨁_{p} H^{p} (X; Q) \otimes K^{- p} (pt) \otimes Q$ , and $K^{q} (pt) = Z$ for even $q$ and $0$ for odd $q$ by Bott periodicity 03.08.07. The Chern character is the natural transformation that realises this identification on cycles. The rational isomorphism is then a corollary of the rational collapse and the bundle-level multiplicativity already established.

The proof has several refinements. Karoubi's variant in K-Theory: An Introduction (Springer 1978) ^[Karoubi] proceeds without the AHSS, using instead the multiplicativity of the Chern character on the product spaces $X \times S^{2 n}$ and Bott periodicity to identify the relative Chern character on each $K^{2 n} (X)$ . Hirzebruch's original computation in Topological Methods in Algebraic Geometry (Springer 1956) ^[Hirzebruch] gave the result for projective algebraic varieties using the Chern character to convert the Hirzebruch-Riemann-Roch formula into an integration-along-the-fibre statement. The three approaches agree.

Extension to higher $K^{*} (X)$ via the suspension isomorphism

The Chern character extends from $K^{0} (X) = K (X)$ to the full $Z /2$ -graded $K^{*} (X) = K^{0} (X) \oplus K^{1} (X)$ as follows. By Bott periodicity, $K^{1} (X) ≅ K^{0} (X \land S^{1}) ≅ K^{0} (Σ X)$ , where $Σ X$ is the unreduced suspension. The Chern character on $K^{0} (Σ X)$ takes values in $H^{even} (Σ X; Q)$ , which is naturally identified with $H^{odd} (X; Q)$ via the suspension isomorphism. The combined map is

ch : K^{*} (X) \otimes Q \sim H^{*} (X; Q),

a ring isomorphism between $Z /2$ -graded rational K-theory and $Z /2$ -graded rational ordinary cohomology. Theorem (full Atiyah-Hirzebruch). The extended Chern character $ch : K^{*} (-) \otimes Q \to H^{*} (-; Q)$ is a natural ring isomorphism between two $Z /2$ -graded cohomology theories on the category of finite CW complexes.

This is the structural form: K-theory and ordinary cohomology agree rationally as multiplicative cohomology theories, with the Chern character as the explicit comparison map. Integrally they disagree — K-theory has the Bott element $β \in K (S^{2})$ generating $K (S^{2}) = Z$ , while ordinary cohomology has $H^{2} (S^{2}; Z) = Z$ but no analogous higher-degree generators from a single suspension. The integer integrality of $ch$ fails by exactly the factorial denominators, which is the obstruction studied by Adams in On the groups $J (X)$ — IV (Topology 5, 1966).

Compatibility with Adams operations

The Adams operations $ψ^{k}$ 03.08.02 interact with the Chern character through a sharp eigenvalue formula. Theorem (Adams-Atiyah compatibility). For every $k \geq 1$ and every $n \geq 0$ ,

ch_{n} (ψ^{k} (x)) = k^{n} ch_{n} (x) for all x \in K (X) .

The proof is short: $ψ^{k}$ is uniquely determined by acting on line bundles as $ψ^{k} (L) = L^{\otimes k}$ , so $c_{1} (ψ^{k} (L)) = k c_{1} (L)$ and $ch (ψ^{k} (L)) = e^{k c_{1} (L)}$ . The degree- $2 n$ component is $(k c_{1} (L))^{n} / n! = k^{n} c_{1} (L)^{n} / n! = k^{n} ch_{n} (L)$ . Extend by additivity and multiplicativity via the splitting principle.

This eigenvalue formula is the cohomological shadow of $ψ^{k}$ acting as a Frobenius-like endomorphism on $K (X) \otimes Q$ . It identifies $ψ^{k}$ with a graded scaling on the rational Chern character image, and it is the source of the K-theoretic proof of the Hopf-invariant-one theorem [03.08.02 Master Theorem 3]. The eigenspace decomposition of $ψ^{k}$ on $K (X) \otimes Q$ recovers the cohomological grading via the Chern character: $V_{n} = ch^{- 1} (H^{2 n} (X; Q))$ is the $k^{n}$ -eigenspace, simultaneously for all $k \geq 2$ .

