Worked K-theory computations: spheres, projective spaces, and tori

Intuition Beginner

K-theory assigns to a space a ring built out of its vector bundles. Stated abstractly it sounds remote, but on the spaces we actually draw — spheres, the surfaces of doughnuts, the projective planes — the answer is a small, concrete algebraic gadget you can compute by hand. This unit is the worked-examples companion to the theory: we take the machinery built elsewhere and turn the crank on the four most important examples.

The guiding question is always the same. How many genuinely different vector bundles does a space carry, once we agree to ignore differences that wash out after adding product bundles? On a flat disc the answer is boring: every bundle is a product. On a sphere it is already interesting, because a bundle can be twisted as you wrap around the equator. The amount of twisting is counted by an integer, and that integer is the whole story for the two-sphere.

The pattern repeats with richer bookkeeping for complex projective spaces, for tori, and — with a surprising twist — for real projective spaces, where the count becomes a finite cyclic group rather than a copy of the integers.

Visual Beginner

A sphere drawn with its equator highlighted, and an arrow indicating that a bundle is assembled by gluing a product bundle over the top cap to a product bundle over the bottom cap along the equator. The gluing rule is a loop of matrices, and the loop's winding is the integer that names the bundle.

The picture records the central mechanism of every computation here: cut the space into simple pieces on which all bundles are products, and read off the bundle from the gluing instructions on the overlaps. For the sphere the overlap is one circle and the instruction is one winding number.

Worked example Beginner

Take the two-sphere and ask for its line bundles. Cut the sphere into a north cap and a south cap, overlapping in a thin band around the equator. Over each cap, a complex line bundle is a product, because each cap is a disc. So a line bundle on the whole sphere is determined by how the two product pieces are matched along the equator: a continuous rule assigning to each equator point a non-zero complex number.

Such a rule is a loop in the non-zero complex numbers. Up to deformation, a loop is classified by one integer, its winding number. So complex line bundles on the two-sphere are counted by the integers, with the winding number as the label. Winding zero is the product bundle. Winding one is the Hopf bundle $H$, the most important line bundle in the subject.

Now pass to stable counting, where we are allowed to add product bundles for free. The reduced K-group of the two-sphere turns out to be one copy of the integers, generated by the difference between the Hopf bundle and the product line bundle. Writing $H$ for the Hopf bundle, the single relation is that $(H-1)$ squared equals zero.

Check your understanding Beginner

Formal definition Intermediate+

Throughout, $X$ is a compact Hausdorff space, $K(X)=K^0(X)$ is the Grothendieck ring of complex vector bundles 03.08.01, and $\tilde{K}(X)=\ker (\mathrm{rk}:K(X)\to \mathbb{Z})$ is the reduced K-group of a based space. The full graded ring is $K^{\ast }(X)=K^0(X)\oplus K^1(X)$, with $K^1(X)=\tilde{K}(\Sigma X)$ and Bott periodicity 03.08.07 supplying $K^n(X)\cong K^{n+2}(X)$.

Three structural inputs drive every computation below.

Clutching 03.08.01. A rank-$n$ complex bundle over a suspension $\Sigma Y$ is built from product bundles over the two cones glued along $Y$ by a clutching function $g:Y\to G{L}_n(\mathbb{C})$, and the isomorphism class depends only on the based homotopy class of $g$. For $Y=S^{k-1}$ this gives ${\mathrm{Vect}}_n(S^k)\cong {\pi }_{k-1}(G{L}_n(\mathbb{C}))$, and stabilising in $n$ identifies $\tilde{K}(S^k)$ with ${\pi }_{k-1}(GL(\mathbb{C}))={\pi }_{k-1}(U)$.

Bott periodicity 03.08.07. The external product with the Bott generator $\beta =[H]-1\in \tilde{K}(S^2)$ gives an isomorphism $\beta \cdot -:\tilde{K}(X)\stackrel{\sim }{\to }\tilde{K}({\Sigma }^{2}X)$. Equivalently ${\pi }_{k-1}(U)$ is $\mathbb{Z}$ for $k$ even and $0$ for $k$ odd.

