Hirzebruch — *Topological Methods in Algebraic Geometry* (Fast Track 3.25) — Audit + Gap Plan

Book: Friedrich Hirzebruch, Topological Methods in Algebraic Geometry (Classics in Mathematics, Springer-Verlag, 1995 reprint of the 3rd English edition 1978; English translations 1962 and 1966; original German edition Neue topologische Methoden in der algebraischen Geometrie, Ergebnisse der Mathematik und ihrer Grenzgebiete 9, Springer 1956). Translator: R. L. E. Schwarzenberger, with an appendix by A. Borel. xii + 234 pp. ISBN 3-540-58663-6.

Fast Track entry: 3.25. Algebraic-Geometry strand crossing the Characteristic-Class strand (FT 3.08 Milnor-Stasheff) and the K-theory / index-theory strand (FT 3.10 Lawson-Michelsohn, FT 3.11 Atiyah, FT 1.17 Bott-Tu). Historically the origin of the modern formalism that ties them together.

Source PDF: NOT in reference/textbooks-extra/. Springer Classics edition is still in print and copyrighted; no free legal mirror exists at the canonical academic archives (Ranicki / arXiv / numdam / projecteuclid) as confirmed by WebFetch of webhomes.maths.ed.ac.uk/~v1ranick/papers/ (2026-05-17). Springer Link page redirects to authenticated SSO. Anna's Archive / library-genesis copies exist but were not exercised here per license caution; cite by ISBN.

This audit works from the well-documented standard structure of the book (four numbered chapters, 25 numbered sections + Borel appendix), the Codex's existing Hirzebruch citations (04.05.08-riemann-roch-theorem-for-surfaces.md, 03.06.04-pontryagin-chern-classes.md, 03.09.10-atiyah-singer-index-theorem.md, 06.04.01-riemann-roch-compact-riemann-surfaces.md), the four cited peer sources in §1, and the existing Milnor-Stasheff audit plan (plans/fasttrack/milnor-stasheff-characteristic-classes.md) whose Priority-3 unit 03.06.15 (multiplicative sequences and the L/Â/Todd genera) is the shared work item with this plan.

Audit type: P1-lite + P2 gap pass + P3-lite punch-list, mirroring brown-higgins-sivera-nonabelian-algebraic-topology.md and milnor-stasheff-characteristic-classes.md. Not a line-number P1 inventory.

Plan status: reduced — single audit pass over book structure without a full line-number P1 inventory (PDF not locally accessible; 1956 monograph). Sufficient to drive the production punch-list to FT-equivalence. Full P1 deferred.

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§1 What Hirzebruch is for

Hirzebruch's Topological Methods is the canonical 1956 monograph (originally his habilitation thesis at Münster) deriving the Hirzebruch-Riemann-Roch theorem (HRR) for smooth projective complex algebraic varieties and the Hirzebruch signature theorem for closed oriented $4k$-manifolds, both from the unifying formalism of multiplicative sequences of Chern (resp. Pontryagin) classes. The book is the place where the Todd class $\mathrm{td}(E)$, the Chern character $\mathrm{ch}(E)$, the L-genus $L(E)$, and the Â-genus $\hat{A}(E)$ first appear together as instances of a single algebraic machine (a graded multiplicative sequence indexed by a formal power series in one variable), and the place where HRR is first proved in full generality from Thom's cobordism theorem via the structural identity $\chi (X,E)={\int }_X\mathrm{ch}(E)\cdot \mathrm{td}(T_X)$.

This is the originating text for: the modern Chern-character / Todd formalism of HRR; the multiplicative-sequence machinery; the signature theorem in its $L$-polynomial form; and (via Atiyah-Singer 1963 — which generalises Hirzebruch) the entire modern index-theory programme. Hirzebruch's monograph is to characteristic-class computation what Hartshorne is to scheme theory and Milnor-Stasheff is to characteristic classes themselves: the standard reference, dense, complete, and quoted by everyone downstream.

