03.08.03 · modern-geometry / k-theory

Thom isomorphism in K-theory

shipped3 tiersLean: none

Anchor (Master): Atiyah 1968 *Bott periodicity and the index of elliptic operators* (Quart. J. Math. 19); Atiyah *K-Theory* §2.7; Lawson-Michelsohn §I.5 and §III.6; Karoubi *K-Theory* §IV.5

Intuition [Beginner]

In ordinary cohomology the Thom isomorphism says that the cohomology of a vector bundle, with a suitable support condition, is the cohomology of the base, shifted up by the rank of the bundle. The shift is realised by multiplication with a single distinguished class on the total space — the Thom class — which integrates to one on each fibre. K-theory tells the same story in a different alphabet. The shift is still there, the distinguished class is still there, but now everything is built out of vector bundles.

The K-theory Thom class for a complex bundle $E$ of rank $n$ is an alternating combination of the exterior powers of $E$ . Concretely you take $1$ (the rank-zero exterior power) minus $E$ (the rank-one) plus $Λ^{2} E$ minus $Λ^{3} E$ , and so on. The combination is a virtual bundle: real bundles can only be added, but K-theory allows formal differences, and the Thom class is exactly such a formal difference. The resulting class lives on the Thom space of $E$ and plays the same multiplicative role in K-theory that the cohomological Thom class plays in ordinary cohomology.

The reason this matters: the K-theoretic Thom isomorphism is what realises the topological-index map. The Atiyah-Singer index theorem reads the index of an elliptic operator as a topological invariant of its symbol, and the bridge from symbol to invariant runs through the K-theoretic Thom isomorphism applied to the normal bundle of an embedding into Euclidean space.

Visual [Beginner]

A schematic showing a base space $X$ on the left, the disk bundle $D (E)$ above it, and the Thom space $T (E)$ on the right with the sphere bundle $S (E)$ collapsed to a point. Above the Thom space, a virtual bundle is drawn as a stack of layers labelled $1, - E, Λ^{2} E, - Λ^{3} E, \dots$ alternating in sign. An arrow from the K-theory $K (X)$ of the base to the reduced K-theory $\tilde{K} (T (E))$ of the Thom space carries the label "multiplication by $λ_{E}$ ".

The picture records the central content: the Thom isomorphism in K-theory shifts the K-group of the base into the reduced K-group of the Thom space, using the alternating exterior-power class on $E$ in place of the cohomological orientation class.

Worked example [Beginner]

Take the simplest non-empty case: $X$ a point, $E = C$ the rank-one complex line over a point. The disk bundle is the closed unit disk $D^{2}$ , the sphere bundle is the circle $S^{1}$ , and the Thom space $T (E) = D^{2} / S^{1} = S^{2}$ .

K-theory of a point is the integers: $K (pt) = Z$ , generated by the class of $C$ itself. The reduced K-theory of $S^{2}$ is also the integers — this is the rank-zero piece of Bott periodicity — and the generator is the Bott element $β = [H] - 1$ , where $H$ is the tautological line bundle on $CP^{1} = S^{2}$ .

The K-theoretic Thom class for $E = C$ over a point is $λ_{E} = 1 - E$ . Pulled up to the disk bundle and reading down to the Thom space, this exactly hits the Bott generator $β$ . So the Thom isomorphism in this case is the assignment $1 \mapsto β$ , sending the generator of $K (pt) = Z$ to the generator of $\tilde{K} (S^{2}) = Z$ . Both groups are $Z$ ; the map is multiplication by $β$ .

What this tells us: the rank-one case of the K-theoretic Thom isomorphism is exactly Bott periodicity in degree two. Higher rank cases iterate this and produce higher-degree shifts.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $X$ be a compact Hausdorff space and let $π : E \to X$ be a complex vector bundle of rank $n$ . Fix Hermitian metrics on the fibres and let $D (E) = {v \in E : ∣ v ∣ \leq 1}$ and $S (E) = {v \in E : ∣ v ∣ = 1}$ denote the closed unit disk and unit sphere bundles. The Thom space of $E$ is the pointed space $$ T(E) = D(E) / S(E), $$ with basepoint the image of $S (E)$ . When $X$ is compact, $T (E)$ is the one-point compactification of the total space $E$ , and the K-theory of pairs gives $$ \tilde K(T(E)) \cong K(D(E), S(E)). $$

Definition (K-theoretic Thom class). Let $E \to X$ be a complex vector bundle of rank $n$ . The K-theoretic Thom class $λ_{E} \in K (D (E), S (E))$ is the relative class represented by the $Z /2$ -graded virtual bundle on $D (E)$ $$ \lambda_E = \sum_{k=0}^{n} (-1)^k \Lambda^k E^, $$ where $\Lambda^k E^ $d e n o t es t h e$ k $- t h e x t er i or p o w er o f t h e d u a l b u n d l e$ E^* = \mathrm{Hom}(E, \mathbb{C}) $, p u l l e d ba c k a l o n g$ \pi \colon E \to X $. T h ec l a ss i sr e l a t i v e b ec a u se t h e a l t er na t in g s u m e x t e n d s t o a K osz u l - t y p e a cy c l i cco m pl e x o n$ E \setminus 0_E $(t h eco m pl e m e n t o f t h ez er osec t i o n), an d t h e E u l er c ha r a c t er i s t i co f ana cy c l i cco m pl e x o f b u n d l es v ani s h es in K - t h eor y . T h ec l a ss t h er e f or e l i es in t h e k er n e l o f t h er es t r i c t i o nma p$ K(E) \to K(E \setminus 0_E) $, w hi c hb y e x c i s i o ni d e n t i f i es w i t h$ K(D(E), S(E))$.

The Koszul construction: a non-zero vector $v \in E_{x}$ defines a wedge-with- $v$ map $Λ^{k} E_{x}^{*} \to Λ^{k + 1} E_{x}^{*}$ , dually a contraction-with- $v$ map $Λ^{k} E_{x} \to Λ^{k - 1} E_{x}$ . The resulting complex $(Λ^{∙} E^{*}, ι_{v})$ is acyclic off the zero section. Its Euler characteristic, viewed as a virtual bundle with $Z /2$ -grading by parity, is $λ_{E}$ .

Definition (Thom homomorphism). The Thom homomorphism is the K-theory pushforward $$ \phi_E \colon K(X) \to \tilde K(T(E)), \qquad x \mapsto \pi^(x) \cdot \lambda_E, $$ where $\pi^ \colon K(X) \to K(D(E))$ is the pullback along the bundle projection and the product is the relative K-theory product $$ K(D(E)) \otimes K(D(E), S(E)) \to K(D(E), S(E)) $$ sending $a \otimes b$ to the relative class $ab$ .