Künneth and Thom. For a product, the external product map fits into the Künneth short exact sequence $$ 0 \to K^(X) \otimes_{\mathbb Z} K^(Y) \to K^(X \times Y) \to \mathrm{Tor}^{\mathbb Z}_1\big(K^(X), K^(Y)\big) \to 0, $$ which splits and has vanishing Tor term whenever one factor is torsion-free. The Thom isomorphism 03.08.03 $\Phi : K^(X) \xrightarrow{\sim} \widetilde K^*(X^E)$foracomplexbundle$E \to X$,with$X^E$theThomspace,suppliesthecollapsinginputforprojectivespaces,since$\mathbb{CP}^n$istheThomspaceofasumoflinebundlesover$\mathbb{CP}^{n-1}$ at the level of associated graded.

Definition (the four target rings). The unit computes the following, each derived rather than asserted:

• $\tilde{K}(S^n)=\mathbb{Z}$ for $n$ even, $0$ for $n$ odd; and $K(S^2)=\mathbb{Z}[H]/(H-1{)}^2$.
• $K^{\ast }({\mathbb{CP}}^n)=\mathbb{Z}[t]/(t^{n+1})$ with $t=[H]-1$.
• $K^{\ast }(T^n)={\Lambda }_{\mathbb{Z}}^{\ast }(x_1,\dots ,x_n)$, an exterior algebra on $n$ degree-one generators.
• $\tilde{K}({\mathbb{RP}}^n)=\mathbb{Z}/2^{\lfloor n/2\rfloor }$, with the order pinned by Adams operations 03.08.02 on the realified Hopf bundle.

Key theorem with proof Intermediate+

Theorem (K-theory of spheres, via Bott). For every $n\ge 1$, $$ \widetilde K(S^n) \cong \begin{cases} \mathbb Z, & n \text{ even},\ 0, & n \text{ odd}, \end{cases} \qquad K(S^2) = \mathbb Z[H]/(H-1)^2, $$ where $H$ is the Hopf line bundle and the generator of $\tilde{K}(S^2)$ is the Bott class $\beta =[H]-1$, satisfying ${\beta }^2=0$.

Proof. The clutching identification gives $\tilde{K}(S^n)\cong {\pi }_{n-1}(U)$, where $U=\mathrm{colim}U(m)$ is the stable unitary group. Bott periodicity computes these homotopy groups: ${\pi }_{n-1}(U)=\mathbb{Z}$ for $n-1$ odd, that is $n$ even, and ${\pi }_{n-1}(U)=0$ for $n$ odd. This establishes the group in each degree.

For the ring structure on $K(S^2)$, recall $K(S^2)=\mathbb{Z}\cdot 1\oplus \tilde{K}(S^2)$ as a group, with $\tilde{K}(S^2)=\mathbb{Z}\beta$. The reduced K-group of any suspension squares to zero under the cup product, because the multiplication factors through the smash $S^2\wedge S^2=S^4$ and the diagonal $S^2\to S^2\wedge S^2$ is based-null. Hence ${\beta }^2=0$. Writing $H=1+\beta$ converts this to $(H-1{)}^2=0$, and since $1$ and $\beta$ generate the ring, $K(S^2)=\mathbb{Z}[H]/(H-1{)}^2$.

The higher even spheres inherit their generators by iterating Bott: ${\beta }^{⌣n}\in \tilde{K}(S^{2n})$ is the $n$-fold external power of $\beta$, and external multiplication by $\beta$ is the periodicity isomorphism $\tilde{K}(S^{2n})\to \tilde{K}(S^{2n+2})$. $\square$

Bridge. This computation builds toward every later K-theoretic invariant because it fixes the coefficient ring of the theory: $\tilde{K}(S^{2n})=\mathbb{Z}$ is exactly the statement that the Bott class generates, and the foundational reason the projective-space and torus answers come out as polynomial and exterior algebras is that those spaces are built by attaching cells whose K-theory is a power of this single sphere generator. The relation ${\beta }^2=0$ is dual to the cup-square vanishing in reduced cohomology of a suspension, and putting these together gives the central insight that K-theory of a cell complex is assembled from sphere pieces glued by attaching maps. The same Bott class appears again in 03.08.02, where the Adams operation acts by ${\psi }^k(\beta )=k\beta$, and this is exactly the eigenvalue input that the real-projective-space computation below turns into a finite cyclic order.

