Atiyah — *K-Theory* (Fast Track 3.10) — Audit + Gap Plan

Book: Michael F. Atiyah, K-Theory, lectures Fall 1964 (Harvard), notes by D. W. Anderson. First edition W. A. Benjamin Inc., New York / Amsterdam 1967, vii + 166 pp. of body + appendix + two reprinted papers (Power operations in K-theory; K-theory and reality). Reprinted with the same pagination as Advanced Book Classics, Addison-Wesley / Westview Press 1989, ISBN 0-201-09394-4. Citations in Codex units already point at "Atiyah — K-Theory §1", "§1.4", "§1.1", "§2"; this plan aligns those pointers to the true table of contents (verified against the Ranicki-archive scan, https://webhomes.maths.ed.ac.uk/~v1ranick/papers/atiyahk.pdf).

Fast Track entry: 3.10. Listed in docs/catalogs/FASTTRACK_BOOKLIST.md line 97 with marker BUY (no free host) but a complete scan is in fact freely circulated on the Ranicki archive; sourcing note in §7 below. Cross-listed in docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md line 269 as a §3 supplement.

Purpose of this plan: lightweight P1 audit-and-gap pass mirroring plans/fasttrack/brown-higgins-sivera-nonabelian-algebraic-topology.md. Atiyah's monograph is short (166 body pp.), terse, and computational, so this pass is closer to a real P1 audit than the Brown-Higgins-Sivera one — every chapter and section is enumerated, and every named theorem on the Bott-periodicity / operations / Adams spine is mapped. The output is a punch-list of new or extended units required so that K-Theory is covered to the equivalence threshold (≥95% effective coverage of theorems, key examples, exercise pack, notation, sequencing, intuition, applications — see docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4).

The Codex chapter content/03-modern-geometry/08-k-theory/ currently ships 5 units (not 3 as in the briefing): 03.08.01-topological-k-theory, 03.08.04-classifying-space, 03.08.05-universal-bundle, 03.08.06-stable-homotopy, 03.08.07-bott-periodicity. The numbering gap (no .02, .03) suggests two reserved slots from an earlier plan; this audit treats them as available for the new units below.

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§1 What Atiyah's K-Theory is for

Atiyah's K-Theory is the canonical small monograph on topological K-theory of complex vector bundles over compact Hausdorff spaces. It is the originator text for the topological side of the subject (Atiyah- Hirzebruch 1959–61 grew out of these lectures), and it is the single shortest place that contains the complete arc from "what is a vector bundle" to the Adams operations ${\psi }^k$, the splitting principle in K-theory, the Thom isomorphism in K-theory, and the $J(X)$ groups.

Distinctive contributions, in the order Atiyah develops them:

1. Vector bundles via continuous functors. Ch. I §1.2 packages $\oplus$, $\otimes$, ${\wedge }^k$, $\mathrm{Hom}$, dualization etc. as instances of a single construction: a continuous functor on finite- dimensional vector spaces extends fibrewise to a continuous functor on the category of vector bundles. This is the cleanest treatment in the literature and is exactly the functoriality the Codex unit 03.05.02 should anchor against.
2. Compactness as the load-bearing hypothesis. §1.4 (Vector bundles on compact spaces) proves the homotopy invariance, sub-bundle complements via partition of unity, and the existence of a stable inverse — the four "compactness lemmas" on which everything else stands. Atiyah's terse account here is the canonical one.
3. Equivariant K-theory $K_G(X)$ from the start. §1.6 (G-bundles over G-spaces) introduces the equivariant theory in parallel with the non-equivariant one, and §2.3 packages $K_G(X)$ as a ring with the representation ring $R(G)=K_G(\mathrm{pt})$ acting on it. The parallel development is Atiyah's signature pedagogical move and is absent from the Codex.
4. Bott periodicity proved via clutching functions. Ch. II §2.2. The proof is the finite-dimensional, Fourier-Laurent-series-of- clutching-function argument (Atiyah-Bott 1964 Topology paper), reduced here from 30 pages to about 20. The proof is via three reduction lemmas: (i) every clutching function is a finite Laurent polynomial, (ii) every Laurent polynomial is a polynomial after shift, (iii) every polynomial clutching reduces to linear via a "linearisation trick." The fundamental class $\beta = [H] - 1 \in \widetilde K(\mathbb{C}P^1)$where$H$ is the tautological line bundle generates everything.
5. The Thom isomorphism in K-theory. §2.7. $K(P(E\oplus 1), P(E)) \cong K(X)$viathecanonicalclass$\lambda_E$ on the projective bundle; this is the K-theoretic shadow of the cohomological Thom isomorphism, and supplies the multiplicative structure needed by the Atiyah-Singer index theorem [03.09.10].
6. Exterior power operations ${\lambda }^k$ and Adams operations ${\psi }^k$. Ch. III §3.1–§3.2. The $\lambda$-ring structure on $K(X)$ via the exterior-power functors; Newton's identities convert ${\lambda }^k$ to ${\psi }^k$, which are ring homomorphisms (the multiplicativity is what makes Adams operations powerful). The Codex has no unit for either ${\lambda }^k$ or ${\psi }^k$.
7. The group $J(X)$ and stable-fibre-homotopy invariants. §3.3 with the Power-operations-in-K-theory reprint. Adams 1962–66 Topology four-paper sequence is the originator citation; Atiyah's lectures give the compact in-book version. $J(X)$ classifies sphere-bundles up to fibre homotopy and is the engine behind the solution of the vector-fields-on-spheres problem.
8. Appendix: the space of Fredholm operators. $\mathrm{Fred}(H)$ classifies $K^0$: $K^0(X)=[X,\mathrm{Fred}(H)]$. Atiyah's geometric / analytic alternative to the Grothendieck-completion definition and the direct bridge to the index theorem.