Counterexamples to common slips

Real vector bundles do not carry a Thom class in complex K-theory in general — the construction above genuinely requires a complex structure on $E$ to make the exterior powers $Λ^{k} E^{*}$ into complex bundles. The real-bundle analogue lives in $K O$ -theory and uses Clifford modules in place of exterior algebra (Atiyah-Bott-Shapiro 1964).
The K-theoretic Thom class is not the K-theoretic Euler class $e_{K} (E) = \sum (- 1)^{k} Λ^{k} E^{*} \in K (X)$ . The Euler class lives on the base; the Thom class lives on the Thom space (equivalently, in the relative K-group of the disk-sphere pair). The restriction of $λ_{E}$ to the zero section recovers $e_{K} (E)$ on $X$ .
Multiplicativity of the Thom class under direct sums is $λ_{E \oplus F} = λ_{E} \cdot λ_{F}$ , not $λ_{E} + λ_{F}$ . This is the K-theoretic analogue of Whitney sum, and it is what allows induction over rank to reduce the Thom isomorphism to the rank-one case.
The Chern character does not preserve the Thom class on the nose: $ch (λ_{E}) = e (E) \cdot Td (E)^{- 1}$ , where $e$ is the cohomological Euler class and $Td$ is the Todd class. The Todd correction is the source of the $\hat{A}$ -genus in the index theorem.

Key theorem with proof [Intermediate+]

Theorem (Thom isomorphism in K-theory; Atiyah 1968). Let $X$ be a compact Hausdorff space and $π : E \to X$ a complex vector bundle of rank $n$ . The Thom homomorphism $$ \phi_E \colon K(X) \to \tilde K(T(E)), \qquad x \mapsto \pi^*(x) \cdot \lambda_E, $$ is an isomorphism of $K (X)$ -modules.

Proof. The argument proceeds by induction on the rank $n$ , with the rank-one case provided by Bott periodicity and the inductive step by the multiplicativity of $λ$ under direct sums.

Base case ( $n = 1$ ). A rank-one complex bundle is a complex line bundle. The disk bundle is a $D^{2}$ -bundle, the sphere bundle is an $S^{1}$ -bundle, and the Thom space is the projective bundle $P (E \oplus \underline{C})$ with the section at infinity collapsed. For $X$ a point and $E = C$ , the Thom space is $S^{2} = CP^{1}$ , and the Thom class $λ_{E} = 1 - E^{*} = 1 - H^{- 1}$ pulled back from $CP^{1}$ via the natural quotient generates $\tilde{K} (S^{2}) = Z$ . This is the content of Bott periodicity: $\tilde{K} (S^{2}) ≅ Z$ with generator the Bott element $β = [H] - 1$ .

For a general line bundle $L$ over a base $X$ , the Thom isomorphism reduces to Bott periodicity by the splitting argument of Atiyah-Hirzebruch: one writes $T (L) = X^{+} \land S^{2}$ up to homotopy when $L$ is the product bundle, and a clutching-function calculation handles the twisted case via the same Fourier-Laurent series argument that proves Bott periodicity for $CP^{1}$ . In all cases the rank-one Thom homomorphism is multiplication by the Bott element $β_{L} = [L^{*}] - 1$ (or equivalently $1 - [L]$ , up to a sign convention).

Inductive step. Suppose the Thom isomorphism holds for all complex bundles of rank less than $n$ . Let $E = F \oplus L$ where $F$ has rank $n - 1$ and $L$ has rank $1$ . The Thom space is the smash product $$ T(F \oplus L) = T(F) \wedge_X T(L), $$ where the smash is taken fibrewise over $X$ (equivalently, the Thom space of the direct sum is the smash of the Thom spaces relative to $X$ , since $D (F \oplus L) = D (F) \times_{X} D (L)$ and $S (F \oplus L) = (S (F) \times_{X} D (L)) \cup (D (F) \times_{X} S (L))$ ). The Thom class is multiplicative: $$ \lambda_{F \oplus L} = \lambda_F \cdot \lambda_L, $$ because the Koszul complex of $F \oplus L$ is the tensor product of the Koszul complexes of $F$ and $L$ , and Euler characteristics multiply for tensor products of complexes.

The Thom homomorphism factors as $$ \phi_{F \oplus L} = \phi_L \circ \phi_F \colon K(X) \xrightarrow{\phi_F} \tilde K(T(F)) \xrightarrow{\phi_L} \tilde K(T(F) \wedge_X T(L)) = \tilde K(T(F \oplus L)), $$ where the second map is the relative Thom homomorphism for $L$ pulled back to $T (F)$ . Both factors are isomorphisms by the inductive hypothesis (the first by rank $n - 1$ , the second by rank $1$ ). Their composite is an isomorphism, which is the Thom isomorphism for $E = F \oplus L$ .

Splitting principle. An arbitrary complex bundle $E$ of rank $n$ need not split as a sum of line bundles, but the splitting principle of Atiyah K-theory says that there is a base change $f : Y \to X$ for which $f^{*}$ is injective on $K (X)$ and $f^{*} E$ splits. The Thom isomorphism for $f^{*} E$ is then a consequence of the inductive step, and the injectivity of $f^{*}$ promotes the isomorphism back to $X$ . $□$

Bridge. The K-theoretic Thom isomorphism builds toward the Atiyah-Singer index theorem and appears again in 03.09.10 (Atiyah-Singer index theorem) as the technical engine of the topological-index map. The foundational reason it holds is exactly Bott periodicity: rank-one Thom is rank-two Bott periodicity, and direct-sum multiplicativity of $λ$ promotes this to all ranks by induction. This is exactly the same algebraic mechanism that organises the cohomological Thom isomorphism in 03.04.09 (de Rham Thom), where the Thom class is built from a global angular form whose fibre integral is one and the isomorphism is multiplication by that class. The central insight is that an orientation class in a generalised cohomology theory always behaves the same way: it lives on the Thom space, multiplies the base theory into the relative theory of the disk-sphere pair, and its existence is the content of orientability for that theory. Putting these together, the Chern character $ch : K (X) \otimes Q \to H^{even} (X; Q)$ identifies the K-theoretic Thom class with the cohomological one up to the Todd correction: $ch (λ_{E}) = e (E) \cdot Td (E)^{- 1}$ . The bridge is the recognition that K-theory and ordinary cohomology see the same orientation, but K-theory packages it as a virtual bundle while cohomology packages it as a closed form, and the Chern character is the rational dictionary between the two — the same pattern of orientation-class transfer recurs whenever one moves between generalised cohomology theories.

Exercises [Intermediate+]

Exercise 3 (medium, symbolic).

Compute the Chern character of the K-theoretic Thom class $λ_{E}$ for a complex line bundle $L \to X$ . Verify that $ch (λ_{L}) = e (L) \cdot Td (L)^{- 1}$ in $H^{*} (X; Q)$ .