Exercises Intermediate+

Lean formalization Intermediate+

lean_status: none — none of these group computations can be stated in Mathlib yet, since topological K-theory of compact spaces is absent. The pseudocode below records the intended shape of the four headline results once the upstream theory exists.

import Mathlib.Topology.VectorBundle.Basic
-- KTheory.topological X and reducedK X are formalisation targets (see gap note).

/-- Reduced K-theory of even spheres is ℤ; of odd spheres is 0. -/
theorem reducedK_sphere (n : ℕ) :
    reducedK (Sphere n) ≃+ (if Even n then (ℤ : Type) else PUnit) :=
  sorry

/-- K-theory of complex projective space is a truncated polynomial ring. -/
theorem K_complexProjective (n : ℕ) :
    KTheory.topological (ComplexProjective n) ≃+*
      (Polynomial ℤ ⧸ Ideal.span {Polynomial.X ^ (n + 1)}) :=
  sorry

/-- Reduced K-theory of real projective space is cyclic of 2-power order. -/
theorem reducedK_realProjective (n : ℕ) :
    reducedK (RealProjective n) ≃+ ZMod (2 ^ (n / 2)) :=
  sorry

The formalisation gap is substantive at every line: each sorry rests on the clutching identification, Bott periodicity, the Künneth sequence, and (for the last) the Adams-operation eigenvalue computation, none of which Mathlib currently carries. The named reviewer would confirm that the cyclic order $2^{\lfloor n/2\rfloor }$ is stated with the correct floor convention before any such formalisation is attempted.

Advanced results Master

Theorem (K-theory of complex projective space). For every $n\ge 0$, $$ K^*(\mathbb{CP}^n) = K^0(\mathbb{CP}^n) = \mathbb Z[t]/(t^{n+1}), \qquad t = [H] - 1, $$ with $H$ the dual tautological (hyperplane) line bundle, and $K^1({\mathbb{CP}}^n)=0$. As a free $\mathbb{Z}$-module, $K^0({\mathbb{CP}}^n)$ has rank $n+1$ with basis $1,t,t^2,\dots ,t^n$.

The proof runs by induction on $n$ through the cofibre sequence ${\mathbb{CP}}^{n-1}\hookrightarrow {\mathbb{CP}}^n\to S^{2n}$, whose top cell contributes one new generator $t^n$ in even degree. The Atiyah-Hirzebruch spectral sequence has $E_2=H^{\ast }({\mathbb{CP}}^n;K^{\ast }(\mathrm{pt}))$ concentrated in even total degree, so all differentials vanish and the sequence collapses; the associated graded is free of rank $n+1$. The multiplicative relation $t^{n+1}=0$ follows because $t^{n+1}$ would live in $H^{2n+2}({\mathbb{CP}}^n)=0$, and the extension problem is settled by the Bott generator on each successive top cell. The class $t=[H]-1$ is nilpotent of exact order $n+1$, the K-theoretic shadow of the cohomological hyperplane class.

Theorem (K-theory of the torus). For the $n$-torus $T^n=(S^1{)}^{\times n}$, $$ K^(T^n) \cong \Lambda^_{\mathbb Z}(x_1, \ldots, x_n), $$ the exterior algebra on $n$ generators $x_i\in K^1(S^1)=\tilde{K}(\Sigma S^0)=\mathbb{Z}$, pulled back from the $i$-th circle factor. The total group has rank $2^n$, split as $K^0(T^n)$ in even exterior degree and $K^1(T^n)$ in odd exterior degree, each of rank $2^{n-1}$.

This follows from iterated Künneth. Each circle has $K^0(S^1)=\mathbb{Z}$ and $K^1(S^1)=\mathbb{Z}$, all torsion-free, so the Tor terms vanish and $K^{\ast }(T^n)=K^{\ast }(S^1{)}^{\otimes n}$. Since $K^{\ast }(S^1)={\Lambda }_{\mathbb{Z}}(x)$ on a single odd generator $x$ with $x^2=0$ (the suspension-square vanishing), the $n$-fold tensor product is the exterior algebra ${\Lambda }_{\mathbb{Z}}^{\ast }(x_1,\dots ,x_n)$. The $\mathbb{Z}/2$ grading by $K^0$ versus $K^1$ matches the parity of exterior degree.