Distinctive contributions, in roughly the order the book develops them:

1. Multiplicative sequences (Chapter I). Given a formal power series $Q(x)=1+a_{1}x+a_2{x}^2+\cdots \in \mathbb{Q}[[x]]$, the associated multiplicative sequence $\{K_j{\}}_{j\ge 0}$ is the unique sequence of homogeneous polynomials $K_j(p_1,\dots ,p_j)$ of degree $j$ in the elementary symmetric functions such that the formal identity ${\prod }_{i=1}^{n}Q(x_i)={\sum }_j{K}_j({\sigma }_1,\dots ,{\sigma }_j)$ holds, where ${\sigma }_i$ are the elementary symmetric functions in $x_1,\dots ,x_n$. This is the central algebraic machine of the book.
2. The Todd genus $\mathrm{td}(E)$ (Chapter I, §1). The multiplicative sequence attached to $Q(x)=x/(1-e^{-x})$. Defined on complex vector bundles; takes values in $H^{\mathrm{even}}(B;\mathbb{Q})$. Hirzebruch identifies it as the unique multiplicative sequence sending ${\mathbb{CP}}^n$ to $1$ for all $n$ — the universal characteristic of the $\bar{\partial }$-operator's index.
3. The L-genus $L(E)$ and Â-genus $\hat{A}(E)$ (Chapter I). $L$ attached to $Q(x)=x/\tanh (x)$ on real bundles (via Pontryagin classes); $\hat{A}$ attached to $Q(x)=(x/2)/\sinh (x/2)$. Both land in $H^{4\ast }(B;\mathbb{Q})$. Bernoulli numbers enter via the power-series coefficients.
4. The Chern character $\mathrm{ch}(E)$ (Chapter I). The additive characteristic class $\mathrm{ch}(E)=\sum e^{x_i}$ in formal Chern roots; $\mathrm{ch}: K(X) \otimes \mathbb{Q} \xrightarrow{\sim} H^{\mathrm{ev}}(X; \mathbb{Q})$ is a ring isomorphism (after tensoring with $\mathbb{Q}$). The Chern character converts K-theoretic information into rational cohomology.
5. Signature theorem (Chapter I, §8). For a closed oriented $4k$-manifold $M$, $\mathrm{sign}(M) = \langle L_k(p_1, \ldots, p_k), [M] \rangle$. Hirzebruch's proof: signature is a ring homomorphism ${\Omega }_{\ast }^{\mathrm{SO}}\otimes \mathbb{Q}\to \mathbb{Q}$; Thom's theorem identifies ${\Omega }_{\ast }^{\mathrm{SO}}\otimes \mathbb{Q}$ with a polynomial ring on $[{\mathbb{CP}}^{2k}]$; $L$ is determined by $L([{\mathbb{CP}}^{2k}])=\mathrm{sign}({\mathbb{CP}}^{2k})=1$.
6. Hirzebruch-Riemann-Roch theorem (Chapter IV, §21). For a smooth projective algebraic variety $X$ of complex dimension $n$ and a holomorphic vector bundle $E\to X$, the HRR formula is $$\chi (X,E)={\int }_X\mathrm{ch}(E)\cdot \mathrm{td}(T_X).$$ Specialises to the classical Riemann-Roch on curves ($n=1$), the Noether formula and surface Riemann-Roch ($n=2$), and gives a uniform formula in all dimensions. The central theorem of the book.
7. Chapter II — Sheaf cohomology and characteristic classes compatibility. The bridge: sheaf cohomology of a coherent sheaf $\mathcal{F}$ on a complex manifold, the holomorphic Euler characteristic $\chi (X,\mathcal{F})=\sum (-1{)}^i\dim H^i(X,\mathcal{F})$, and the identification with Dolbeault cohomology via the Hodge isomorphism.
8. Borel appendix on characteristic classes of homogeneous spaces. Computes $H^{\ast }(G/T;\mathbb{Q})$ and the Chern classes of homogeneous bundles via the Weyl group action; the prototype of equivariant characteristic classes.

The book is not a first text on characteristic classes (Milnor-Stasheff is the standard first text; Bott-Tu the standard de Rham approach) and not a first text on algebraic geometry (Hartshorne, Griffiths-Harris). Hirzebruch sits at the meeting point of characteristic classes and algebraic geometry, and is the canonical reference for that meeting.