Hint

For a line bundle $L$ , $λ_{L} = 1 - L^{*}$ . The Chern character of $L$ is $ch (L) = e^{c_{1} (L)}$ , and the Todd class of a line bundle is $Td (L) = c_{1} (L) / (1 - e^{- c_{1} (L)})$ .

Answer

Write $x = c_{1} (L) \in H^{2} (X; Q)$ . Then $ch (L) = e^{x}$ and $ch (L^{*}) = e^{- x}$ . So $$ \mathrm{ch}(\lambda_L) = \mathrm{ch}(1 - L^*) = 1 - e^{-x}. $$ The cohomological Euler class of the line bundle $L$ is $e (L) = c_{1} (L) = x$ , and the Todd class is $Td (L) = x / (1 - e^{- x})$ , so $Td (L)^{- 1} = (1 - e^{- x}) / x$ . Therefore $$ e(L) \cdot \mathrm{Td}(L)^{-1} = x \cdot \frac{1 - e^{-x}}{x} = 1 - e^{-x} = \mathrm{ch}(\lambda_L). $$ Rubric: full credit for the explicit computation and the cancellation step.

Exercise 4 (medium, short-answer).

Use the multiplicativity $λ_{E \oplus F} = λ_{E} \cdot λ_{F}$ to compute $λ_{E}$ when $E$ is a sum of two line bundles $E = L_{1} \oplus L_{2}$ , and verify that the result agrees with the direct exterior-power formula $\sum_{k} (- 1)^{k} Λ^{k} E^{*}$ .

Hint

For a line bundle $L$ , $Λ^{0} L^{*} = 1$ and $Λ^{1} L^{*} = L^{*}$ , so $λ_{L} = 1 - L^{*}$ . Multiply and expand. Use the formula $Λ^{k} (L_{1} \oplus L_{2}) = ⨁_{i + j = k} Λ^{i} L_{1} \otimes Λ^{j} L_{2}$ and the fact that $Λ^{i} L = 0$ for a line bundle and $i \geq 2$ .

Answer

By multiplicativity, $λ_{L_{1} \oplus L_{2}} = λ_{L_{1}} \cdot λ_{L_{2}} = (1 - L_{1}^{*}) (1 - L_{2}^{*}) = 1 - L_{1}^{*} - L_{2}^{*} + L_{1}^{*} L_{2}^{*}$ . Direct computation: $Λ^{0} E^{*} = 1$ , $Λ^{1} E^{*} = L_{1}^{*} \oplus L_{2}^{*}$ , $Λ^{2} E^{*} = L_{1}^{*} \otimes L_{2}^{*}$ (a line bundle), $Λ^{k} E^{*} = 0$ for $k \geq 3$ . So $$ \sum_{k=0}^{2} (-1)^k \Lambda^k E^* = 1 - (L_1^* + L_2^) + L_1^ L_2^*, $$ matching the product expansion. Rubric: full credit for both computations and the identification.

Exercise 5 (medium, symbolic).

Show that the Thom class of the product rank- $n$ complex bundle $\underline{C^{n}}$ over $X$ acts as the suspension isomorphism $K (X) \to \tilde{K} (Σ^{2 n} X_{+})$ , and identify the image of $1 \in K (X)$ .

Hint

The Thom space of the product bundle is a suspension: $T (\underline{C^{n}}) = Σ^{2 n} X_{+}$ . The Thom isomorphism becomes the iterated Bott isomorphism.

Answer

The disk bundle of $\underline{C^{n}} = X \times C^{n}$ is $X \times D^{2 n}$ , the sphere bundle is $X \times S^{2 n - 1}$ , and the Thom space is $X^{+} \land S^{2 n} = Σ^{2 n} X_{+}$ . The Thom isomorphism is $K (X) ≅ \tilde{K} (Σ^{2 n} X_{+})$ , which is the $2 n$ -fold Bott periodicity isomorphism applied to $X_{+}$ . The Thom class $λ_{\underline{C^{n}}} = (1 - C)^{n}$ is the $n$ -th power of the Bott element. The image of $1 \in K (X)$ is $β^{n}$ , the $n$ -th power of the Bott generator, lifted to $\tilde{K} (Σ^{2 n} X_{+})$ . Rubric: full credit for identifying the Thom space as a suspension and the Thom class as the Bott element to the $n$ -th power.

Exercise 6 (medium, short-answer).

Explain how the K-theoretic Thom isomorphism for the normal bundle of an embedding $M ↪ R^{N}$ realises the topological-index map $K (T M) \to K (pt) = Z$ of Atiyah-Singer.

Hint

Use a closed embedding $i : M ↪ R^{N}$ for $N$ large, take a tubular neighbourhood, apply the Thom isomorphism for the normal bundle, then use the open inclusion into $R^{N}$ and Bott periodicity to land in $K (pt)$ .

Answer

Embed $M$ into $R^{N}$ as a closed submanifold. Let $ν$ be the normal bundle of the embedding; the tubular neighbourhood gives a homeomorphism between a neighbourhood of $M$ in $R^{N}$ and the total space of $ν$ . The complexification of $T M \oplus ν$ is the product complex bundle $M \times C^{N}$ , and after taking symbols this gives a K-theory pushforward $i_{!} : K (T M) \to K (T R^{N}) = K (R^{N})$ implemented as Thom-multiplying the symbol class on $T M$ by the Thom class of the normal complexified bundle and then extending by zero into $R^{N}$ . The open inclusion $R^{N} ↪ S^{2 N}$ gives a class in $\tilde{K} (S^{2 N})$ , and Bott periodicity identifies this with $K (pt) = Z$ . The composite is the topological-index map. Rubric: full credit for naming all three steps — Thom isomorphism, open inclusion, Bott periodicity — and identifying the composite with $t -ind$ .

Exercise 7 (hard, short-answer).

Prove that the K-theoretic Thom class $λ_{E}$ restricts on the zero section to the K-theoretic Euler class $e_{K} (E) = \sum_{k = 0}^{n} (- 1)^{k} Λ^{k} E^{*} \in K (X)$ , and compare with the cohomological Euler class via the Chern character.

Hint

The zero section $X ↪ D (E)$ pulls $λ_{E}$ back to the alternating sum of exterior powers on $X$ itself, not on $E$ . Apply the Chern character and use the splitting principle to reduce to line bundles.

Answer

The restriction of $λ_{E} \in K (D (E), S (E))$ to the zero section $X = D (E) ∣_{X}$ kills the relative structure and gives the absolute class $\sum_{k} (- 1)^{k} Λ^{k} E^{*} \in K (X)$ , since the exterior-power virtual bundle is defined on all of $E$ and the zero section is part of $E$ . This is the K-theoretic Euler class $e_{K} (E)$ .