The book is not an introduction to algebraic topology; it assumes basic point-set topology and linear algebra plus willingness to manipulate clutching functions. It is the canonical entry point to topological K-theory; Hatcher's Vector Bundles and K-Theory is the standard modern counterpart; Karoubi's K-Theory: An Introduction (1978) is the next step up and the canonical algebraic-K-theory bridge.

Peer sources for cross-checking (all consulted in §2 below):

• Allen Hatcher, Vector Bundles and K-Theory, in-progress draft, Cornell, ~2017 revision. Free at https://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html.
• Max Karoubi, K-Theory: An Introduction, Grundlehren 226, Springer
  1978. The category-theoretic / Banach-category treatment; covers Bott via Wood-Wood and the Karoubi tower.
• Dale Husemoller, Fibre Bundles, 3rd ed., GTM 20, Springer 1994. §10 (K-theory) is the bundle-theoretic peer text.
• Raoul Bott, Loring Tu, Differential Forms in Algebraic Topology (FT 1.17). §23 (universal bundle, $H^{\ast }(BU(k))$) is the cohomological peer; not a K-theory text per se but the Codex 03.08.05 unit is Bott-Tu-anchored.

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§2 Coverage table (Codex vs Atiyah)

Cross-referenced against the current corpus (5 shipped K-theory units + ancillaries in 03.05, 03.06, 03.09, 03.13). Pagination given is Benjamin 1967 / Westview 1989 (identical). ✓ = covered, △ = partial / different framing, ✗ = not covered.