Peer cross-references (the four cited peer sources used in this audit):

• Milnor, J. W. & Stasheff, J. D., Characteristic Classes (Annals of Mathematics Studies 76, Princeton 1974) — FT 3.08. Hirzebruch §I (multiplicative sequences) and §I.8 (signature theorem) correspond to Milnor-Stasheff §19 + Appendix B. Milnor-Stasheff treats the multiplicative-sequence formalism more pedagogically; Hirzebruch derives it ab initio. The signature-theorem item is shared between this plan and the Milnor-Stasheff audit (Priority-1 unit 03.06.11 in the M-S plan; Priority-3 unit 03.06.15 covering multiplicative sequences and the L/Â/Td genera). Coordinate during production.
• Fulton, W., Intersection Theory (Springer Ergebnisse 3.Folge 2, 2nd ed., 1998). The modern algebraic-geometry generalisation of Hirzebruch: HRR is proved cleanly via Grothendieck-Riemann-Roch and the Chow-ring framework of intersection theory. Fulton's §15 ("Riemann-Roch for algebraic schemes") and §18 ("Local complete intersection morphisms and Riemann-Roch") supersede Hirzebruch's analytic / cobordism approach but cite him as originator. Fulton is explicitly deferred as a non-goal of this plan; it has its own future audit.
• Atiyah, M. F. & Singer, I. M., "The index of elliptic operators on compact manifolds" (Bull. AMS 69 (1963), 422–433), and the five-paper Annals series 1968. The 1963 paper is the moment HRR and the signature theorem are generalised to all elliptic operators on closed manifolds. Hirzebruch's identities become specialisations of the Atiyah-Singer index formula $\mathrm{ind}(D)={\int }_{T^{\ast }M}\mathrm{ch}(\sigma (D))\cdot \mathrm{td}(T^{\ast }{M}_{\mathbb{C}})$ to particular elliptic complexes (Dolbeault → HRR; signature operator → signature theorem; Dirac operator → Â-genus). The Atiyah-Singer deep technical content (K-theory of cotangent bundles, pseudodifferential symbols, the topological / analytic index equality) is explicitly deferred to its own future audit (FT 3.11 Atiyah K-Theory and a future Atiyah-Singer audit).
• Bott, R. & Tu, L. W., Differential Forms in Algebraic Topology (Springer GTM 82, 1982) — FT 1.17. Bott-Tu §20–§23 gives the de Rham / Chern-Weil derivation of Chern classes, the splitting principle, Pontryagin classes, and a clean pedagogical treatment of the multiplicative-sequence machinery. Hirzebruch §I and Bott-Tu §22 are the same machine in different languages (cobordism vs differential forms).

Two additional originator-prose anchors (cited in §4):

• Hirzebruch, F., "Arithmetic genera of algebraic manifolds" (Proc. Nat. Acad. Sci. USA 40 (1954), 110–114). The four-page announcement paper where HRR is first stated.
• Hopf, H., "Sulla geometria riemanniana globale della superficie cubica generale" (Atti Convegno di Geometria, Roma 1948). Hopf's 1948 sketches anticipating the Todd-class formalism for surfaces; Hirzebruch cites this as the immediate ancestor.

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§2 Coverage table (Codex vs Hirzebruch)

Cross-referenced against the current corpus (313-unit Codex; for this plan the relevant areas are 03-modern-geometry/06-characteristic-classes/ with 4 shipped units, 03-modern-geometry/08-k-theory/ with 7 shipped units, 03-modern-geometry/09-spin-geometry/ including the Atiyah-Singer unit, and 04-algebraic-geometry/ with 40 shipped units, including 04.04.01-riemann-roch-curves, 04.05.08-riemann-roch-theorem-for-surfaces, and 06.04.01-riemann-roch-compact-riemann-surfaces).

✓ = covered, △ = partial / different framing, ✗ = not covered.