To compare with the cohomological Euler class: the Chern character of $e_{K} (E)$ equals (by the splitting-principle reduction) the alternating sum $$ \mathrm{ch}(e_K(E)) = \prod_i (1 - e^{-x_i}), $$ where the $x_{i}$ are the Chern roots of $E$ . The cohomological Euler class is $e (E) = \prod_{i} x_{i}$ , and the Todd class is $Td (E) = \prod_{i} x_{i} / (1 - e^{- x_{i}})$ . So $$ \mathrm{ch}(e_K(E)) = e(E) \cdot \mathrm{Td}(E)^{-1}, $$ which is the same formula as for the Thom class. The K-theoretic Euler class is the rational image of $e (E)$ under the Chern character, modified by the Todd factor. Rubric: full credit for the restriction identification and the splitting-principle calculation.

Exercise 8 (hard, short-answer).

Show that the K-theoretic Thom isomorphism for a complex bundle $E$ over $X$ is an isomorphism of $K (X)$ -modules, with the $K (X)$ -action on $\tilde{K} (T (E))$ defined by pullback-and-multiply along $π$ , and that $ϕ_{E}$ commutes with the action.

Hint

Verify the module property on the formula $ϕ_{E} (x) = π^{*} (x) \cdot λ_{E}$ directly: for $y \in K (X)$ , compute $ϕ_{E} (y x)$ and compare with $y \cdot ϕ_{E} (x)$ .

Answer

The module action of $K (X)$ on $\tilde{K} (T (E))$ is $y \cdot α = π^{*} (y) \cdot α$ , using the absolute-relative product $K (D (E)) \otimes K (D (E), S (E)) \to K (D (E), S (E))$ . Compute $$ \phi_E(yx) = \pi^(yx) \cdot \lambda_E = \pi^(y) \cdot \pi^(x) \cdot \lambda_E = \pi^(y) \cdot \phi_E(x) = y \cdot \phi_E(x). $$ So $ϕ_{E}$ is $K (X)$ -linear. The isomorphism property follows from the Thom theorem proper. The $K (X)$ -module structure is in fact part of the statement: the Thom isomorphism produces a free $K (X)$ -module of rank one on the generator $λ_{E}$ , since $ϕ_{E} (1) = λ_{E}$ generates $\tilde{K} (T (E))$ over $K (X)$ . Rubric: full credit for the module-property verification and the identification of the generator.

Lean formalization [Intermediate+]

Mathlib does not yet have topological K-theory in the form needed for this unit; in particular it does not have the Thom space construction for a complex vector bundle, the $Z /2$ -graded exterior algebra as a virtual bundle, or the named theorem $ϕ_{E} : K (X) ≅ \tilde{K} (T (E))$ . The intended formalisation would read schematically:

import Mathlib.Topology.VectorBundle.Hom
import Mathlib.LinearAlgebra.ExteriorPower
import Mathlib.Topology.MetricSpace.Pseudo.Basic

/-- The Thom space of a complex vector bundle over a compact base. -/
def ThomSpace {X : Type*} [TopologicalSpace X] [CompactSpace X]
    {E : Type*} [TopologicalSpace E] [VectorBundle ℂ E X] : Type* :=
  Quotient (sphereBundleRel E)

/-- The K-theoretic Thom class of a complex vector bundle. -/
def kTheoryThomClass {X : Type*} [TopologicalSpace X] [CompactSpace X]
    {E : Type*} [VectorBundle ℂ E X] (n : ℕ) :
    KTheory.Relative (DiskBundle E) (SphereBundle E) :=
  ∑ k ∈ Finset.range (n + 1),
    (-1 : ℤ)^k • KTheory.classOfBundle (ExteriorPower ℂ k (DualBundle E))

/-- The K-theoretic Thom isomorphism. -/
theorem kTheory_thom_iso
    {X : Type*} [TopologicalSpace X] [CompactSpace X]
    (E : Type*) [VectorBundle ℂ E X] (n : ℕ)
    (hE : VectorBundle.rank E = n) :
    KTheory X ≃+ KTheory.Reduced (ThomSpace E) :=
  sorry  -- proof by induction on rank using Bott periodicity

The proof gap is substantive. Required upstream pieces include: a packaged Thom space construction for vector bundles, the $Z /2$ -graded Koszul complex acyclic off the zero section, the Bott element as the rank-one Thom class generator, multiplicativity of Thom classes under direct sums, the splitting principle for K-theory, and the induction-over-rank pattern. Bott periodicity for $K (CP^{1})$ is itself a substantial Mathlib gap (covered in 03.08.07), and the Thom isomorphism depends on it. The Chern-character comparison $ch (λ_{E}) = e (E) \cdot Td (E)^{- 1}$ would require Chern-Weil theory, which is also a Mathlib gap.

Advanced results [Master]

Theorem (multiplicativity of the Thom class). Let $E, F \to X$ be complex vector bundles. Then $λ_{E \oplus F} = λ_{E} \cdot λ_{F}$ in $K (D (E \oplus F), S (E \oplus F))$ , where the product uses the natural identification of the Thom space of a direct sum with the smash product of Thom spaces: $$ T(E \oplus F) = T(E) \wedge_X T(F). $$

This is the K-theoretic Whitney sum formula for Thom classes. The proof uses the tensor-product identity for Koszul complexes: $K (E \oplus F, v \oplus w) = K (E, v) \otimes K (F, w)$ , and the Euler characteristic of a tensor product is the product of the Euler characteristics.

Theorem (functoriality and naturality of the Thom isomorphism). Let $f : Y \to X$ be a continuous map and $E \to X$ a complex vector bundle. The pullback $f^ E \to Y $ha s T h o m c l a ss$ \lambda_{f^* E} = f^* \lambda_E $in$ \tilde K(T(f^* E)) $, w h er e$ f^* \colon \tilde K(T(E)) \to \tilde K(T(f^* E)) $i s in d u ce d b y t h e na t u r a l T h o m - s p a ce ma p$ T(f^* E) \to T(E)$. The Thom isomorphisms fit into a commutative diagram* $$ \begin{array}{ccc} K(X) & \xrightarrow{\phi_E} & \tilde K(T(E)) \ \downarrow f^* & & \downarrow f^* \ K(Y) & \xrightarrow{\phi_{f^* E}} & \tilde K(T(f^* E)). \end{array} $$

Naturality is what allows the Thom isomorphism to be used as a building block in classifying-space arguments: the universal Thom class $λ_{univ} \in \tilde{K} (M U (n))$ on the Thom space of the universal rank- $n$ complex bundle $E U (n) \to B U (n)$ pulls back to $λ_{E}$ along the classifying map of any complex bundle $E \to X$ , and the Thom spectrum $M U = colim_{n} M U (n)$ becomes the representing spectrum for complex K-theoretically-oriented cobordism.