Role in Hirzebruch-Riemann-Roch and Atiyah-Singer

The most consequential application is in index theory. Theorem (Hirzebruch-Riemann-Roch). Let $X$ be a compact complex manifold of complex dimension $n$ and $E$ a holomorphic vector bundle on $X$ . The holomorphic Euler characteristic satisfies

χ (X, E) = \int_{X} ch (E) \cdot Td (T X),

where $Td (T X) = \prod_{j} \frac{x _{j}}{1 - e ^{- x_{j}}}$ is the Todd class of the holomorphic tangent bundle.

Hirzebruch proved this in 1953 ^[Hirzebruch] using the cobordism approach: both sides are additive on direct sums and multiplicative on tensor products, so it suffices to verify the formula on a generating set of compact complex manifolds with their canonical bundles, and the Todd class is precisely the cobordism multiplier that makes the formula universal. The Chern character is the universal coefficient that converts the K-theoretic bundle datum into a cohomological-degree datum integrable against the fundamental class.

Theorem (Atiyah-Singer index formula). Let $D : Γ (E) \to Γ (F)$ be an elliptic differential operator on a compact oriented manifold $X$ with symbol class $[σ (D)] \in K (T^{*} X)$ . The analytical index of $D$ equals

index (D) = \int_{T^{*} X} ch ([σ (D)]) \cdot Td (T^{*} X \otimes C) .

Atiyah-Singer 1963 (Bull. Amer. Math. Soc. 69, 422-433) ^{[Atiyah-Singer Bull]} announced this with a sketch; the full proof in The index of elliptic operators I (Annals 87, 1968, 484-530) ^{[Atiyah-Singer Annals]} uses an axiomatic K-theoretic argument plus the Chern-character identification. The formula generalises Hirzebruch-Riemann-Roch from the Dolbeault complex to every elliptic operator. The Chern character is the indispensable bridge: without the multiplicativity and the rational K-cohomology comparison, the integer-valued analytical index cannot be expressed as a cohomological integral.

Synthesis. The Chern character is exactly the natural ring map identifying rational K-theory with rational even cohomology. The central insight is that the splitting principle reduces every Chern-character identity to a symmetric-function statement about formal Chern roots, and the exponential converts line-bundle tensor products (sums on roots) into multiplicative cohomology classes (products on $e^{x_{j}}$ ) — this is exactly the structural reason ring-homomorphism property holds. Putting these together, the Atiyah-Hirzebruch rational isomorphism identifies $K (X) \otimes Q$ with $H^{even} (X; Q)$ as graded rings, generalises to a $Z /2$ -graded rational isomorphism between full K-theory and full ordinary cohomology, and the bridge to index theory is exactly the integration $\int_{X} ch (E) \cdot Td (T X)$ in Hirzebruch-Riemann-Roch and Atiyah-Singer. The foundational reason the Chern character is needed at all is that integer K-theory has torsion the Chern classes cannot see; the rationalisation is what makes the comparison clean, and the factorial denominators in $ch_{k}$ are exactly the price. The same pattern recurs in every rational cohomology theory comparison: the natural map from a generalised cohomology theory to ordinary rational cohomology is a ring isomorphism after rationalisation, identifying the generalised theory's universal Bott-like generator with the ordinary fundamental class. This is dual to the Adams-operation eigenspace decomposition: $ψ^{k}$ acts as $k^{n}$ on the Chern-character-preimage of $H^{2 n} (X; Q)$ , and the bridge between the K-theoretic Frobenius and the cohomological grading is the rational Chern character identification.

Full proof set [Master]

Proposition 1 (additivity). For complex vector bundles $E, F$ on a compact base $X$ , $ch (E \oplus F) = ch (E) + ch (F)$ in $H^{even} (X; Q)$ .

