Math 2.0

K-theory Topological K-theory Theorem

Atiyah–Jänich theorem

Builds on Fredholm operatorFredholm indexHilbert spaceOperator normComplex K-theory KUKuiper's theoremHomotopyCompact space

Statement

Let HH be a separable infinite-dimensional complex Hilbert space. Let Fred⁡(H)\operatorname{Fred}(H) be the space of Fredholm operators on HH with the operator norm topology. Let XX be a compact Hausdorff space, and let [X,Fred⁡(H)][X, \operatorname{Fred}(H)] be the set of homotopy classes of continuous maps X→Fred⁡(H)X \to \operatorname{Fred}(H). For a continuous map f ⁣:X→Fred⁡(H)f \colon X \to \operatorname{Fred}(H), there is a closed subspace V⊆HV \subseteq H of finite codimension with V∩ker⁡f(x)=0V \cap \ker f(x) = 0 for all x∈Xx \in X. The spaces H/f(x)(V)H / f(x)(V) then form a vector bundle H/f(V)H/f(V) over XX, and the index bundle of ff is ind⁡(f)=[X×(H/V)]−[H/f(V)]∈K0(X).\operatorname{ind}(f) = [X \times (H/V)] - [H/f(V)] \in K^0(X). The class ind⁡(f)\operatorname{ind}(f) does not depend on VV, and it depends only on the homotopy class of ff. The theorem states that ind⁡ ⁣:[X,Fred⁡(H)]→K0(X)\operatorname{ind} \colon [X, \operatorname{Fred}(H)] \to K^0(X) is a bijection, where K0K^0 is complex K-theory.

Remarks

For XX a point, the index bundle is the Fredholm index dim⁡ker⁡T−dim⁡coker⁡T\dim \ker T - \dim \operatorname{coker} T. The theorem then says that the connected components of Fred⁡(H)\operatorname{Fred}(H) are exactly the sets of operators of a given index.

The set [X,Fred⁡(H)][X, \operatorname{Fred}(H)] is a monoid under pointwise composition of operators, and ind⁡\operatorname{ind} turns composition into addition, so the bijection is an isomorphism of groups. Surjectivity is proved by realizing a class [E]−[ℂn][E] - [\C^n] through an explicit family of operators. Injectivity uses Kuiper's theorem, that the group of invertible operators on HH is contractible, to deform a family of index zero to a family of invertible operators.

The theorem says that Fred⁡(H)\operatorname{Fred}(H) is a classifying space for complex K-theory, homotopy equivalent to ℤ×BU\Z \times BU. It was proved independently by Atiyah and by Jänich.

Going deeper

For a student

The index bundle is the natural home for the index of a family of elliptic operators, as in the families index theorem. The theorem gives an analytic model of K-theory, which makes it possible to define twisted K-theory by twisting bundles of Fredholm operators. It also explains why index theory and K-theory are so closely linked.

Reading path

To understand this page from the ground up, read these 20 pages in order. The foundations come first.

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Compact space, concept, no evidence levelCompact spaceconceptComplex K-theory KU, concept, no evidence levelComplex K-theory KUconceptContractible space, concept, no evidence levelContractible spaceconceptFredholm index, concept, no evidence levelFredholm indexconceptFredholm operator, concept, no evidence levelFredholm operatorconceptGeneral linear group, object, no evidence levelGeneral linear groupobjectHilbert space, concept, no evidence levelHilbert spaceconceptHilbert space adjoint, concept, no evidence levelHilbert space adjointconceptHomotopy, concept, no evidence levelHomotopyconceptIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionOperator norm, concept, no evidence levelOperator normconceptUnitary operator, concept, no evidence levelUnitary operatorconceptAlgebra of bounded operators on a Hilbert space, object, no evidence levelAlgebra of bounded op…objectTranspose, definition, no evidence levelTransposedefinitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionUnitary matrix, definition, no evidence levelUnitary matrixdefinitionUnitary group, object, no evidence levelUnitary groupobjectKuiper's theorem, statement, no evidence levelKuiper's theoremstatementAtiyah–Jänich theorem, statement, no evidence levelAtiyah–Jänich theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Compact space concept
  2. Complex K-theory KU concept
  3. Contractible space concept
  4. Fredholm index concept
  5. Fredholm operator concept
  6. General linear group object
  7. Hilbert space concept
  8. Hilbert space adjoint concept
  9. Homotopy concept
  10. Identity matrix object
  11. Matrix definition
  12. Operator norm concept
  13. Unitary operator concept
  14. Algebra of bounded operators on a Hilbert space object
  15. Transpose definition
  16. Conjugate transpose definition
  17. Unitary matrix definition
  18. Unitary group object
  19. Kuiper's theorem statement
  20. Atiyah–Jänich theorem statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Fredholm_operator
About this page
TagH353HJA3E5
URIm2://H353HJA3E5
Kindstatement, theorem
WritingW0 stub
MSC19K56, 47A53, 55N15
StatusThis copy of the site has no build data. Run m2 status H353HJA3E5.
AboutFredholm operator, Complex K-theory KU
UsesFredholm operator, Fredholm index, Hilbert space, Operator norm, Complex K-theory KU, Kuiper's theorem, Homotopy, Compact space
Changed3 November 2026

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