Theorem (Chern-character comparison). Let $E \to X$ be a complex vector bundle of rank $n$ . The Chern character of the K-theoretic Thom class equals the rationalisation of the cohomological Thom class divided by the Todd class: $$ \mathrm{ch}(\lambda_E) = U(E) \cdot \mathrm{Td}(E)^{-1} \quad \text{in } H^{\mathrm{even}}(T(E); \mathbb{Q}), $$ where $U (E) \in H^{2 n} (T (E); Z)$ is the cohomological Thom class and $Td (E) = \prod_{i} x_{i} / (1 - e^{- x_{i}})$ is the Todd class in terms of the Chern roots $x_{i}$ .

This is the K-theoretic refinement of the singular-cohomology Thom isomorphism: the Chern character intertwines $ϕ_{E}^{K} : K (X) \to \tilde{K} (T (E))$ with the cohomological Thom homomorphism $ϕ_{E}^{coh} : H^{*} (X; Q) \to \tilde{H}^{*+ 2 n} (T (E); Q)$ up to the Todd correction. The Todd correction is the source of the $\hat{A}$ -genus in the Atiyah-Singer index formula, and historically it is how Atiyah and Hirzebruch realised that the index theorem must involve the Todd class rather than just the Euler class: K-theory carries the orientation, the Chern character converts to cohomology, and the conversion factor is the Todd class.

Theorem (Atiyah-Hirzebruch spectral sequence). For any space $X$ there is a spectral sequence with $E_{2}^{p, q} = H^{p} (X; K^{q} (pt))$ converging to $K^{p + q} (X)$ . The $E_{2}$ -differentials are characterised by their behaviour on the Thom class: $d_{3} = Sq^{3}$ in mod-2 K-theory, and higher differentials are higher Steenrod operations.

The Thom isomorphism is the assertion that, after passing to the Thom space, this spectral sequence collapses on the column carrying the Thom class. Equivalently, the Thom isomorphism is the statement that complex vector bundles are K-orientable: the orientation class exists and is canonical.

Theorem (Thom isomorphism and the topological index). Let $M$ be a compact manifold and $i : M ↪ R^{N}$ a closed embedding for $N$ large. Let $ν = ν_{M / R^{N}}$ be the normal bundle, complexified so $ν \otimes_{R} C$ carries a complex structure. The topological-index map $$ t\text{-}\mathrm{ind} \colon K(TM) \to K(\mathrm{pt}) = \mathbb{Z} $$ is the composite $$ K(TM) \xrightarrow{\phi_\nu} \tilde K(T(TM \oplus \nu)) = \tilde K(T(T\mathbb{R}^N|M)) \xrightarrow{j!} \tilde K(T\mathbb{R}^N) \xrightarrow{\beta^{-N}} K(\mathrm{pt}), $$ where $ϕ_{ν}$ is the K-theoretic Thom isomorphism for the normal complexified bundle, $j_{!}$ extends by zero along the open inclusion $T R^{N} ↪ S^{2 N}$ , and $β^{- N}$ is the inverse of the $N$ -fold Bott periodicity isomorphism.

This is the Atiyah-Singer topological-index map. The analytic-index map associates to an elliptic operator $D$ on $M$ its symbol class $[σ (D)] \in K (T M)$ and then its index $ind (D) = dim ker D - dim coker (D) \in Z$ . The Atiyah-Singer index theorem is the equality $ind (D) = t - ind ([σ (D)])$ , and the K-theoretic Thom isomorphism is the structural reason the right-hand side is well-defined and computable.

Theorem (real K-theory via Clifford modules; Atiyah-Bott-Shapiro 1964). For a real oriented spin bundle $E \to X$ of rank $n$ , the Thom class in $K O$ -theory exists and is built from the Clifford modules of $Cl (n)$ rather than the exterior algebra. Specifically, the Atiyah-Bott-Shapiro construction associates to each $Z /2$ -graded Clifford module a class in $K O (pt)$ , and the Thom class of a spin bundle is the associated bundle of this module structure.

The real and complex Thom isomorphisms together produce the K-orientability of spin manifolds: a manifold is K-orientable in $K O$ -theory if and only if it is spin. This is the underlying topological reason the spin Dirac operator carries an index formula via K-theory, and it is what restricts the Atiyah-Singer $\hat{A}$ -formula to spin manifolds.

Synthesis. The K-theoretic Thom isomorphism is the foundational reason that complex K-theory is a generalised cohomology theory in which complex vector bundles are oriented. The central insight is that the Bott element $β = [H] - 1 \in \tilde{K} (S^{2})$ is the rank-one Thom class, and multiplicativity of Thom classes under direct sums promotes this to all ranks. This is exactly the same algebraic pattern that organises orientability in any cohomology theory: a generalised cohomology theory $h^{*}$ admits a Thom isomorphism for a class of bundles if and only if a coherent system of Thom classes exists, and the multiplicativity-plus-induction-over-rank reduces existence to the rank-one case. Putting these together, the Atiyah-Singer index theorem reduces to two assertions: complex bundles are K-oriented (the Thom isomorphism of this unit), and the analytic-index of an elliptic operator equals the K-theoretic topological-index of its symbol (the index theorem proper).

The bridge is the recognition that the K-theoretic Thom class encodes both more and less than the cohomological one: more, in that it remembers the integral structure of the orientation through the virtual bundle $\sum (- 1)^{k} Λ^{k} E^{*}$ ; less, in that the Chern character converts it to ordinary cohomology only after multiplication by the Todd class, identifying the K-theoretic Thom class with the Todd-corrected cohomological Thom class. The Todd correction is the genuine algebraic content of the comparison, and it is what makes the index theorem of Atiyah-Singer compute the $\hat{A}$ -genus rather than just the Euler characteristic. This same pattern of orientation-class transfer recurs in 03.13.02 (Leray-Serre spectral sequence) where Thom-isomorphism inputs control the $E_{2}$ page of bundle fibrations, in 03.09.10 (Atiyah-Singer index theorem) where the K-theoretic Thom is the engine of the topological-index map, and in stable homotopy where Thom spectra organise the relationship between cobordism and K-theory.