Proof. By the splitting principle 03.06.04, there is a flag-bundle pullback $f : Y \to X$ with $f^{*} : H^{*} (X) \to H^{*} (Y)$ injective such that $f^{*} E$ and $f^{*} F$ split as direct sums of line bundles: $f^{*} E = ⨁_{i} L_{i}$ with Chern roots $x_{i}$ , $f^{*} F = ⨁_{j} M_{j}$ with Chern roots $y_{j}$ . Then $f^{*} (E \oplus F) = ⨁_{i} L_{i} \oplus ⨁_{j} M_{j}$ , with Chern root multiset ${x_{i}} \cup {y_{j}}$ . By definition,

ch (f^{*} (E \oplus F)) = i \sum e^{x_{i}} + j \sum e^{y_{j}} = ch (f^{*} E) + ch (f^{*} F) .

Injectivity of $f^{*}$ transports the identity back to $X$ . $□$

Proposition 2 (multiplicativity). For complex vector bundles $E, F$ on $X$ , $ch (E \otimes F) = ch (E) \cdot ch (F)$ in $H^{even} (X; Q)$ .

Proof. With the same splitting cover, $f^{*} (E \otimes F) = (⨁_{i} L_{i}) \otimes (⨁_{j} M_{j}) = ⨁_{i, j} L_{i} \otimes M_{j}$ . By the line-bundle tensor formula $c_{1} (L_{i} \otimes M_{j}) = c_{1} (L_{i}) + c_{1} (M_{j}) = x_{i} + y_{j}$ , the Chern roots of $f^{*} (E \otimes F)$ are the pairwise sums ${x_{i} + y_{j}}$ . So

ch (f^{*} (E \otimes F)) = i, j \sum e^{x_{i} + y_{j}} = i, j \sum e^{x_{i}} e^{y_{j}} = (i \sum e^{x_{i}}) (j \sum e^{y_{j}}) = ch (f^{*} E) \cdot ch (f^{*} F) .

The second equality uses the exponential identity $e^{a + b} = e^{a} e^{b}$ in the cohomology ring, valid because $a, b$ are nilpotent (every cohomology class on a finite-dimensional base is nilpotent above some degree). Injectivity of $f^{*}$ on cohomology gives the identity on $X$ . $□$

Proposition 3 (Newton-Girard explicit formulas). For a complex vector bundle $E$ of rank $n$ , the components of the Chern character are

ch_{k} (E) = \frac{p _{k} ( c _{1} , \dots , c _{k} )}{k !},

where $p_{k}$ is the $k$ -th power-sum polynomial expressed in elementary symmetric functions via the Newton-Girard recursion $p_{k} = e_{1} p_{k - 1} - e_{2} p_{k - 2} + \dots + (- 1)^{k - 1} k e_{k}$ , and $c_{i} = e_{i} (x_{1}, \dots, x_{n})$ are the Chern classes.

Proof. By definition, $ch_{k} (E) = \sum_{j} x_{j}^{k} / k! = p_{k} (x_{1}, \dots, x_{n}) / k!$ . Newton-Girard expresses $p_{k}$ as a polynomial in the elementary symmetric functions $e_{1}, \dots, e_{k}$ with integer coefficients; the recursion is the classical Newton identity proved by computing $\prod_{i} (t - x_{i})$ and taking logarithmic derivatives. Substituting $e_{i} = c_{i} (E)$ gives the formula. $□$

Proposition 4 (ring-homomorphism extension to $K (X)$ ). The bundle-level Chern character extends uniquely to a ring homomorphism $ch : K (X) \to H^{even} (X; Q)$ .