Atiyah §	Topic	Codex unit(s)	Status	Note
§1.1	Basic defs of vector bundle	03.05.02-vector-bundle	✓	Atiyah's def ≡ Codex's; align notation in Master tier.
§1.2	Operations via continuous functors	03.05.02 (notes), 03.05.08-complex-vector-bundle	△	Codex treats $\oplus$, $\otimes$, ${\wedge }^k$ ad hoc; Atiyah's continuous-functor packaging is not present. Minor gap — add a Master-tier note to 03.05.02 rather than a new unit.
§1.3	Sub-bundles and quotient bundles	—	△	Stated implicitly in 03.05.02/03.05.10 but the complement-via-partition-of-unity lemma (Atiyah 1.4.1) is not its own theorem. Add to 03.05.02 Master.
§1.4	Vector bundles on compact spaces (homotopy invariance, stable inverse, classification by $[X,BU(n)]$)	03.05.02, 03.08.04-classifying-space	△	Classification is in 03.08.04; the four compactness lemmas (1.4.1 complement, 1.4.3 homotopy invariance, 1.4.7 stable inverse, 1.4.9 finite-dim classification) are implicit in Codex but not theorem-numbered. Add a §"Compactness lemmas" block to 03.05.02 Master.
§1.5	Additional structures (Hermitian, orientation)	03.05.08-complex-vector-bundle	△	Hermitian metric structure mentioned; orientation as $\det$-bundle reduction not its own item.
§1.6	$G$-bundles over $G$-spaces	03.05.01-principal-bundle	△	Principal-bundle treatment exists; equivariant vector bundle structure (the precursor to $K_G(X)$) is not isolated. Gap.
§2.1	Definitions of $K(X)$, $K^0$, reduced $\tilde{K}$, relative $K(X,Y)$	03.08.01-topological-k-theory	✓	Covered; the reduced and relative versions are in the Master tier of 03.08.01.
§2.2	Periodicity theorem (Atiyah-Bott clutching-function proof)	03.08.07-bott-periodicity	△	Statement covered, but the clutching-function proof itself (the Laurent-polynomial reduction lemmas) is not in 03.08.07 — that unit anchors instead on the Morse-theoretic Milnor-Bott proof. Gap (Master tier of 03.08.07 should record both proofs or split out a "Bott periodicity — clutching-function proof" deepening unit).
§2.3	$K_G(X)$, equivariant K-theory, $R(G)$ action	03.09.21-family-equivariant-index	△	Family / equivariant index appears in Spin chapter, but the foundational equivariant K-theory unit ($K_G(X)$ as a ring, $R(G)=K_G(\mathrm{pt})$, induced rep ${\mathrm{Ind}}_H^G$ etc.) is missing. Gap.
§2.4	Cohomology properties of $K^{\ast }$ (long exact sequence of a pair, exactness, Mayer-Vietoris in K-theory)	—	✗	Gap. The fact that $K^{\ast }$ is a generalised cohomology theory (homotopy, excision, exactness, additivity) is implicit in 03.08.01 and 03.08.07 but not stated as such.
§2.5	Computations of $K^{\ast }(X)$ for $X=S^n$, $\mathbb{C}{P}^n$, $T^n$, Stiefel manifolds, $\mathbb{R}{P}^n$ (the latter via ${\psi }^k$ obstruction)	—	✗	Gap. Worked computations are absent from the Codex. The $\mathbb{C}{P}^n$ computation $K^{\ast }(\mathbb{C}{P}^n)=\mathbb{Z}[H]/(H-1{)}^{n+1}$ is a load-bearing example used throughout the index-theorem chapter.
§2.6	External / internal product $K(X)\otimes K(Y)\to K(X\times Y)$, multiplicative structure on $K^{\ast }$	—	△	The ring structure on $K^0(X)$ is stated in 03.08.01; the external product and the suspension-Bott combinatorics are not. Add to 03.08.01 Master or split as a unit.
§2.7	Thom isomorphism in K-theory	03.08.07 (passing mention)	✗	Gap (high priority). Atiyah's ${\lambda }_E\in K(P(E\oplus 1),P(E))$ construction and the resulting $K(X)\cong K_c(E)$ for $E\to X$ a complex bundle is the K-theoretic ingredient of Atiyah-Singer. Mentioned in 03.08.07 and 03.09.10, never developed.
§3.1	Exterior powers ${\lambda }^k$, $\lambda$-ring structure	—	✗	Gap (high priority). No unit exists. The continuous functor ${\Lambda }^k$ extends to operations ${\lambda }^k:K(X)\to K(X)$ satisfying ${\lambda }^k(x+y)={\sum }_{i+j=k}{\lambda }^i(x){\lambda }^j(y)$; this is the input for §3.2.
§3.2	Adams operations ${\psi }^k$	—	✗	Gap (high priority). No unit exists. Defined via Newton's identities from ${\lambda }^k$; ${\psi }^k$ are ring homomorphisms (vs ${\lambda }^k$ which are only additive after correction). The single most important K-theory operations after Bott. Codex references AHSS but never introduces ${\psi }^k$.
§3.3	The group $J(X)$, stable fibre-homotopy classification of sphere-bundles, $J$-homomorphism	—	✗	Gap. No unit. Connects to Adams' 1962–66 solution of the vector-fields-on-spheres problem and to stable homotopy [03.08.06].
Appx	Space of Fredholm operators, $K^0(X)=[X,\mathrm{Fred}(H)]$	03.09.06-fredholm-operators	△	The Fredholm-operator unit exists; the classification statement "$\mathrm{Fred}(H)$ represents $K^0$" (Atiyah-Jänich) is not in 03.09.06. Add to 03.09.06 Master.
Reprint 1	Power operations in K-theory (Atiyah 1966 Quart. J. Math.)	—	✗	Gap (Master-tier only). Externalisation of ${\lambda }^k$ to symmetric-group equivariant operations $P^k:K(X)\to K_{{\Sigma }_k}(X^k)$. Underlies the Atiyah-Hirzebruch Steenrod operations in K-theory programme.
Reprint 2	K-theory and reality (Atiyah 1966 Quart. J. Math.)	—	✗	Gap (Master-tier only). $KR(X)$ unifying $K$, $KO$, $KSp$; eight-fold periodicity via the Clifford volume element. Already partially anchored in 03.09.15-clk-dirac and the CONCEPT_CATALOG $K{R}^{p,q}$ entry; needs a dedicated stub at the K-theory chapter.
AHSS	Atiyah-Hirzebruch spectral sequence (not in K-Theory the book itself, but the naming-author topic)	—	✗	Gap. Mentioned in 03.13.01–.03 and in 03.12.13 as the K-theoretic shadow of cellular / Leray-Serre; never its own unit. Originator: Atiyah-Hirzebruch 1961 Vector bundles and homogeneous spaces (Proc. Sympos. Pure Math. III). Out-of-scope for the book but in-scope for "Atiyah's K-theory programme as a whole." Add as a separate unit; book-attribution note in §4.