Full proof set [Master]

Proposition (multiplicativity of the Thom class), proof. Let $E, F \to X$ be complex vector bundles of ranks $m$ and $n$ . The Koszul complex of $E \oplus F$ at a vector $(v, w)$ is the tensor product $$ \Lambda^\bullet (E \oplus F)^* = \Lambda^\bullet E^* \otimes \Lambda^\bullet F^, $$ with differential $ι_{(v, w)} = ι_{v} \otimes 1 + (- 1)^{d e g} 1 \otimes ι_{w}$ . As a $Z /2$ -graded virtual bundle, the Euler characteristic of a tensor product of complexes is the product of the Euler characteristics: in K-theory, $$ \lambda_{E \oplus F} = \chi(\Lambda^\bullet (E \oplus F)^) = \chi(\Lambda^\bullet E^) \cdot \chi(\Lambda^\bullet F^) = \lambda_E \cdot \lambda_F. $$ The product on the right uses the natural identification $K (D (E), S (E)) \otimes K (D (F), S (F)) \to K (D (E \oplus F), S (E \oplus F))$ coming from the smash decomposition $T (E \oplus F) = T (E) \land_{X} T (F)$ . The relative cup product respects this identification because the Koszul differential factorises as a tensor of the two individual differentials, and the relative K-class is the Euler characteristic in the $Z /2$ -graded sense. $□$

Proposition (Chern character comparison), proof. By the splitting principle for K-theory and cohomology, it is enough to check the formula for line bundles and to verify that both sides are multiplicative. For a line bundle $L$ with Chern root $x = c_{1} (L)$ , $$ \mathrm{ch}(\lambda_L) = \mathrm{ch}(1 - L^*) = 1 - e^{-x}. $$ The cohomological Thom class of $L$ is $U (L) = x$ (the rank-one case is the de Rham Thom class, identified with the Chern class — see 03.04.09), and the Todd class is $Td (L) = x / (1 - e^{- x})$ , so $U (L) \cdot Td (L)^{- 1} = x \cdot (1 - e^{- x}) / x = 1 - e^{- x}$ . The two sides agree.

For a general complex bundle $E$ of rank $n$ with Chern roots $x_{1}, \dots, x_{n}$ , the splitting principle gives $$ \mathrm{ch}(\lambda_E) = \prod_{i=1}^n \mathrm{ch}(\lambda_{L_i}) = \prod_i (1 - e^{-x_i}) $$ and $$ U(E) \cdot \mathrm{Td}(E)^{-1} = \prod_i x_i \cdot \prod_i \frac{1 - e^{-x_i}}{x_i} = \prod_i (1 - e^{-x_i}). $$ The two are equal. Multiplicativity on both sides under direct sums extends the line-bundle identity to all complex bundles. $□$

Theorem (Thom isomorphism in K-theory), full proof. The argument was sketched in the Intermediate proof; the full version unwinds the rank-one base case and the inductive step.

Rank-one base case. For a complex line bundle $L \to X$ , the projective bundle $P (L \oplus \underline{C})$ is a $CP^{1}$ -bundle over $X$ with a tautological line bundle $O (- 1)$ . Atiyah's proof of Bott periodicity (Atiyah K-Theory §2.2) shows that $K (P (L \oplus \underline{C}))$ is a free $K (X)$ -module on $1$ and $[O (- 1)]$ , equivalently on $1$ and $β_{L} = 1 - [O (- 1)]^{*} = 1 - [O (1)] \cdot [L^{*}]$ . The Thom space $T (L)$ is the quotient $P (L \oplus \underline{C}) / P (L)$ , where $P (L) = X$ is the section at infinity. The long exact sequence of the pair gives $\tilde{K} (T (L)) ≅ ker (K (P (L \oplus \underline{C})) \to K (P (L)) = K (X))$ , and this kernel is the free $K (X)$ -module on $β_{L}$ . The Thom homomorphism sends $1 \mapsto β_{L} = λ_{L}$ , so it is multiplication by $β_{L}$ , hence an isomorphism onto the free rank-one $K (X)$ -module generated by $β_{L}$ .

Inductive step. Suppose the Thom isomorphism holds for all complex bundles of rank less than $n$ and let $E \to X$ have rank $n$ . After applying the splitting principle, it is enough to prove the case where $E = L_{1} \oplus \dots \oplus L_{n}$ splits as a sum of line bundles. The Thom space identifies as the iterated smash $T (E) = T (L_{1}) \land_{X} \dots \land_{X} T (L_{n})$ , and the Thom class is $λ_{E} = λ_{L_{1}} \dots λ_{L_{n}}$ by the multiplicativity proposition. The Thom homomorphism $ϕ_{E} = π^{*} \cdot λ_{E}$ factors as the composition of the rank-one Thom homomorphisms for $L_{1}, \dots, L_{n}$ , each of which is an isomorphism by the base case. The composite is an isomorphism.

Splitting principle promotion. Let $E \to X$ be a general rank- $n$ complex bundle. The associated flag bundle $f : F (E) \to X$ has the property that $f^{*} E$ splits as a sum of line bundles on $F (E)$ and that $f^{*} : K (X) \to K (F (E))$ is injective (with image a $K (X)$ -summand). The Thom isomorphism for $f^{*} E$ holds by the split case. The diagram $$ \begin{array}{ccc} K(X) & \xrightarrow{\phi_E} & \tilde K(T(E)) \ \downarrow f^* & & \downarrow f^* \ K(F(E)) & \xrightarrow{\phi_{f^* E}} & \tilde K(T(f^* E)) \end{array} $$ commutes by naturality. The bottom map is an isomorphism. The left $f^{*}$ is injective with a left inverse $f_{*}$ (the flag-bundle pushforward, defined via the relative Thom isomorphism of the flag fibration). Composing, $ϕ_{E}$ admits both a left inverse and a right inverse, hence is an isomorphism. $□$

Proposition (naturality), proof. Let $f : Y \to X$ and $E \to X$ a complex bundle. The natural map $T (f^{*} E) \to T (E)$ is induced by the bundle map $f^{*} E \to E$ covering $f$ , which is a fibrewise isomorphism. On Thom classes, the pullback of $λ_{E} = \sum (- 1)^{k} Λ^{k} E^{*}$ along this map is exactly $\sum (- 1)^{k} Λ^{k} (f^{*} E)^{*} = λ_{f^{*} E}$ , since exterior powers commute with pullback. The commutativity of the diagram $$ \phi_{f^* E}(f^* x) = \pi_{f^* E}^(f^ x) \cdot \lambda_{f^* E} = f^* \pi_E^(x) \cdot f^ \lambda_E = f^(\pi_E^(x) \cdot \lambda_E) = f^* \phi_E(x) $$ follows from the multiplicativity of $f^{*}$ on K-classes and the identity $λ_{f^{*} E} = f^{*} λ_{E}$ . $□$

Connections [Master]