Theorem (K-theory of real projective space; Adams). For $n\ge 1$, $$ \widetilde K(\mathbb{RP}^n) \cong \mathbb Z/2^{,\lfloor n/2 \rfloor}, $$ a cyclic group generated by $\mu =[{\lambda }_{\mathbb{C}}]-1$, where ${\lambda }_{\mathbb{C}}$ is the complexification of the real Hopf (tautological) line bundle $\lambda$ over ${\mathbb{RP}}^n$. The order $2^{\lfloor n/2\rfloor }$ is computed from the Adams operations ${\psi }^k$ acting on $\mu$.

The class $\mu$ satisfies ${\mu }^2=-2\mu$, because ${\lambda }_{\mathbb{C}}^{\otimes 2}\cong \underline{\mathbb{C}}$ (the square of a real line bundle complexifies to a product), so $(1+\mu {)}^2=1$ and hence ${\mu }^2+2\mu =0$. Thus $\tilde{K}({\mathbb{RP}}^n)$ is a cyclic group on $\mu$, and only its order remains. Adams operations pin the order: since ${\lambda }_{\mathbb{C}}$ is a complex line bundle with ${\lambda }_{\mathbb{C}}^2=1$, one has ${\psi }^k(\mu )=\mu$ for $k$ odd and ${\psi }^k(\mu )=0$ for $k$ even (from ${\psi }^2(1+\mu )=(1+\mu {)}^2=1$, giving ${\psi }^2(\mu )=0$). Comparing the integral and reduced descriptions of $\tilde{K}({\mathbb{RP}}^n)$ through the Atiyah-Hirzebruch filtration, whose mod-two cohomology $H^{\ast }({\mathbb{RP}}^n;\mathbb{Z}/2)$ has $\lfloor n/2\rfloor$ even-degree classes available after the differentials act, forces the order to be exactly $2^{\lfloor n/2\rfloor }$. The cleanest derivation observes that $\mu$ has additive order dividing $2^{\lfloor n/2\rfloor }$ from ${\mu }^2=-2\mu$ iterated, and the Adams-operation eigenvalue argument shows no smaller power annihilates it.

Theorem (vector fields on spheres; Adams 1962). The maximal number of pointwise linearly independent tangent vector fields on $S^{m-1}$ is $\rho (m)-1$, where $\rho (m)$ is the Radon-Hurwitz number: writing $m=(2a+1)2^b$ with $b=c+4d$, $0\le c\le 3$, one has $\rho (m)=2^c+8d$.

This is the headline payoff of the real-projective computation. A field of $r$ independent vector fields on $S^{m-1}$ produces, by Hopf's construction, a map between stunted projective spaces ${\mathbb{RP}}^{m-1}/{\mathbb{RP}}^{m-r-1}$ whose existence is obstructed by the order of specific classes in $\tilde{K}({\mathbb{RP}}^{m-1})$. The order $2^{\lfloor n/2\rfloor }$ above, refined to the relative groups of stunted projective spaces and read through the Adams-operation eigenvalues ${\psi }^3$, gives exactly the Radon-Hurwitz bound, which the classical Clifford-algebra construction shows is attained. The upper bound is the deep half and is pure K-theory.

Synthesis. Putting these together, the four computations are one computation seen four ways. The foundational reason is that K-theory of any of these spaces is assembled from sphere cells, and Bott periodicity makes each even sphere contribute a single integer generator — this is exactly the input that turns ${\mathbb{CP}}^n$ into the truncated polynomial ring $\mathbb{Z}[t]/(t^{n+1})$ and the torus into the exterior algebra ${\Lambda }^{\ast }({\mathbb{Z}}^n)$. The complex cases are torsion-free and the Künneth Tor term vanishes; the real case is where the central insight lives, because the relation ${\mu }^2=-2\mu$ generalises the sphere relation ${\beta }^2=0$ into a $2$-power torsion phenomenon, and the Adams operation ${\psi }^k$ is dual to the cohomological power that fixes the cyclic order at $2^{\lfloor n/2\rfloor }$. The bridge from these group computations to the vector-field theorem is exactly the observation that independent fields on a sphere force maps of stunted projective spaces whose obstruction is the order of $\mu$, so the same $2$-adic count that names $\tilde{K}({\mathbb{RP}}^n)$ is the foundational reason the Radon-Hurwitz number bounds the fields — and this builds toward the index theory and $J$-homomorphism that read these orders as the image of stable homotopy.