Aggregate coverage estimate: ~35% of K-Theory maps cleanly to existing Codex units. The Ch. I foundations (§1.1–§1.5) are ~80% covered via 03.05.02 and 03.05.08; Ch. II §2.1 is ✓, §2.2 is △ (the proof gap), §2.3–§2.7 are 0–20%; Ch. III is 0% (the $\lambda$-ring, ${\psi }^k$, $J(X)$ story); Appendix and reprints are ~10% covered via 03.09.06 and 03.09.15. The principal gaps are concentrated in Ch. III (operations) and §2.4–§2.7 (cohomology properties, computations, external product, Thom isomorphism). Closing those gaps is independent of any other Fast Track book — Atiyah-internal work only.

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§3 Gap punch-list (units to write / extend, priority-ordered)

Priority 1 — high-leverage, captures Atiyah's central content:

1. 03.08.02 Adams operations ${\psi }^k$. New unit. Definition via ${\psi }^k=N_k({\lambda }^1,\dots ,{\lambda }^k)$ with $N_k$ the Newton polynomial; ring-homomorphism property; ${\psi }^k{\psi }^l={\psi }^{kl}$; action on line bundles ${\psi }^k(L)=L^{\otimes k}$; computation on $K(\mathbb{C}{P}^n)$ and $K(S^{2n})$. Atiyah §3.2 anchor; Hatcher VB&K §2.3 anchor; Karoubi §IV.7 anchor. Three-tier, ~1800 words. Master tier records the Adams 1962 Topology originator paper plus the Atiyah lectures' Newton-polynomial derivation. Foundational for the vector-fields-on-spheres application and for AHSS differentials.
2. 03.08.03 Thom isomorphism in K-theory. New unit. Statement $K(X)\cong \tilde{K}(X^E)$ for $E\to X$ a complex vector bundle of rank $n$; construction of the Thom class ${\lambda }_E=\sum (-1{)}^i{\Lambda }^{i}E\in K(X^E)$; cup-product isomorphism; reduction to Bott's theorem when $X=\mathrm{pt}$. Atiyah §2.7 anchor; Karoubi §IV.5; Lawson-Michelsohn §I.9 for the Spin variant. Three-tier, ~1800 words. Master tier proves the isomorphism via the projective-bundle theorem. Load-bearing for the Atiyah-Singer index theorem [03.09.10] symbol-class argument.
3. 03.08.08 Exterior-power operations ${\lambda }^k$ and the $\lambda$-ring structure on $K(X)$. New unit. The continuous-functor ${\Lambda }^k$ from §1.2, its extension to a well-defined operation ${\lambda }^k:K(X)\to K(X)$, the $\lambda$-ring axioms $\lambda^k(x+y) = \sum \lambda^i x \cdot \lambda^j y$,$\lambda_t(x) = \sum \lambda^k(x) t^k$ as a homomorphism to $1+tK(X)[[t]]$ under multiplication. Atiyah §3.1 anchor; Karoubi §IV.7; Fulton-Lang Riemann-Roch Algebra Ch. I. Prerequisite for 03.08.02. Three-tier, ~1500 words.
4. 03.08.09 Computations of $K^{\ast }(X)$: spheres, projective spaces, tori. New unit. Worked computations $K^(S^{2n}) = \mathbb{Z} \oplus \mathbb{Z}$,$K^(\mathbb{C}P^n) = \mathbb{Z}[H]/(H-1)^{n+1}$,$K^(T^n) = \Lambda^ \mathbb{Z}^n$, and Hopf-bundle generators. Atiyah §2.5 anchor. Three-tier, ~1500 words. Beginner tier covers $S^2$ and $\mathbb{C}{P}^1$; Master tier records the Adams obstruction computation for $K^{\ast }(\mathbb{R}{P}^n)$ via ${\psi }^k$ (the input for 03.08.02's vector-fields application). Hatcher VB&K §2.1 uses these as canonical examples.
5. 03.08.10 Equivariant K-theory $K_G(X)$ and $R(G)$. New unit. $G$-vector bundles on $G$-spaces; Grothendieck construction $K_G(X)$; the case $X=\mathrm{pt}$ recovers the representation ring $R(G)=K_G(\mathrm{pt})$; functoriality, induction ${\mathrm{Ind}}_H^G$, restriction ${\mathrm{Res}}_H^G$. Atiyah §1.6 + §2.3 anchor; Segal 1968 Equivariant K-theory (Publ. IHES 34) as the definitive originator paper. Master-only initially, ~2000 words. Prerequisite for the family-equivariant index in 03.09.21.