Proof. The Grothendieck-completion universal property says any additive function on the abelian monoid of isomorphism classes of bundles under direct sum extends uniquely to an abelian-group homomorphism from $K (X)$ to any abelian group. By Proposition 1, $ch$ is additive on direct sums; by the universal property, $ch$ extends uniquely to $ch : K (X) \to H^{even} (X; Q)$ . The multiplicativity statement (Proposition 2) on the bundle level lifts to multiplicativity on the extension by the bilinearity of the tensor product over direct sums:

([E] - [F]) \otimes ([E^{'}] - [F^{'}]) = [E \otimes E^{'}] - [F \otimes E^{'}] - [E \otimes F^{'}] + [F \otimes F^{'}]

in $K (X)$ , and applying $ch$ term by term gives $(ch (E) - ch (F)) \cdot (ch (E^{'}) - ch (F^{'})) = ch (([E] - [F]) \cdot ([E^{'}] - [F^{'}]))$ . The unit relation $ch (1) = 1$ holds because $1 \in K (X)$ is the class of the rank-one product bundle, with first Chern class $0$ , so $ch = e^{0} = 1$ . $□$

Proposition 5 (Adams-character compatibility). For every $k \geq 1$ , every $n \geq 0$ , and every $x \in K (X)$ ,

ch_{n} (ψ^{k} (x)) = k^{n} ch_{n} (x) in H^{2 n} (X; Q) .

Proof. Both $ψ^{k}$ and $ch$ are natural ring homomorphisms on $K (-)$ , determined by their action on line bundles via the splitting principle. On a line bundle $L$ , $ψ^{k} (L) = L^{\otimes k}$ has first Chern class $k c_{1} (L)$ , so

ch (ψ^{k} (L)) = e^{k c_{1} (L)} = n \sum \frac{( k c _{1} ( L ) ) ^{n}}{n !} = n \sum k^{n} \frac{c _{1} ( L ) ^{n}}{n !} = n \sum k^{n} ch_{n} (L) .

Hence $ch_{n} (ψ^{k} (L)) = k^{n} ch_{n} (L)$ . Both sides are ring homomorphisms on $K (X)$ , agree on the line-bundle generators of $f^{*} E$ for any splitting cover, and the injectivity of $f^{*}$ transports the identity to $X$ . $□$

Proposition 6 (Atiyah-Hirzebruch rational isomorphism). For any finite CW complex $X$ , the rationalised Chern character

ch \otimes Q : K (X) \otimes Q \sim H^{even} (X; Q)

is an isomorphism of $Q$ -algebras.

Proof sketch. The proof in the originator paper of Atiyah-Hirzebruch (1959) goes via the Atiyah-Hirzebruch spectral sequence. The AHSS has $E_{2}$ page $H^{p} (X; K^{q} (pt))$ converging to $K^{p + q} (X)$ . By Bott periodicity 03.08.07, $K^{q} (pt) = Z$ for even $q$ and $0$ for odd $q$ . The lowest non-vanishing differential is $d_{3}$ , identified with the Steenrod operation $Sq^{3}$ on integer cohomology by a theorem of Atiyah. Steenrod operations are torsion-valued — they act as zero on $Q$ -coefficient classes — so $d_{3} \otimes Q = 0$ . Similarly, all higher differentials are torsion. Therefore $E_{2}^{p, q} \otimes Q = E_{\infty}^{p, q} \otimes Q$ , and the rational filtration on $K (X) \otimes Q$ has associated graded $⨁_{p} H^{p} (X; Q) \otimes K^{- p} (pt) \otimes Q$ . The Chern character realises this identification on cycles by sending the K-theory class of a bundle to its Chern-character polynomial in cohomology. The map is multiplicative by Proposition 2 and injective on $Q$ -rationalisation by the AHSS-collapse argument. Surjectivity follows from the fact that every cohomology class is rationally in the image (the Atiyah-Hirzebruch filtration has surjective associated-graded map onto $H^{*} (X; Q)$ ). The map is then a $Q$ -algebra isomorphism. The full argument is in Atiyah, K-Theory, Ch. III, with a streamlined treatment in Karoubi, K-Theory: An Introduction, Ch. V. $□$

Proposition 7 (HRR via Chern character). Let $X$ be a compact complex manifold of complex dimension $n$ and $E$ a holomorphic vector bundle on $X$ . Then $χ (X, E) = \int_{X} ch (E) \cdot Td (T X)$ .

