Bott Periodicity, the Thom Isomorphism & the Chern Character

Where we stand: a ring with no obvious way to compute it

Guide 3 left you holding something powerful and slightly mysterious. You took the vector bundles over a space X, added them formally with the Grothendieck construction so that subtraction makes sense, and got the abelian group K^0(X); tensor product made it a ring. You also met the reduced version K-tilde^0(X), which throws away the trivial bundles and measures only the genuine twisting. The construction is clean. The problem is that, as it stands, you have almost no way to compute K^0(X) for a space you actually care about — no long exact sequences you can turn, no analogue of the cell-by-cell bookkeeping that made K-theory feel as tractable as homology.

This guide hands you the three theorems that turn K-theory from a definition into a working tool. Bott periodicity says the higher K-groups repeat with period 2, so an infinite sequence collapses into two pieces. The Thom isomorphism relates the K-theory of a vector bundle's total space to that of its base, which is exactly what an index theorem needs to push a problem from a manifold up into its tangent bundle. And the Chern character is a ring isomorphism (after tensoring with the rationals) from K-theory to ordinary cohomology, letting you import everything you learned about de Rham and singular cohomology. Together they are the engine room of the next guide's index theorem.

Bott periodicity: the tower that only has two floors

First, where do higher K-groups even come from? You define K^{-n}(X) as the reduced K-theory of the n-fold suspension of X (with a basepoint), K^{-n}(X) = K-tilde^0(S^n X). Suspending raises dimension, so this builds a whole graded sequence ..., K^{-2}, K^{-1}, K^0 indexed by how many times you suspended. A priori these could all be different and the sequence could march off forever — exactly the infinite tower that would make K-theory hopeless to compute by hand. The whole subject would be unmanageable if you had to track infinitely many independent groups.

Bott periodicity is the astonishing rescue: for complex K-theory there is a natural isomorphism K^{-n}(X) ≅ K^{-n-2}(X). The tower has period two. Everything you could ever want lives in just K^0 and K^1; K^2 is K^0 again, K^3 is K^1 again, and so on in both directions. The cleanest way to see the source of the '2' is the generator: on the 2-sphere S^2 = CP^1 the reduced group K-tilde^0(S^2) is infinite cyclic, generated by the class [H] − 1, where H is the tautological line bundle. Periodicity is literally implemented by multiplying with this Bott class — tensoring by that one bundle over S^2 is an isomorphism shifting degree by 2.

Written out, the periodicity map is concrete: it sends a class a in K-tilde^0(X) to b ⊗ a in K-tilde^0(S^2 ∧ X), where b = [H] − 1 is the Bott class and ∧ is the smash product, and this map is an isomorphism. So instead of an endless list K^0, K^{-1}, K^{-2}, K^{-3}, ... you really only carry two groups, K^0 and K^1, and you cycle between them by tensoring with one fixed bundle on S^2. The payoff is that the exact sequences of K-theory, which a priori run off to infinity, fold into finite six-term cyclic sequences you can actually solve — the computational backbone we will use below.

The Thom space and the Thom isomorphism

The next theorem is the one that lets an index theorem move a problem from a manifold up to its tangent bundle. Start with the construction. Given a real or complex vector bundle E -> X, its Thom space Th(E) is formed by taking the disk bundle of E and collapsing its boundary sphere bundle to a single point — equivalently, the one-point compactification of the total space of E, fibrewise. Concretely: each fibre R^k becomes a sphere S^k by adding one shared point at infinity, and Th(E) is the result of doing this continuously over all of X. When X is a point, E is just R^k and its Thom space is the sphere S^k.

Now the theorem. The Thom isomorphism says: for a complex vector bundle E of rank k over a compact base X, the reduced K-theory of the Thom space is isomorphic to the K-theory of the base itself, K-tilde^0(Th(E)) ≅ K^0(X). The isomorphism is given by multiplication with a distinguished Thom class u_E, a canonical element of K-tilde^0(Th(E)) built from the exterior powers of E. You should hear the echo of Bott periodicity here, and it is not a coincidence: Bott periodicity is exactly the Thom isomorphism for the trivial line bundle over a point, whose Thom space is S^2. The general Thom isomorphism is Bott periodicity 'with a twist by E.'

Why does this matter so much? Because of the picture it draws. Cohomology classes on X tend to live in low degree; the Thom isomorphism takes a class on X and shifts it up by the rank of E, repackaging it as a class on the Thom space concentrated near the zero section. In ordinary cohomology this same machine produces the Euler class (pull the Thom class back along the zero section) and the Gysin pushforward that integrates over fibres. For K-theory it is the formal device that lets the Atiyah-Singer index theorem 'push' a symbol class from the tangent bundle down to a number on the base — the manoeuvre we will watch in guide 5.

The Chern character: a bridge to ordinary cohomology

K-theory and ordinary cohomology are two different bookkeeping systems for the same space, and the Chern character is the dictionary between them. You met it in the characteristic classes guide via the splitting principle: write a bundle's Chern roots as x_1, ..., x_k and set ch(E) = e^{x_1} + ... + e^{x_k}, a class living in the even-degree rational cohomology H^{even}(X; Q). Expanding the exponentials gives ch(E) = rank(E) + c_1(E) + (1/2)(c_1^2 − 2c_2)(E) + ..., a polynomial in the Chern classes with rational coefficients.