Priority 2 — proof completions and clarifications:

6. Extend 03.08.07-bott-periodicity Master tier with Atiyah-Bott clutching-function proof. The existing unit anchors on the Milnor-Bott Morse-theoretic proof. Add a §"Atiyah-Bott proof via clutching functions" Master block: Laurent-polynomial reduction, linearisation trick, and the fundamental class $\beta =[H]-1\in \tilde{K}(\mathbb{C}{P}^1)$. Atiyah §2.2 anchor; Atiyah-Bott 1964 Topology 3, 1–18. No new unit, in-place edit.
7. Extend 03.08.01-topological-k-theory Master tier with cohomology properties. Long exact sequence of a pair, Mayer-Vietoris, excision, suspension (i.e. the proof that $K^{\ast }$ is a generalised cohomology theory in the Eilenberg-Steenrod sense modulo dimension). Atiyah §2.4. No new unit.
8. Extend 03.05.02-vector-bundle Master tier with Atiyah's continuous-functor packaging. Add a §"Operations via continuous functors" Master block citing Atiyah §1.2 and the four compactness lemmas of §1.4. No new unit, in-place edit.
9. Extend 03.09.06-fredholm-operators Master tier with the Atiyah-Jänich classification. Statement $\mathrm{Fred}(H) \simeq \mathbb{Z} \times BU$, with citation to Atiyah Appendix and Jänich 1965 Math. Ann. 161, 129–142. No new unit, in-place edit.

Priority 3 — Master-tier deepenings (Atiyah-internal):

10. 03.08.11 The group $J(X)$ and the $J$-homomorphism. Atiyah §3.3 + reprint 1. Master-only, ~1500 words. Connects to Adams' 1962–66 Topology four-paper sequence and the vector-fields-on- spheres theorem. Prereqs: 03.08.02 ${\psi }^k$, 03.08.06 stable homotopy.
11. 03.08.12 $KR$-theory (K-theory with reality). Atiyah 1966 Quart. J. Math. 17, 367–386 (reprint 2). Master-only, ~1500 words. The bigraded $K{R}^{p,q}$ theory unifying $K$, $KO$, $KSp$; eight-fold periodicity via the Clifford volume element. Already partially anchored in 03.09.15-clk-dirac historical section and the CONCEPT_CATALOG $K{R}^{p,q}$ entry; needs the dedicated stub here.

Priority 4 — extension beyond the book (Atiyah's K-theory programme, originator-attributed elsewhere):

12. 03.13.04 Atiyah-Hirzebruch spectral sequence. Originator Atiyah-Hirzebruch 1961, not in K-Theory the book per se but a standard companion topic; currently referenced in 03.13.01–.03 and 03.12.13 but not its own unit. Three-tier, ~1800 words. Pairs with 03.13.02-leray-serre as the K-theoretic analogue. Add to Ch. 13 spectral-sequences chapter rather than Ch. 8 K-theory.

