One Number, Two Ways · Book XXVIII

Chapters 1–6 computed the index of one operator at a time and met it twice under other names: the winding number of a symbol, and the dimension count of Riemann–Roch. The Atiyah–Singer index theorem says they are the same kind of statement. The analytic index of an elliptic operator, a difference of kernel and cokernel dimensions, equals a topological index computed from the operator's symbol. This chapter states it from the sources and checks it in the one family where both sides can be computed by hand.

1 · The statement

Two indices and an equality

Blackadar opens the section on index theorems: the Atiyah–Singer theorem, “which asserts that the Fredholm index of an elliptic pseudodifferential operator can be calculated from purely topological data”, “is regarded as one of the great achievements of modern mathematics”. Zois, in the same tone, called the index theorems “arguably the most central result in mathematics during the second half of the 20th century”. CITED

For an elliptic operator D on a compact manifold M, “the analytic index of D is defined to be” Ind_a(D) = dim ker D − dim coker D. That is Chapter 2's number. If M is oriented, “the topological index of D is defined as” Ind_t(D) = ⟨τ_!(ch(σ_D)) ∪ Td(T*M ⊗ ℂ), [M]⟩. It is built only from the Chern character of the symbol σ_D and the Todd class of the cotangent bundle, so no analysis enters. Theorem 24.1.1 (Atiyah–Singer Index Theorem). If M is compact and oriented, then Ind_a(D) = Ind_t(D). CITED

2 · The two examples are this book's

Odd and even

Blackadar gives “Two special cases” that “illustrate the typical content of the Index Theorem in the odd- and even-dimensional cases respectively”. The first is the circle: with T_f the compression of multiplication by f to the Hardy space, “The Index Theorem then says that Ind_a(D_f) = −(winding number of f)”. That is Noether's 1921 theorem, Chapter 3. The second is a compact Riemann surface: “The Index Theorem in this case gives the Riemann–Roch Theorem”, Ind_a(∂̄) = d − g + 1, Chapter 4. CITED

The topological side of the second example is easy to check. For a line bundle L of degree d on a surface of genus g, ch(L) = 1 + d[pt], and the Todd class of the tangent bundle is 1 + c₁/2 with c₁[M] = 2 − 2g. The integral is d + (2 − 2g)/2 = d − g + 1. The script confirms this for g = 0 to 6 and d = −5 to 12. The circle needs no computation, because there is no Todd class to compute and the winding number is all there is. COMPUTED

3 · Where both sides can be computed

Line bundles on projective space

Dugger's book has a case where the analytic side is a table of cohomology groups and the topological side is a residue. On complex projective space ℂPⁿ, the line bundle O(k) has Euler characteristic χ = Σ(−1)ⁱ dim Hⁱ. His Table 27.2 gives the cohomology. For k ≥ 0 only H⁰ = S^kℂⁿ⁺¹ is non-zero, the homogeneous polynomials of degree k. For −n ≤ k < 0 there is no cohomology at all. For k < −n only Hⁿ = S^−k−n−1 survives, in degree n. He then notes: “We leave the reader to check that all three cases in the above formula can be unified into the simple statement” χ = C(n + k, n). CITED

The script builds the analytic side from that table, for n = 1 to 8 and k = −25 to 25. For n = 3 the values from k = −8 to 4 are: COMPUTED

k	−8	−7	−6	−5	−4	−3	−2	−1	0	1	2	3	4
χ(O(k))	−35	−20	−10	−4	−1	0	0	0	1	4	10	20	35

The topological side is Dugger's Proposition 27.4, “Td-num_X(E) = χ(X; E)”, where the Todd number is the coefficient of xⁿ in (x/(1 − e^−x))ⁿ⁺¹e^kx, a residue computation. The script expands the series in exact fractions. The Todd series starts 1 + x/2 + x²/12 + 0·x³ − x⁴/720. In all 8 × 51 cases the topological number is an integer, and it equals the analytic count. Both equal the polynomial (k+1)(k+2)…(k+n)/n!, which vanishes at k = −1, …, −n, so it explains the empty middle range. The values also satisfy the Serre-duality shape χ(O(−k−n−1)) = (−1)ⁿχ(O(k)). Two very different calculations, one counting monomials and one expanding a power series, give the same integer every time. COMPUTED

IndexTwoWays.card_monomials proves the counting under the analytic side: the number of monomials of degree d in n + 1 variables is C(n + d, n). choose_zero_between proves the vanishing in the middle range, and P1_count recovers ch 4's count k + 1 for the sphere. KERNEL-CHECKED

4 · What the theorem became

The same statement, with the numbers replaced

Blackadar then lists the generalisations, and each one changes what the index is allowed to be. The first difficulty is always “to make sense of the analytic index”, so “usually the analytic index is instead interpreted as an element of a certain K-group”, and a general index theorem says an analytically defined K-group element equals one “defined in purely topological terms from the symbol of the operator”. CITED

Families (24.1.3). For a family D_y of elliptic operators, “the index (both analytic and topological) takes values in K⁰(Y)”. It is the difference of two vector bundles, the situation of Chapter 5. CITED
Mishchenko–Fomenko (24.1.4). Replace the complex numbers by a C*-algebra B, and the index takes values in K₀(B). Blackadar: “The Atiyah–Singer Index Theorem is the case B = C, and the Index Theorem for Families is the case B = C₀(Y).” CITED
Equivariant (24.1.5). With a group acting, the index lands in the representation ring, or in K₀ of the group's C*-algebra (Kasparov; Connes–Moscovici). CITED
Foliations (24.1.6). “The longitudinal index theorem for foliations of Connes and Skandalis”, with values in the K-theory of the foliation's C*-algebra. CITED

Every item after the first replaces the integers by a richer group, and the number Ind = dim ker − dim coker of Chapter 1 becomes an element of K-theory. That is the direction of the reserved volume on noncommutative geometry (rung 33).

For the curious reader

The two examples run in opposite directions. In the circle case the analysis is the hard part, and the topology is one integer, the winding number. In the projective-space case the topology is the hard part (a residue of a power series), and the analysis is a table of counts. The theorem says the easy side of each equals the hard side of the other. Dugger's “We leave the reader to check”, said of the unification of three cases into one binomial coefficient, is the theorem in miniature. MODEL

What is not hereBlackadar's Theorem 24.1.1 for pseudodifferential operators on general manifolds is quoted, not proved, and no operator here is an elliptic operator on a manifold. The projective-space test uses Dugger's cohomology table, which is quoted, with the Todd side computed. Agreement for n ≤ 8 and |k| ≤ 25 is a test of that instance, not a proof of the theorem. Atiyah and Singer's own papers of 1968 are cited by Blackadar and not held. Mathlib at the pinned revision has none of the Todd class, the Chern character or an elliptic operator. OPEN

Sources

B. Blackadar, K-Theory for Operator Algebras, §24.1 (pp. 248–251; PDF 262–265), Theorem 24.1.1 and Examples 24.1.2. Held.
D. Dugger, A Geometric Introduction to K-Theory, §27 (Table 27.2, Proposition 27.4; PDF 214–216). Held.
I. P. Zois, 18 Lectures on K-Theory, arXiv:1008.1346, Part II intro, p. 56. Held.
M. F. Atiyah and I. M. Singer, The index of elliptic operators I–V (1968–1971), cited in Blackadar. Not held.