Fredholm Index and the Atiyah-Singer Index Theorem

Overview

A Fredholm operator is a bounded operator with finite-dimensional kernel and cokernel. Its Fredholm index ind(T) = dim ker(T) - dim coker(T) is a stable integer invariant: it is invariant under compact perturbations and continuous deformations. The Atiyah-Singer index theorem (1963) computes the analytical index of an elliptic differential operator on a compact manifold purely in terms of topological data, unifying analysis, topology, and geometry. It subsumes the Gauss-Bonnet, Hirzebruch signature, and Riemann-Roch theorems as special cases.

Intuition

The index measures the 'net number of solutions' of the equation Tu = f. If ind(T) > 0, the equation Tu = 0 has more solutions than Tu = f has obstructions (cokernel), giving a surplus of solutions. The remarkable fact is that this integer is stable: adding a compact operator or deforming T continuously does not change it. This allows topological methods to compute it -- the Atiyah-Singer theorem says the index equals a topological integral over the manifold, computed from the symbol of the operator.

Formal Definition

Definition

An operator T in B(X,Y) is Fredholm if ker(T) is finite-dimensional and range(T) is closed with finite-dimensional cokernel coker(T) = Y/range(T). The Fredholm index is ind(T) = dim ker(T) - dim coker(T). For an elliptic differential operator D on sections of vector bundles E and F over a compact manifold M, the analytical index equals the topological index computed from the symbol sigma(D): ind(D) = integral_M ch(sigma(D)) td(TM_C).

\mathrm{ind}(T) = \dim \ker(T) - \dim \mathrm{coker}(T)

Fredholm index

\mathrm{ind}(T + K) = \mathrm{ind}(T) \text{ for compact } K

Stability under compact perturbations

\mathrm{ind}(ST) = \mathrm{ind}(S) + \mathrm{ind}(T)

Index is a homomorphism

\mathrm{ind}(D) = \int_M \mathrm{ch}(\sigma(D)) \cdot \mathrm{td}(TM_{\mathbb{C}})

Atiyah-Singer index theorem

Notation

Notation	Meaning
$\mathrm{ind}(T)$	Fredholm index of operator T
$\Phi(X,Y)$	Set of Fredholm operators from X to Y
$\mathrm{ch}$	Chern character in cohomology
$\mathrm{td}$	Todd class of the complexified tangent bundle
$\sigma(D)$	Principal symbol of differential operator D

Theorems

Theorem 13.1: Fredholm Alternative

Let T = I + K where K is compact on a Banach space X. Then exactly one of the following holds: (i) Tu = 0 has only the trivial solution (hence T is bijective and T^{-1} is bounded); (ii) Tu = 0 has a finite-dimensional nontrivial solution space (and the equation Tu = f is solvable iff f is orthogonal to all solutions of T*v = 0).

Theorem 13.2: Stability of Fredholm Index

The Fredholm index ind: Phi(X,Y) -> Z is locally constant (continuous as a function into the integers) and satisfies ind(T + K) = ind(T) for every compact operator K. Moreover, ind(ST) = ind(S) + ind(T) when both S and T are Fredholm.

Theorem 13.3: Atiyah-Singer Index Theorem (Statement)

Let D: Gamma(E) -> Gamma(F) be an elliptic differential operator on sections of Hermitian vector bundles E, F over a compact Riemannian manifold M without boundary. Then the analytical index ind(D) = dim ker(D) - dim ker(D*) equals the topological index, an integer computed by integrating characteristic classes of E, F, and TM over M.

Theorem 13.4: Atkinson's Theorem

T in B(X,Y) is Fredholm if and only if it is invertible modulo compact operators: there exist S in B(Y,X) and compact operators K_1, K_2 such that ST = I - K_1 and TS = I - K_2. The set Phi(X) of Fredholm operators on X is an open subset of B(X) stable under compact perturbations.

Worked Examples

1
Compute the kernel: Tx = 0 means x_n = 0 for n >= 2, so x = (x_1, 0, 0, ...). Thus ker(T) = span{e_1}, dim ker(T) = 1.
2
Compute the range: T is surjective (for any (y_1, y_2, ...) in l^2, take x = (0, y_1, y_2, ...) which has Tx = y). So coker(T) = 0.
3
Index:
$\mathrm{ind}(T) = \dim \ker(T) - \dim \mathrm{coker}(T) = 1 - 0 = 1$

✓ Answer

The unilateral left shift has Fredholm index 1.

Practice Problems

Hardproof writing

Prove that the Fredholm index is stable under compact perturbations: ind(T + K) = ind(T) for T Fredholm and K compact.

Hardfree response

State one classical theorem that is a corollary of the Atiyah-Singer index theorem and explain the connection.

MediumMultiple choice

For the right shift operator R(x_1, x_2, ...) = (0, x_1, x_2, ...) on l^2, what is the Fredholm index?

ind(R) = 0ind(R) = 1ind(R) = -1R is not Fredholm

Common Mistakes

Common Mistake

Thinking the Fredholm index can be changed by adding small (non-compact) perturbations.

The index is stable under compact perturbations but NOT under arbitrary small perturbations. Adding a small non-compact operator can change the Fredholm index.

Common Mistake

Confusing Fredholm operators with invertible operators.

A Fredholm operator of index 0 is 'close to invertible' (invertible modulo compacts) but need not be invertible. Index 0 means dim ker = dim coker but both can be positive.

Quiz

The Fredholm index of an operator T is:

dim ker(T) + dim coker(T)dim ker(T) - dim coker(T)||T||The number of nonzero eigenvalues

The Atiyah-Singer index theorem computes the index of an elliptic operator as:

The number of eigenvalues of TThe spectral radius of TA topological integral of characteristic classes over the manifoldThe trace of T

Historical Background

Erik Ivar Fredholm studied integral equations around 1900, observing that certain operators behave like finite-rank perturbations of the identity. Frigyes Riesz developed the abstract Fredholm theory. The index as a topological invariant was recognised by Gelfand in 1960. Michael Atiyah and Isadore Singer proved their landmark index theorem in 1963, for which Singer received the Abel Prize in 2004 (Atiyah had received it in 2004 jointly for other work).

1900
Fredholm studies integral equations, founding Fredholm operator theory
Erik Ivar Fredholm
1960
Gelfand proposes that the index of elliptic operators is a topological invariant
Israel Gelfand
1963
Atiyah and Singer prove the index theorem for elliptic operators on compact manifolds
Michael Atiyah, Isadore Singer
1971
Atiyah, Patodi, and Singer extend the theorem to manifolds with boundary
Michael Atiyah, Vijay Patodi, Isadore Singer

Summary

A Fredholm operator has finite-dimensional kernel and cokernel; its index is dim ker - dim coker.
The index is stable under compact perturbations and continuous deformations (it is the connecting map K_1(Calkin algebra) -> Z).
Atkinson's theorem: T is Fredholm iff T is invertible modulo compact operators.
The Atiyah-Singer index theorem computes the analytical index of an elliptic differential operator as a topological integral.
Classical results -- Gauss-Bonnet, Riemann-Roch, Hirzebruch signature -- are special cases of the index theorem.

References

BookAtiyah, M. & Singer, I. — The Index of Elliptic Operators I-V, Annals of Mathematics, 1968-1971
BookLawson, H.B. & Michelsohn, M.-L. — Spin Geometry, Princeton University Press, 1989
WebsiteWikipedia: Atiyah-Singer index theorem