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§4 Implementation sketch (P3 → P4)

For a full Atiyah coverage pass, priority-1 items 1–5 (new units) plus the priority-2 in-place extensions are the minimum set. Realistic production estimate (mirroring earlier K-theory and Bott-Tu batches):

• ~3–4 hours per new unit (three-tier with worked computations and Master-tier proofs).
• 5 priority-1 new units × ~3.5 hours = ~17–18 hours of focused production.
• 4 priority-2 in-place edits × ~1 hour = ~4 hours.
• 2 priority-3 Master-only units × ~2.5 hours = ~5 hours.
• 1 priority-4 unit 03.13.04 × ~3.5 hours = ~3.5 hours.
• Total ~30 hours focused production, plus integration / weaving / cross-link audit. Fits a focused 4-day window.

Originator-prose targets. Atiyah is the originator of topological K-theory (jointly with Hirzebruch 1959 Bull. AMS); Adams is the originator of ${\psi }^k$; Grothendieck originated the algebraic side (1957 Sur quelques points d'algèbre homologique, Tôhoku); Bott originated periodicity (1959 Bull. AMS; 1959 Ann. Math. 70, 313–337). Units 1–5 + 6 (extension) should carry originator-prose treatment per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §10, citing:

• A. Grothendieck, Sur quelques points d'algèbre homologique, Tôhoku Math. J. 9 (1957), 119–221 — algebraic K-theory of coherent sheaves and the original Grothendieck completion.
• M. F. Atiyah, F. Hirzebruch, Riemann-Roch theorems for differentiable manifolds, Bull. AMS 65 (1959), 276–281 — originator of topological K-theory; Vector bundles and homogeneous spaces, Proc. Sympos. Pure Math. III (1961), 7–38 — AHSS.
• R. Bott, The stable homotopy of the classical groups, Ann. Math. 70 (1959), 313–337 — originator of periodicity.
• M. F. Atiyah, R. Bott, On the periodicity theorem for complex vector bundles, Topology 3 (1964), Suppl. 1, 1–18 — the clutching-function proof reproduced in §2.2 of the book.
• J. F. Adams, Vector fields on spheres, Ann. Math. 75 (1962), 603–632 — originator of ${\psi }^k$ and the $J(X)$ application.
• M. F. Atiyah, K-theory and reality, Quart. J. Math. Oxford 17 (1966), 367–386 — originator of $KR$-theory.
• G. Segal, Equivariant K-theory, Publ. IHES 34 (1968), 129–151 — definitive equivariant treatment (Atiyah §1.6 + §2.3 are the in-book version; Segal is the standard citation).

Notation crosswalk. Atiyah writes $K(X)$ for unreduced K-theory and $\tilde{K}(X)$ for reduced; the Codex (per 03.08.01) writes $K^0(X)$ and ${\tilde{K}}^0(X)$ with explicit grading. Atiyah uses $H$ for the tautological / Hopf line bundle on $\mathbb{C}{P}^n$ and $\beta =[H]-1$ for the Bott class; the Codex should adopt the same. Atiyah's ${\lambda }_E$ for the K-theory Thom class clashes with the spinor-symbol $\Lambda (E)$ in 03.08.07 line 466 — record the decision in 03.08.03 (new Thom unit) §Notation: use ${\Lambda }_E$ for the K-theoretic Thom class throughout, reserving ${\lambda }^k$ for the exterior-power operation. Atiyah's $J(X)$ is the standard symbol; keep. Atiyah's $K_G(X)$ for equivariant K-theory is universal; keep. Record all decisions in a §Notation paragraph of 03.08.02, 03.08.03, 03.08.08, 03.08.10.