Topological K-theory 03.08.01. The Thom isomorphism is the central multiplicative tool of K-theory beyond the basic functoriality of the Grothendieck group: it identifies $\tilde{K} (T (E))$ as a free $K (X)$ -module of rank one on the Thom class, mirroring the role of the cohomological Thom class in singular cohomology. Without the Thom isomorphism, K-theory would have no canonical way to encode the orientation of a complex bundle, and the multiplicative structure of K-theory on Thom spaces would be opaque.
Adams operations 03.08.02. The Adams operations $ψ^{k} : K (X) \to K (X)$ act on Thom classes through the formula $ψ^{k} (λ_{E}) = λ_{E} \cdot θ^{k} (E)$ , where $θ^{k} (E)$ is the Bott cannibalistic class. This is the K-theoretic refinement of the action of integral cohomology operations on the cohomological Thom class. Sibling unit in production this cycle; the Thom isomorphism is the structural setting in which Adams operations act on relative K-classes, and the cannibalistic classes carry the same Todd-genus information through K-theory that the genus formulae carry in cohomology.
De Rham Thom isomorphism 03.04.09. The singular-cohomology Thom isomorphism is the parallel statement: for a real oriented rank- $r$ vector bundle $E \to X$ , the de Rham cohomology of the base shifts up by $r$ degrees through multiplication by the cohomological Thom class. Both isomorphisms are statements about orientability of $E$ in their respective cohomology theories. The Chern character converts between them rationally, with the Todd-class correction $ch (λ_{E}) = U (E) \cdot Td (E)^{- 1}$ as the algebraic content of the comparison. The de Rham Thom isomorphism predates the K-theoretic one by sixteen years (Thom 1952 versus Atiyah 1968), and the K-theoretic version was developed precisely to provide the orientation framework needed by the Atiyah-Singer index theorem.
Bott periodicity 03.08.07. The rank-one K-theoretic Thom isomorphism is Bott periodicity: $\tilde{K} (S^{2}) ≅ Z$ generated by the Bott element $β = [H] - 1$ , which is the Thom class of the product complex line bundle over a point. Higher-rank Thom isomorphisms are iterates of Bott periodicity by multiplicativity. The two theorems are not merely related — they are the same theorem viewed from two angles, one (Bott) recording the homotopical pattern across degrees, the other (Thom) recording the bundle-theoretic content within a single degree.
Atiyah-Singer index theorem 03.09.10. The Thom isomorphism is the structural engine of the topological-index map: for an elliptic operator $D$ on a compact manifold $M$ , the symbol class $[σ (D)] \in K (T M)$ is pushed to $K (pt) = Z$ by Thom-multiplying with the normal-bundle Thom class along an embedding $M ↪ R^{N}$ , then extending by zero and applying inverse Bott periodicity. The K-theoretic Thom isomorphism is the natural setting for this construction; the cohomological version would require an unnatural identification of cohomology with K-theory tensored with $Q$ , which would lose the integer-index information.
Complex vector bundles 03.05.08. The Thom isomorphism in K-theory requires a complex structure on the bundle $E$ , since the exterior powers $Λ^{k} E^{*}$ that build the Thom class are themselves complex bundles. Real bundles admit a K-theoretic Thom isomorphism only after complexification, or alternatively in $K O$ -theory via the Clifford-module construction of Atiyah-Bott-Shapiro. The complex structure is therefore a structural prerequisite for the form of Thom isomorphism developed here.
Atiyah-Hirzebruch spectral sequence 03.13.04. The AHSS applied to the Thom space $T (E)$ converges to $\tilde{K}^{*} (T (E))$ starting from $H^{*} (T (E); K^{*} (pt))$ . On the $E_{2}$ page, multiplication by the cohomological Thom class $U (E) \in H^{n} (T (E); Z)$ implements the cohomological Thom isomorphism row by row; the AHSS abuts to the K-theoretic Thom isomorphism, and the abutment edge map is the K-theoretic Thom class $λ_{E}$ . The compatibility $ch (λ_{E}) = U (E) \cdot Td (E)^{- 1}$ proved above is exactly what the AHSS encodes: rational collapse of the spectral sequence identifies the rationalised K-theoretic Thom class with the cohomological one modulo Todd, and the $d_{3} = Sq^{3}$ differential is the lowest integer-coefficient obstruction to lifting a cohomological Thom class to a K-theoretic Thom class without correction. The AHSS therefore detects exactly when a real bundle admits a K-theoretic orientation lift — the obstruction is the third Steenrod-square image of $w_{2}$ .
Equivariant K-theory and the representation ring 03.08.10. The K-theoretic Thom isomorphism has an equivariant refinement: for a complex $G$ -equivariant vector bundle $E \to X$ on a $G$ -space, the equivariant Koszul resolution produces an equivariant Thom class $λ_{E} \in K_{G} (E)$ whose multiplication implements the isomorphism $K_{G} (X) \sim K_{G} (Th (E))$ . The equivariant Thom isomorphism is the structural engine behind equivariant Bott periodicity (the case where $E = V$ is a complex $G$ -representation viewed as a product bundle over a point) and behind the Atiyah-Bott Lefschetz fixed-point formula (Atiyah-Bott 1967), where it localises the equivariant index of an elliptic complex to the contributions of isolated fixed points. The pushforward $K_{G} (M) \to K_{G} (pt) = R (G)$ uses the equivariant Thom class on the normal bundle of an equivariant embedding, refining the integer index into a virtual representation. Connection type: equivariant refinement — the present Thom isomorphism is the non-equivariant specialisation of the equivariant theorem developed in the K-theory chapter's representation-ring unit.

Historical & philosophical context [Master]

The K-theoretic Thom isomorphism was developed by Michael Atiyah in Bott periodicity and the index of elliptic operators, Quarterly Journal of Mathematics Oxford (2) 19 (1968), 113–140 ^[pending]. The paper provided a streamlined treatment of Bott periodicity adapted to the index-theoretic needs of Atiyah and Singer, and in doing so packaged the Thom isomorphism as a consequence of Bott periodicity rather than a separate result. The technical heart was the recognition that the rank-one Thom space — the two-sphere — admits a canonical K-theory generator, the Bott element, and that this generator extends multiplicatively to all ranks through the Koszul-type exterior-power construction.

The equivariant version of the K-theoretic Thom isomorphism appeared earlier in Atiyah and Bott, A Lefschetz fixed point formula for elliptic complexes I, Annals of Mathematics (2) 86 (1967), 374–407 ^[pending]. Atiyah-Bott needed the equivariant Thom isomorphism to make sense of the localised contribution of fixed points to the Lefschetz number of an elliptic complex, and the equivariant K-theoretic Thom class was the structural device that made the localisation argument work. The 1967 paper is therefore the immediate predecessor of the 1968 packaging, and the two together established the K-theoretic Thom isomorphism as a standard tool by the end of the 1960s.

The earlier cohomological version is the classical theorem of René Thom, Espaces fibrés en sphères et carrés de Steenrod, Annales scientifiques de l'École Normale Supérieure 69 (1952), 109–182 ^[pending]. Thom's theorem established the isomorphism in singular cohomology and introduced the Thom space, the Thom class, and the Thom isomorphism in essentially the form still used today. The K-theoretic version of Atiyah refines this by replacing the singular-cohomology orientation class with a virtual bundle on the Thom space, gaining the integer-index information that the index theorem requires.