Proof sketch. Both sides are additive on direct sums and multiplicative on tensor products with respect to flat-twist operations. By the cobordism argument due to Hirzebruch, both sides agree on a generating set of compact complex manifolds with their canonical line bundles — explicitly the projective spaces $CP^{n}$ with $O (k)$ — and the universal Todd class is precisely the multiplier that makes the identity hold across the generating set. The Chern character is the natural ring homomorphism that promotes the K-theoretic datum $[E] \in K (X)$ to a cohomology class $ch (E) \in H^{even} (X; Q)$ , multiplicatively compatible with the Todd genus. The full proof is in Hirzebruch's Topological Methods in Algebraic Geometry (1956), with subsequent K-theoretic proofs in Atiyah, K-Theory, Ch. III, and Borel-Serre, Le théorème de Riemann-Roch (Bull. Soc. Math. France 86, 1958, 97-136). $□$

Connections [Master]

Pontryagin and Chern classes 03.06.04. The Chern character is the universal exponential combinator of Chern classes. Built from formal exponentials of Chern roots, it converts the additive structure of cohomology into multiplicative bundle data: $ch (E \oplus F) = ch (E) + ch (F)$ contrasts with $c (E \oplus F) = c (E) c (F)$ for the total Chern class. The Chern roots are the line-bundle quotients of the splitting principle, and the Newton-Girard identities convert the power-sum polynomials $\sum_{j} x_{j}^{k}$ into polynomials in the elementary symmetric functions, recovering the explicit formulas $ch_{k}$ as rational polynomials in $c_{1}, \dots, c_{k}$ .
Topological K-theory 03.08.01. The Chern character is the natural transformation between the two cohomology theories $K (-)$ and $H^{even} (-; Q)$ that identifies them rationally. K-theory of a finite CW complex tensored with $Q$ becomes ordinary rational even cohomology, with $ch$ as the explicit comparison map. The integer-coefficient versions disagree by torsion phenomena, controlled by the Atiyah-Hirzebruch spectral sequence differentials $d_{3} = Sq^{3}$ and higher Steenrod operations.
Adams operations $ψ^{k}$ 03.08.02. The eigenvalue formula $ch_{n} (ψ^{k} (x)) = k^{n} ch_{n} (x)$ identifies the Chern-character preimage $V_{n} = ch^{- 1} (H^{2 n} (X; Q))$ as the simultaneous $k^{n}$ -eigenspace of all Adams operations on $K (X) \otimes Q$ . The Adams operations act on K-theory as the cohomological Frobenius scaling $Ψ^{k} = k^{n}$ on degree $2 n$ , and the Chern character is the universal coefficient that converts the K-theoretic eigenvalue datum into the cohomological degree datum.
Complex vector bundle 03.05.08. The Chern character is defined on complex vector bundles via the splitting principle, requiring complex structure on the bundle (and on its Chern roots, which live in integral cohomology) — for real vector bundles the analogous construction yields the Pontryagin character, valued in even cohomology, but built from $\sum_{j} cosh (y_{j})$ over Pontryagin roots rather than $\sum_{j} e^{x_{j}}$ over Chern roots, an honest cosine series in degree- $2$ multiples to respect the $H^{4 k}$ -grading of Pontryagin classes. The unit on complex vector bundles supplies the natural source category for the Chern character.
Bott periodicity 03.08.07. Under the Bott periodicity isomorphism $K (X) ≅ K (X \land S^{2})$ , the Chern character intertwines the K-theoretic Bott element $β \in K (S^{2})$ with the cohomological fundamental class $ω \in H^{2} (S^{2}; Z)$ . Specifically, $ch (β) = ω$ in $H^{*} (S^{2}; Q)$ , and the multiplicative behaviour $ch (β^{n}) = ω^{n} / n!$ on $K (S^{2 n})$ records the factorial denominators that arise from the exponential series.
Atiyah-Singer index theorem 03.09.10. The Chern character appears centrally in the index formula: $index (D) = \int_{T^{*} X} ch ([σ (D)]) \cdot Td (T^{*} X \otimes C)$ for an elliptic operator $D$ on a compact manifold $X$ . The K-theory class $[σ (D)]$ of the symbol lives in $K (T^{*} X)$ , and the Chern character is the indispensable bridge that converts this K-theoretic datum into a cohomology class integrable against the fundamental class of the cotangent space.
Atiyah-Hirzebruch spectral sequence 03.13.04. The AHSS converges from $H^{p} (X; K^{q} (pt))$ to $K^{p + q} (X)$ , and rationally collapses at $E_{2}$ because all differentials are torsion-valued (notably $d_{3} = Sq^{3}$ ). The Chern character is the natural identification of the rational $E_{\infty}$ page with $H^{even} (X; Q)$ . The integral AHSS, by contrast, captures the obstructions that prevent $ch$ from being an integral isomorphism; these obstructions vanish only after rationalisation.
Hirzebruch-Riemann-Roch and Grothendieck-Riemann-Roch [04.05.10, 04.05.12]. HRR expresses the holomorphic Euler characteristic of a holomorphic vector bundle on a compact complex manifold as $χ (X, E) = \int_{X} ch (E) \cdot Td (T X)$ . Grothendieck's relative version replaces the integral by a pushforward in K-theory, with the Chern character converting K-pushforward into cohomological pushforward modulo a Todd-class twist. The Chern character is the universal coefficient appearing on the right-hand side of every Riemann-Roch theorem of this lineage, and its multiplicativity is essential for the cobordism argument that establishes each.
Multiplicative sequences and $L$ / $A$ /Todd genera 03.06.15. The Todd class $Td (T X)$ is the multiplicative sequence associated to the power series $f (z) = z / (1 - e^{- z})$ , and lands in the Chern-character context exactly because the Hirzebruch-Riemann-Roch integrand $ch (E) \cdot Td (T X)$ pairs the Chern character of the present unit with the Todd genus of 03.06.15. The two units form the canonical input pair to every HRR-style theorem, with Chern character supplying the exponential bundle datum and Todd supplying the universal coefficient.
Borel-Hirzebruch and the cohomology of $G / T$ 03.06.20. The Chern character is defined via the splitting principle, which is exactly the Borel-Hirzebruch restriction-to-maximal-torus argument: pulling back along the flag bundle $G / T \to B G$ formally splits the bundle into a sum of line bundles whose first Chern classes are the Chern roots $x_{j}$ , and $ch (E) = \sum_{j} e^{x_{j}}$ then becomes a symmetric power series in the $x_{j}$ that descends to $H^{*} (B G; Q)^{W}$ . The splitting-principle / flag-bundle pullback of 03.06.20 is the structural foundation for the well-definedness of the Chern character.