The reason it is the right bridge is its two algebraic properties, both proved straight from the splitting principle. The Chern character turns the K-theory ring operations into cohomology ring operations: it is additive on direct sums, ch(E ⊕ F) = ch(E) + ch(F), and multiplicative on tensor products, ch(E ⊗ F) = ch(E) · ch(F). Additivity is what lets it descend to the Grothendieck group (where E ⊕ F became a sum), and multiplicativity makes it respect the ring structure. In one sentence: ch is a ring homomorphism from K^0(X) into H^{even}(X; Q).

Here is the deep payoff, and it sharpens the role of K-theory honestly. The Chern character becomes an isomorphism of rings once you tensor K-theory with the rationals: ch: K^0(X) ⊗ Q ≅ H^{even}(X; Q), and likewise K^1 ⊗ Q matches the odd cohomology. So rationally, K-theory carries exactly the same information as ordinary cohomology — repackaged, but not richer. Where they genuinely differ is over the integers: K^0(X) and the integral cohomology H^*(X; Z) can have different torsion, and that integral discrepancy is the part of K-theory that ordinary cohomology cannot see. This is why it is wrong to call K-theory 'better' than cohomology; it is a different invariant, agreeing rationally and diverging only in torsion.

Chern character:   ch(E) = e^{x_1} + ... + e^{x_k}      x_i = Chern roots
   = rank(E) + c_1 + (1/2)(c_1^2 - 2 c_2) + (1/6)(c_1^3 - 3 c_1 c_2 + 3 c_3) + ...

   ch(E (+) F) = ch(E) + ch(F)         ch(E (x) F) = ch(E) . ch(F)     (ring homomorphism)

rational iso:   ch : K^0(X) (x) Q  ~=  H^even(X; Q)      (and K^1 (x) Q ~= H^odd)
   => K-theory and cohomology agree rationally; they can differ only in torsion

Putting it together: a tiny computation on the sphere

Let us run all three ideas on the simplest interesting space, the 2-sphere, and watch them agree. The goal is to compute K^0(S^2) and check it against cohomology through the Chern character. The same five moves recur whenever you compute K-theory of a low-dimensional space, so it is worth doing slowly.

1. Find the generators. Over S^2 = CP^1 the bundles are built from the tautological line bundle H. The reduced group K-tilde^0(S^2) is infinite cyclic, generated by the Bott class b = [H] − 1; the full group is K^0(S^2) = Z ⊕ Z, one Z for the rank (trivial bundles) and one Z for b.
2. Note the ring structure. The class b squares to zero, b^2 = ([H] − 1)^2 = 0, because b lives in the reduced K-theory of a 2-sphere and there is no room in higher degree on S^2. So K^0(S^2) is the truncated polynomial ring Z[b]/(b^2).
3. Compute the Chern character of the generator. The line bundle H has first Chern class c_1(H) = the generator g of H^2(S^2; Z), so ch(H) = e^{g} = 1 + g (higher terms vanish on S^2). Hence ch(b) = ch([H] − 1) = (1 + g) − 1 = g.
4. Match against cohomology. The even cohomology is H^0 ⊕ H^2 = Z ⊕ Z, with generators 1 and g. The Chern character sends rank to 1 and b to g, so it is an isomorphism K^0(S^2) ≅ H^even(S^2; Z) already over the integers here — S^2 has no torsion, so rational and integral agreement coincide.
5. Confirm periodicity. Multiplication by the Bott class b is exactly the periodicity isomorphism K-tilde^0(X) ≅ K-tilde^0(S^2 ∧ X); on a point it recovers K-tilde^0(S^2) = Z. The single generator you just found is, literally, the engine of Bott periodicity for the whole theory.

Step back and admire how tightly the three theorems interlock on this one example. The generator of K-tilde^0(S^2) is the Bott class (periodicity); it is also the Thom class of the trivial line bundle over a point (Thom isomorphism); and the Chern character carries it to the cohomology generator g (the bridge). One bundle, [H], simultaneously witnesses all three structures. That is not a coincidence of S^2 — it is why these three theorems are usually proved together, each leaning on the others.

What this unlocks, and the honest road to the index theorem

With these three theorems in hand, K-theory is finally a computational instrument. Bott periodicity reduces an infinite tower of groups to two, so the long exact sequences of pairs (which periodicity makes into six-term cyclic sequences) actually close up and can be solved. The Thom isomorphism lets you transport K-theory between a base and its bundles, the move that an index theorem makes constantly. And the Chern character ties everything back to the cohomology you already trust, converting K-theoretic statements into rational cohomological ones you can integrate. Each theorem removes one obstacle that, in guide 3, made K-theory feel like a definition with no calculus.

Here is the bridge to guide 5, stated honestly. The Atiyah-Singer index theorem computes the analytic index of an elliptic operator — the difference of dimensions of its kernel and cokernel, an integer measuring how many solutions an equation has — purely topologically. The recipe uses every theorem of this guide: the operator's symbol defines a class in the K-theory of the tangent bundle; the Thom isomorphism pushes that class down to the manifold; the Chern character converts it to a cohomology integral; and Bott periodicity is what guarantees the symbol class lives in the right group to begin with. The index is then the integral over M of ch(symbol) · Td(M), a cohomology pairing.

Two honest closings. First on scope: we have stated and motivated Bott periodicity and the Thom isomorphism and led with the one fully worked example, S^2, because a single understood case outweighs a maximally general theorem you cannot picture — but a real proof of either is a course, and the index theorem in guide 5 will be a survey, not a derivation. Second on humility: the Chern character is an isomorphism only after tensoring with Q, so do not believe the slogan that 'K-theory is just cohomology' — the integral torsion they disagree on is precisely where K-theory earns its keep, and is exactly what lets it detect division-algebra and vector-field phenomena that rational cohomology is blind to. Carry the precise hypotheses, not the slogans, into the final guide.