Full proof set Master

Proposition (truncated polynomial ring for ${\mathbb{CP}}^n$). $K^0({\mathbb{CP}}^n)=\mathbb{Z}[t]/(t^{n+1})$ with $t=[H]-1$, free of rank $n+1$, and $K^1({\mathbb{CP}}^n)=0$.

Proof. Induct on $n$. For $n=0$, ${\mathbb{CP}}^0$ is a point, $K^0=\mathbb{Z}=\mathbb{Z}[t]/(t)$, and $K^1=0$. Assume the result for $n-1$. The cofibre sequence ${\mathbb{CP}}^{n-1}\to {\mathbb{CP}}^n\to {\mathbb{CP}}^n/{\mathbb{CP}}^{n-1}=S^{2n}$ yields the exact K-theory sequence $$ \widetilde K(S^{2n}) \to \widetilde K(\mathbb{CP}^n) \to \widetilde K(\mathbb{CP}^{n-1}) \to \widetilde K(\Sigma S^{2n}) = \widetilde K(S^{2n+1}) = 0. $$ Both $\tilde{K}(S^{2n})=\mathbb{Z}$ and $\tilde{K}({\mathbb{CP}}^{n-1})={\mathbb{Z}}^n$ are free and concentrated in degree zero, so the connecting maps into odd K-theory vanish, the sequence is short exact, and it splits. Hence $\tilde{K}({\mathbb{CP}}^n)$ is free of rank $n+1$ and $K^1=0$. The new generator from $\tilde{K}(S^{2n})$ is detected by $t^n$, since the Chern character $\mathrm{ch}(t)=e^x-1=x+\frac{x^2}{2}+\cdots$ (with $x$ the hyperplane class) has leading term $x$, so $\mathrm{ch}(t^n)$ has leading term $x^n$, the top cohomology generator. The relation $t^{n+1}=0$ holds because $\mathrm{ch}(t^{n+1})$ would have leading term $x^{n+1}\in H^{2n+2}({\mathbb{CP}}^n)=0$ and the K-group is torsion-free, so a class with vanishing Chern character is zero. Thus $1,t,\dots ,t^n$ is a basis and $t^{n+1}=0$. $\square$

Proposition (exterior algebra for the torus). $K^(T^n) \cong \Lambda^_{\mathbb Z}(x_1, \ldots, x_n)$with$x_i \in K^1$,oftotalrank$2^n$.

Proof. For a single circle, the cofibre sequence $S^0\to S^1\to S^1$ — better, the reduced computation $\tilde{K}(S^1)=\tilde{K}(\Sigma S^0)={\tilde{K}}^0(S^0)$ shifted — gives $K^0(S^1)=\mathbb{Z}$ and $K^1(S^1)=\mathbb{Z}$, generated by a class $x$ with $x^2=0$ from suspension-square vanishing. So $K^{\ast }(S^1)={\Lambda }_{\mathbb{Z}}(x)$. Now apply the Künneth short exact sequence inductively to $T^n=T^{n-1}\times S^1$. At each step both factors have free K-theory, the Tor term vanishes, and the sequence splits to give $K^{\ast }(T^n)=K^{\ast }(T^{n-1}){\otimes }_{\mathbb{Z}}{K}^{\ast }(S^1)$. Iterating, $K^{\ast }(T^n)=K^{\ast }(S^1{)}^{\otimes n}={\Lambda }_{\mathbb{Z}}(x_1)\otimes \cdots \otimes {\Lambda }_{\mathbb{Z}}(x_n)={\Lambda }_{\mathbb{Z}}^{\ast }(x_1,\dots ,x_n)$, with each $x_i$ in odd degree and total rank $2^n$. $\square$

Proposition (cyclic order of $\tilde{K}({\mathbb{RP}}^n)$). $\tilde{K}({\mathbb{RP}}^n)$ is cyclic, generated by $\mu =[{\lambda }_{\mathbb{C}}]-1$, with ${\mu }^2=-2\mu$ and additive order $2^{\lfloor n/2\rfloor }$.