Historical & philosophical context [Master]

The Chern character was introduced by Atiyah and Hirzebruch in 1959 in Riemann-Roch theorems for differentiable manifolds (Bull. Amer. Math. Soc. 65, 276-281) ^{[Atiyah-Hirzebruch]}, the same paper that introduced topological K-theory as a tool for generalising Hirzebruch's algebraic Riemann-Roch theorem. The original motivation was to express the analytical index of an elliptic operator on a differentiable manifold as a topological integral; the Chern character was the natural multiplicative coefficient that made the K-theory-to-cohomology comparison work over the rationals. The follow-up paper, Vector bundles and homogeneous spaces (Proc. Symp. Pure Math. 3, 1961, 7-38) ^{[Atiyah-Hirzebruch]}, gave the full Atiyah-Hirzebruch spectral sequence and the rational isomorphism theorem in its modern form.

The bundle-level definition through Chern-Weil curvature representatives was given by Atiyah in his 1965 lectures at Harvard, published as K-Theory (Benjamin 1967, reissued Addison-Wesley 1989) ^{[Atiyah K-Theory]}; the explicit formulas through low degrees first appeared in Hirzebruch's earlier monograph Neue topologische Methoden in der algebraischen Geometrie (Springer 1956; English translation Topological Methods in Algebraic Geometry, Springer 1966) ^[Hirzebruch], where they were used to convert the Hirzebruch-Riemann-Roch formula into an integration-along-the-fibre statement.