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§5 What this plan does NOT cover

• Algebraic K-theory ($K_0$ of a ring, Whitehead's $K_1$, Milnor's $K_2$, Quillen $K_n$). Atiyah's K-Theory is topological throughout; the algebraic side is a separate book (Weibel K-Book, Rosenberg, Srinivas) and a separate Codex chapter would be required. Not on the Fast Track booklist at this priority.
• The full Atiyah-Singer index theorem proof. That is 03.09.10's job and is independent of this audit. Cross-link only.
• Equivariant Bott periodicity (Atiyah-Segal 1969 Equivariant K-theory and completion). Out of scope; pointer in 03.08.10 Master only.
• Exercise pack production. Atiyah's book has no formal exercise sets; problems live inside the prose. Exercise pack production is a P3-priority-3 follow-up after priority-1 units ship.
• The two reprinted papers' full content beyond the dedicated Master-only stubs 03.08.11 (J(X)) and 03.08.12 (KR-theory).
• Connections to motivic K-theory, twisted K-theory, K-homology and Kasparov KK-theory. Beyond the book's scope; pointer-only in 03.08.12 if at all.

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§6 Acceptance criteria for FT equivalence (Atiyah K-Theory)

Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4, the book is at equivalence-coverage when:

• ≥95% of Atiyah's named theorems (compactness lemmas of §1.4, periodicity theorem §2.2, Thom isomorphism §2.7, Adams operation identities §3.2, $J(X)$ exactness §3.3) map to Codex units. Current: ~35%. After priority-1 (units 1–5): ~85%. After priority-1+2 (units 1–9): ~95%. Priority-3+4 close the residual deepening gap.
• ≥90% of Atiyah's worked computations ($K^{\ast }(S^n)$, $K^{\ast }(\mathbb{C}{P}^n)$, $K^{\ast }(\mathbb{R}{P}^n)$ via ${\psi }^k$, $K^{\ast }(\mathrm{Stiefel})$) have a direct Codex unit or are referenced from a unit that covers them (closed by 03.08.09).
• The clutching-function proof of Bott periodicity (Atiyah §2.2) is recorded in 03.08.07 Master alongside the Milnor-Bott Morse-theoretic proof. Currently only the Morse proof anchors.
• The $\lambda$-ring / Adams-operations story (Atiyah Ch. III) has at least one dedicated Codex unit (closed by 03.08.02 + 03.08.08).
• Notation decisions (§4) are recorded in 03.08.03, 03.08.02, 03.08.08, 03.08.10.
• Pass-W weaving connects the new units to 03.05.02, 03.08.04, 03.08.06, 03.08.07, 03.09.06, 03.09.10, 03.13.02, and the AHSS unit 03.13.04.

The 5 priority-1 units close most of the equivalence gap with no external prerequisites. Priority-2 closes the proof / cohomology-property residual. Priority-3 covers the appendix and reprints. Priority-4 closes the AHSS gap which is Atiyah-attributed but lives in 03.13.

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§7 Sourcing

• Status on FT booklist: marked BUY (Westview 1989 reprint, ISBN 0-201-09394-4, list price ~$60).
• Free scan available. A complete OCR'd PDF of the Benjamin 1967 edition (220 pp., 7.7 MB, Adobe Acrobat 8 paper-capture, dated 2008) is freely hosted on Andrew Ranicki's archive at https://webhomes.maths.ed.ac.uk/~v1ranick/papers/atiyahk.pdf (redirected from the older www.maths.ed.ac.uk/~v1ranick/ URL). The Ranicki archive is a long-standing, university-hosted, citation-grade mathematical resource and was used as the authoritative source for the §2 TOC reconstruction in this audit.
• License. Strictly the 1967 Benjamin / 1989 Westview copyright; the Ranicki archive hosts it under fair-use educational provision. Cite as Atiyah, K-Theory, Benjamin 1967 / Westview Advanced Book Classics 1989.
• Local copy. Recommended: add to reference/fasttrack-texts/03-modern-geometry/ as Atiyah-KTheory.pdf to mirror the pattern of other free FT texts (Bott Lectures on Morse Theory, May Concise Algebraic Topology, etc.). The booklist line 3.10 ... BUY should be revised to FREE (Ranicki archive scan) once the file is mirrored locally.
• Peer sources for cross-checking (used in §2 coverage table): Hatcher, Vector Bundles and K-Theory, Cornell in-progress draft, free at https://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html; Karoubi, K-Theory: An Introduction, Grundlehren 226, Springer 1978 (paywalled, on Anna's Archive); Husemoller, Fibre Bundles, GTM 20, Springer 1994 (paywalled, on Anna's Archive); Bott-Tu, Differential Forms in Algebraic Topology, GTM 82, Springer 1982 (paywalled, FT 1.17, already in reference/fasttrack-texts/01-fundamentals/).