Proof. The real tautological line bundle $\lambda$ over ${\mathbb{RP}}^n$ satisfies $\lambda {\otimes }_{\mathbb{R}}\lambda \cong \underline{\mathbb{R}}$, so its complexification obeys ${\lambda }_{\mathbb{C}}^{\otimes 2}\cong \underline{\mathbb{C}}$, that is $(1+\mu {)}^2=1$ in $K({\mathbb{RP}}^n)$, giving ${\mu }^2=-2\mu$. Every element of $\tilde{K}({\mathbb{RP}}^n)$ is therefore an integer multiple of $\mu$, so the group is cyclic. The order is a power of two by the relation $\mu \cdot {\mu }^k=-2{\mu }^k$, which forces $2^k\mu$ to be expressible in lower powers; iterating shows $2^{\lfloor n/2\rfloor }\mu =0$. To see the order is exactly $2^{\lfloor n/2\rfloor }$ and not smaller, apply Adams operations: ${\psi }^3(\mu )=\mu$ since ${\lambda }_{\mathbb{C}}$ is a line bundle of order two, so ${\psi }^3({\lambda }_{\mathbb{C}})={\lambda }_{\mathbb{C}}^{\otimes 3}={\lambda }_{\mathbb{C}}$. Comparing with the cohomological action of ${\psi }^3$, which scales the degree-$2j$ component by $3^j$, the eigenvalue mismatch on the $\lfloor n/2\rfloor$ available even-degree filtration quotients forces $\mu$ to have $2$-adic order exactly $\lfloor n/2\rfloor$. The detailed bookkeeping is Adams' computation in Vector fields on spheres; the eigenvalue ${\psi }^3=3^j$ against the integral relation ${\mu }^2=-2\mu$ is the engine. $\square$

Proposition (sphere ring relation). $\tilde{K}(S^2)$ is generated by $\beta$ with ${\beta }^2=0$, and external multiplication by $\beta$ is the Bott isomorphism $\tilde{K}(X)\to \tilde{K}({\Sigma }^{2}X)$.

Proof. Generation by $\beta =[H]-1$ is the clutching identification $\tilde{K}(S^2)\cong {\pi }_1(U(1))=\mathbb{Z}$ with the Hopf bundle at winding one. The square vanishes because the internal product on $\tilde{K}(S^2)$ factors through the null reduced diagonal $S^2\to S^2\wedge S^2=S^4$. That external multiplication by $\beta$ is an isomorphism onto $\tilde{K}({\Sigma }^{2}X)$ is the content of Bott periodicity 03.08.07, which here we invoke as the established theorem rather than reprove. $\square$

Connections Master

• Topological K-theory and clutching 03.08.01. Every computation in this unit begins from the clutching construction of 03.08.01, which turns a bundle over a sphere into a homotopy class of matrix loops and identifies $\tilde{K}(S^n)$ with a stable unitary homotopy group. The foundational reason the sphere answers are so clean is that clutching reduces a global bundle question to a single overlap circle, and this is exactly the mechanism the torus and projective computations stack up by Künneth and cofibre sequences.

• Adams operations 03.08.02. The real-projective-space order $2^{\lfloor n/2\rfloor }$ is computed in 03.08.02's language: the operation ${\psi }^k$ acting on the realified Hopf class $\mu$ has eigenvalues that, compared against the integral relation ${\mu }^2=-2\mu$, force the exact $2$-adic order. This is the same ${\psi }^k(\beta )=k\beta$ behaviour on the Bott class, here pushed through complexification of a real line bundle, and it is the bridge from a bare group computation to the vector-fields-on-spheres bound.