The Atiyah-Hirzebruch spectral sequence machinery was developed jointly by the authors throughout 1959-1962, with the rational-collapse argument relying on Atiyah's 1962 identification of the lowest differential $d_{3}$ with the Steenrod operation $Sq^{3}$ on integer cohomology. The full theory is consolidated in Atiyah's K-Theory and in Karoubi's 1978 textbook K-Theory: An Introduction (Springer Grundlehren 226) ^{[Karoubi K-Theory]}, where the Chern character receives its canonical pedagogical treatment.

The Chern character entered index theory directly with Atiyah-Singer's 1963 Bull. Amer. Math. Soc. 69 announcement ^{[Atiyah-Singer Bull]} and the full 1968 Ann. Math. 87 proof of the index theorem ^{[Atiyah-Singer Annals]}. The index formula's right-hand side is $\int_{T^{*} X} ch ([σ (D)]) \cdot Td (T^{*} X \otimes C)$ , and the Chern character is the unique universal coefficient that makes this integral over the cotangent space equal the integer-valued analytical index. The compatibility with Adams operations was established by Atiyah in 1966 Power operations in K-theory (Quart. J. Math. 17, 165-193) ^{[Atiyah Power operations]}, showing that $ch_{n} (ψ^{k} (x)) = k^{n} ch_{n} (x)$ identifies the K-theoretic Frobenius with the cohomological grading scaling. The relative pushforward version was the Grothendieck-Riemann-Roch theorem, due to Grothendieck and first published in Borel-Serre's 1958 Bull. Soc. Math. France 86 paper ^{[Borel-Serre]}; Grothendieck's original argument used Chern characters on Grothendieck groups of coherent sheaves, predating Atiyah-Hirzebruch's topological-K-theory formulation by a year.

Bibliography [Master]

@article{AtiyahHirzebruch1959,
  author  = {Atiyah, M. F. and Hirzebruch, F.},
  title   = {Riemann-{R}och theorems for differentiable manifolds},
  journal = {Bulletin of the American Mathematical Society},
  volume  = {65},
  year    = {1959},
  pages   = {276--281}
}

@inproceedings{AtiyahHirzebruch1961,
  author    = {Atiyah, M. F. and Hirzebruch, F.},
  title     = {Vector bundles and homogeneous spaces},
  booktitle = {Proceedings of Symposia in Pure Mathematics},
  volume    = {3},
  publisher = {American Mathematical Society},
  year      = {1961},
  pages     = {7--38}
}

@book{AtiyahKTheory,
  author    = {Atiyah, M. F.},
  title     = {{K}-Theory},
  publisher = {Benjamin},
  address   = {New York},
  year      = {1967},
  note      = {Reissued Addison-Wesley, 1989}
}

@article{AtiyahSinger1963,
  author  = {Atiyah, M. F. and Singer, I. M.},
  title   = {The index of elliptic operators on compact manifolds},
  journal = {Bulletin of the American Mathematical Society},
  volume  = {69},
  year    = {1963},
  pages   = {422--433}
}

@article{AtiyahSinger1968,
  author  = {Atiyah, M. F. and Singer, I. M.},
  title   = {The index of elliptic operators {I}},
  journal = {Annals of Mathematics. Second Series},
  volume  = {87},
  year    = {1968},
  pages   = {484--530}
}

@article{Atiyah1966PowerOperations,
  author  = {Atiyah, M. F.},
  title   = {Power operations in {K}-theory},
  journal = {Quarterly Journal of Mathematics, Oxford. Second Series},
  volume  = {17},
  year    = {1966},
  pages   = {165--193}
}

@book{Hirzebruch1956,
  author    = {Hirzebruch, F.},
  title     = {Topological Methods in Algebraic Geometry},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {131},
  publisher = {Springer-Verlag},
  year      = {1966},
  note      = {Translation of the 1956 German original}
}

@article{BorelSerre1958,
  author  = {Borel, A. and Serre, J.-P.},
  title   = {Le th{\'e}or{\`e}me de {R}iemann-{R}och},
  journal = {Bulletin de la Soci{\'e}t{\'e} Math{\'e}matique de France},
  volume  = {86},
  year    = {1958},
  pages   = {97--136}
}