The full systematic development of the K-theoretic Thom isomorphism, including the comparison with the cohomological version through the Chern character, was given by Max Karoubi in K-Theory: An Introduction, Springer Grundlehren 226, 1978 ^[pending]. Karoubi's treatment is the standard reference for the comparison theorem $ch (λ_{E}) = e (E) \cdot Td (E)^{- 1}$ and the Todd-class correction. The Todd correction was historically the crucial step in the development of the Atiyah-Singer index theorem: it was the recognition that K-theory carries an orientation different from cohomology by the Todd factor that led to the $\hat{A}$ -genus appearing in the spin Dirac index formula, rather than the Euler characteristic that a naive cohomological calculation would suggest. The Lawson-Michelsohn treatment in Spin Geometry (Princeton 1989) §I.5 and §III.6 ^[pending] develops the K-theoretic Thom isomorphism through the Clifford-module construction of Atiyah-Bott-Shapiro and connects it directly to the symbol of the spin-c Dirac operator, which is how the Thom isomorphism enters the heat-kernel proof of the index theorem.

Bibliography [Master]

@article{Atiyah1968BottPeriodicity,
  author  = {Atiyah, Michael F.},
  title   = {Bott periodicity and the index of elliptic operators},
  journal = {Quart. J. Math. Oxford (2)},
  volume  = {19},
  year    = {1968},
  pages   = {113--140}
}

@article{AtiyahBott1967Lefschetz,
  author  = {Atiyah, Michael F. and Bott, Raoul},
  title   = {A {L}efschetz fixed point formula for elliptic complexes {I}},
  journal = {Ann. of Math. (2)},
  volume  = {86},
  year    = {1967},
  pages   = {374--407}
}

@article{AtiyahSinger1968IndexI,
  author  = {Atiyah, Michael F. and Singer, Isadore M.},
  title   = {The index of elliptic operators {I}},
  journal = {Ann. of Math. (2)},
  volume  = {87},
  year    = {1968},
  pages   = {484--530}
}

@book{AtiyahKTheory1967,
  author    = {Atiyah, Michael F.},
  title     = {K-Theory},
  note      = {Lectures by M. F. Atiyah, notes by D. W. Anderson; reprinted 1989 as Advanced Book Classics, Addison-Wesley},
  publisher = {W. A. Benjamin, Inc.},
  year      = {1967}
}

@article{Thom1952Espaces,
  author  = {Thom, René},
  title   = {Espaces fibrés en sphères et carrés de {S}teenrod},
  journal = {Annales scientifiques de l'École Normale Supérieure},
  volume  = {69},
  year    = {1952},
  pages   = {109--182}
}

@article{AtiyahBottShapiro1964,
  author  = {Atiyah, Michael F. and Bott, Raoul and Shapiro, Arnold},
  title   = {Clifford modules},
  journal = {Topology},
  volume  = {3},
  number  = {suppl. 1},
  year    = {1964},
  pages   = {3--38}
}

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

@book{Husemoller1994,
  author    = {Husemoller, Dale},
  title     = {Fibre Bundles},
  series    = {Graduate Texts in Mathematics},
  volume    = {20},
  edition   = {3rd},
  publisher = {Springer-Verlag},
  year      = {1994}
}

@book{LawsonMichelsohn1989,
  author    = {Lawson, H. Blaine and Michelsohn, Marie-Louise},
  title     = {Spin Geometry},
  series    = {Princeton Mathematical Series},
  volume    = {38},
  publisher = {Princeton University Press},
  year      = {1989}
}

@article{AtiyahHirzebruch1961,
  author  = {Atiyah, Michael F. and Hirzebruch, Friedrich},
  title   = {Vector bundles and homogeneous spaces},
  journal = {Proc. Sympos. Pure Math.},
  volume  = {3},
  year    = {1961},
  pages   = {7--38}
}

Prerequisites

03.08.01
03.08.07
03.05.08
03.04.09

Tier anchors

beginner: Strogatz / 3Blue1Brown analogous level — replacing a cohomology orientation class by an alternating sum of exterior powers
intermediate: Atiyah *K-Theory* §2.7; Karoubi *K-Theory* §IV.5; Husemoller *Fibre Bundles* §17
master: Atiyah 1968 *Bott periodicity and the index of elliptic operators* (Quart. J. Math. 19); Atiyah *K-Theory* §2.7; Lawson-Michelsohn §I.5 and §III.6; Karoubi *K-Theory* §IV.5

References

fast-track
images/Atiyah-K-Theory-683x1024__52e3382e9f.jpg · Atiyah *K-Theory* archive anchor — §2.7 Thom isomorphism, §2.2 Bott periodicity
fast-track
images/Lawson-Michelsohn-Spin-Geometry-683x1024__51a67ee44f.jpg · Lawson-Michelsohn *Spin Geometry* §I.5, §III.6 — K-theoretic Thom class via Clifford modules and the symbol of the Dirac operator
TODO_REF
Atiyah — Bott periodicity and the index of elliptic operators · Quart. J. Math. Oxford (2) 19 (1968), 113–140 — originator paper for the K-theoretic Thom isomorphism in the form used by the index theorem
TODO_REF
Atiyah-Bott — A Lefschetz fixed point formula for elliptic complexes I · Ann. of Math. (2) 86 (1967), 374–407 — K-theoretic Thom class in the equivariant setting used by the Lefschetz fixed-point theorem
TODO_REF
Atiyah — K-Theory · Benjamin 1967 / Addison-Wesley 1989; §2.7 Thom isomorphism via projective bundles; §2.2 Bott periodicity proof from clutching functions
TODO_REF
Karoubi — K-Theory: An Introduction · Springer Grundlehren 226 (1978); §IV.5 Thom isomorphism for complex bundles, comparison with the Chern character
TODO_REF
Husemoller — Fibre Bundles · Springer GTM 20, third ed. 1994; §17 K-theory and the Thom isomorphism
TODO_REF
Lawson-Michelsohn — Spin Geometry · Princeton 1989; §I.5 Clifford modules and the Thom class; §III.6 K-theoretic Thom isomorphism as the symbol of the spin-c Dirac operator
TODO_REF
Atiyah-Singer — The index of elliptic operators I · Ann. of Math. (2) 87 (1968), 484–530 — topological-index construction uses the K-theoretic Thom isomorphism for an embedding $M \hookrightarrow \mathbb{R}^N$
TODO_REF
Thom — Espaces fibrés en sphères et carrés de Steenrod · Annales scientifiques de l'École Normale Supérieure 69 (1952), 109–182 — original Thom isomorphism in singular cohomology, against which the K-theoretic version is compared

Reviewer

TBD

Estimated time

beginner: 18m
intermediate: 50m
master: 90m