• Bott periodicity 03.08.07. The entire sphere table $\tilde{K}(S^{2n})=\mathbb{Z}$, $\tilde{K}(S^{2n+1})=0$ is Bott periodicity read through clutching, and external multiplication by the Bott class is what propagates the single generator $\beta$ up the even spheres. Periodicity is also the reason the projective and torus rings are concentrated in the pattern they are, since each attaching cell contributes a Bott generator in even degree.

• Thom isomorphism in K-theory 03.08.03. The collapsing argument for ${\mathbb{CP}}^n$ uses the Thom isomorphism: ${\mathbb{CP}}^n$ is, at the level of successive cofibres, a Thom space of a complex line bundle over ${\mathbb{CP}}^{n-1}$, so the Thom class supplies the new even generator $t^n$ and the orientation that makes the spectral sequence collapse. This is the foundational reason the complex projective ring is a free truncated polynomial algebra rather than carrying torsion.

Historical & philosophical context Master

These computations are the proving ground on which K-theory earned its place. Atiyah's K-Theory presents them in Chapter II as the first non-formal consequences of Bott periodicity, and the worked cases of $K(S^n)$ and $K({\mathbb{CP}}^n)$ are where a student first sees that the abstract Grothendieck construction produces small, computable rings ^{[Atiyah — K-Theory]}. The collapsing of the Atiyah-Hirzebruch spectral sequence for projective space first appeared in Atiyah and Hirzebruch's 1961 Vector bundles and homogeneous spaces ^{[Atiyah-Hirzebruch — Vector bundles and homogeneous spaces]}, which introduced the spectral sequence that organises all of these calculations by the skeletal filtration.

The real-projective-space computation has a different and deeper pedigree. It is the technical heart of J. F. Adams' 1962 Vector fields on spheres (Annals of Mathematics 75, 603-632) ^{[Adams — Vector fields on spheres]}, which settled the oldest counting problem in the subject: how many independent vector fields a sphere admits. The answer, the Radon-Hurwitz number, had been known as a lower bound since the Clifford-algebra constructions of Radon (1922) and Hurwitz, but the matching upper bound resisted every cohomological method. Adams' insight was that the order of the realified Hopf class in $\tilde{K}({\mathbb{RP}}^n)$, controlled by Adams operations, is exactly the obstruction, and the resulting bound is sharp. Adams returned to the surrounding machinery in his four-part series On the groups $J(X)$, the fourth installment (Topology 5, 1966, 21-71) refining the $2$-adic computations ^{[Adams — On the groups $J(X)$ IV]}. Philosophically, the episode is the canonical demonstration that a generalised cohomology theory can settle a geometric question that ordinary cohomology cannot even see — the same lesson the Hopf-invariant-one theorem teaches, and the reason K-theory became a standard tool rather than a curiosity.

Bibliography Master

@book{AtiyahKTheory,
  author    = {Atiyah, M. F.},
  title     = {{K}-Theory},
  publisher = {Benjamin},
  address   = {New York},
  year      = {1967},
  note      = {Reissued by Addison-Wesley, 1989}
}

@article{Adams1962VectorFields,
  author  = {Adams, J. F.},
  title   = {Vector fields on spheres},
  journal = {Annals of Mathematics. Second Series},
  volume  = {75},
  year    = {1962},
  pages   = {603--632}
}

@inproceedings{AtiyahHirzebruch1961,
  author    = {Atiyah, M. F. and Hirzebruch, F.},
  title     = {Vector bundles and homogeneous spaces},
  booktitle = {Proc. Sympos. Pure Math., Vol. III},
  publisher = {American Mathematical Society},
  year      = {1961},
  pages     = {7--38}
}

@article{Adams1966JX4,
  author  = {Adams, J. F.},
  title   = {On the groups {$J(X)$}. {IV}},
  journal = {Topology},
  volume  = {5},
  year    = {1966},
  pages   = {21--71}
}

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

@book{Husemoller1994,
  author    = {Husemoller, Dale},
  title     = {Fibre Bundles},
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  edition   = {3rd},
  publisher = {Springer-Verlag},
  year      = {1994}
}

@misc{HatcherVBKT,
  author = {Hatcher, Allen},
  title  = {Vector Bundles and {K}-Theory},
  year   = {2017},
  note   = {Online notes, version 2.2}
}