@book{Karoubi1978,
  author    = {Karoubi, Max},
  title     = {{K}-Theory: {A}n Introduction},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {226},
  publisher = {Springer-Verlag},
  year      = {1978}
}

@book{BottTu1982,
  author    = {Bott, R. and Tu, L. W.},
  title     = {Differential Forms in Algebraic Topology},
  series    = {Graduate Texts in Mathematics},
  volume    = {82},
  publisher = {Springer-Verlag},
  year      = {1982}
}

@book{MilnorStasheff1974,
  author    = {Milnor, J. and Stasheff, J.},
  title     = {Characteristic Classes},
  series    = {Annals of Mathematics Studies},
  volume    = {76},
  publisher = {Princeton University Press},
  year      = {1974}
}

Prerequisites

03.06.04
03.08.01
03.08.02
03.05.08

Tier anchors

beginner: Bott-Tu §21 (informal exposition); 3Blue1Brown / Strogatz analogous level for exponential bookkeeping of bundle twisting
intermediate: Bott-Tu *Differential Forms in Algebraic Topology* §21; Atiyah *K-Theory* §2.4
master: Atiyah-Hirzebruch *Riemann-Roch theorems for differentiable manifolds* (Bull. AMS 65, 1959, 276-281); Atiyah *K-Theory* §2-§3; Milnor-Stasheff *Characteristic Classes* §16; Bott-Tu §21

References

TODO_REF
Atiyah-Hirzebruch — Riemann-Roch theorems for differentiable manifolds · Bull. Amer. Math. Soc. 65 (1959) 276-281 — originator paper introducing the Chern character as the rational K-to-cohomology isomorphism
TODO_REF
Atiyah — K-Theory · §2.4 Chern character; §3 Adams operations and the comparison with cohomology
TODO_REF
Milnor-Stasheff — Characteristic Classes · §16, Chern character and Pontryagin character
TODO_REF
Bott-Tu — Differential Forms in Algebraic Topology · §21, Chern character via Chern-Weil and the splitting principle
TODO_REF
Hirzebruch — Topological Methods in Algebraic Geometry · Ch. III, Chern character in the Riemann-Roch theorem
TODO_REF
Karoubi — K-Theory: An Introduction · Springer Grundlehren 226 (1978), Ch. V — Chern character and Adams operations
TODO_REF
Atiyah-Singer — The index of elliptic operators on compact manifolds · Bull. Amer. Math. Soc. 69 (1963) 422-433 — Chern character in the index formula
TODO_REF
Atiyah-Singer — The index of elliptic operators I · Annals of Mathematics (2) 87 (1968) 484-530 — full proof of the index theorem

Lean module

Codex.Modern.CharClasses.ChernCharacter

Mathlib gap

Mathlib does not yet provide bundled topological K-theory of compact
Hausdorff spaces, the Chern character as a natural transformation
from K-theory to rational even cohomology, or the Atiyah-Hirzebruch
rational isomorphism. The required upstream pieces include: complex
topological vector bundles as a packaged category, Grothendieck
completion of the direct-sum monoid to the K-theory ring K(X),
rational singular cohomology with the cup-product ring structure,
Chern classes as natural cohomology operations, the exponential
series in H^{even}(X; Q), the splitting principle for vector
bundles over a CW base, the resulting multiplicativity statement
ch(E (x) F) = ch(E) ch(F), and the Atiyah-Hirzebruch spectral
sequence converging from H^p(X; K^q(pt)) to K^{p+q}(X) with the
rational collapse argument identifying the abutment with even
cohomology. The Lean stub records the four anchor theorems
(additivity, multiplicativity, ring-homomorphism property,
Atiyah-Hirzebruch isomorphism) with placeholder bundle and ring
carriers and sorry-proofs; promotion to lean_status: full requires
the bundle-level characteristic-class library plus the rational
cohomology and AHSS work.

Reviewer

TBD

Estimated time

beginner: 20m
intermediate: 50m
master: 90m