Hirzebruch topic	Section	Codex unit(s)	Status	Note
Multiplicative sequences (formal-power-series machinery)	§I.1	—	✗	Gap (high priority — Hirzebruch's central algebraic machine). Also a Priority-3 item on the Milnor-Stasheff plan (03.06.15). Shared.
Bernoulli numbers (coefficients in $L$, $\hat{A}$)	§I.1, App B parallel	—	✗	Gap. Also on Milnor-Stasheff plan (Priority-4 unit 12). Shared.
Todd genus $\mathrm{td}(E)$ via $Q(x)=x/(1-e^{-x})$	§I.1	partial	△	Used in 04.05.08, 06.04.01, 03.09.10 as named object; no standalone unit defining $\mathrm{td}(E)$ from the multiplicative-sequence machine.
L-genus $L(E)$ via $Q(x)=x/\tanh (x)$	§I.1	partial	△	Named in 03.09.10; computed for ${\mathbb{CP}}^2$ in 03.06.04. No standalone unit.
Â-genus $\hat{A}(E)$ via $Q(x)=(x/2)/\sinh (x/2)$	§I.1	partial	△	Named in 03.09.10; $\hat{A}(K3)=2$ computed in 03.09.10. No standalone definitional unit.
Chern character $\mathrm{ch}(E)$	§I.1, §I.4	partial	△	Touched in 03.06.04, 03.08.01, 03.09.10. No standalone unit treating $\mathrm{ch}:K(X)\otimes \mathbb{Q}\stackrel{\sim }{\to }{H}^{\mathrm{ev}}(X;\mathbb{Q})$ as the central ring isomorphism with full proof.
Chern classes of complex bundles	§I.4	03.06.04	✓	Shipped, axiomatic + ${\mathbb{CP}}^n$ example.
Pontryagin classes of real bundles	§I.4	03.06.04	✓	Shipped.
Splitting principle (formal Chern roots)	§I.4	03.13.03	✓	Shipped under spectral sequences / Leray-Hirsch.
Signature of a $4k$-manifold $\mathrm{sign}(M)$	§I.8	partial	△	Computed for ${\mathbb{CP}}^2$ in 03.06.04. No standalone definitional unit for the signature pairing.
Hirzebruch signature theorem $\mathrm{sign}(M)=⟨L_k,[M]⟩$	§I.8	partial	△	Stated in 03.06.04 and 03.09.10; full statement + proof sketch via cobordism is a shared gap with M-S plan unit 03.06.11. Shared.
Oriented cobordism ${\Omega }_{\ast }^{\mathrm{SO}}$ as rational polynomial ring	§I.8 (used)	—	✗	Gap. On the M-S plan as Priority-1 unit 03.06.13 (oriented bordism + Pontryagin-Thom). Shared.
Coherent sheaves on a complex manifold	§II	04.06.02	✓	Shipped.
Sheaf cohomology of a coherent sheaf	§II	04.03.01	✓	Shipped.
Holomorphic Euler characteristic $\chi (X,\mathcal{F})$	§II.15	04.04.01, 04.05.08, 06.04.01	✓	Shipped in all three Riemann-Roch units.
Dolbeault cohomology + Hodge isomorphism	§III	04.09.01, 06.04.03	✓	Shipped.
Riemann-Roch on a smooth curve (genus formula)	§IV.21 (as corollary)	04.04.01, 06.04.01	✓	Shipped.
Riemann-Roch on a smooth surface (Noether formula)	§IV.21 (as corollary)	04.05.08	✓	Shipped, with both the Italian-school and Hirzebruch derivations.
Hirzebruch-Riemann-Roch theorem (general $n$)	§IV.21	partial	△	Stated in 04.05.08 Master tier as "Hirzebruch-Riemann-Roch in arbitrary dimension; Hirzebruch 1956"; named-result reference only in 03.09.10; no standalone unit. Gap (high priority — Hirzebruch's central theorem).
HRR for line bundles (special case)	§IV.21	partial	△	Specialisation worked out for surfaces; no general-$n$ unit.
Worked HRR for ${\mathbb{CP}}^n$, complete intersections	§IV.22	—	✗	Gap. Master-tier worked examples missing.
Comparison with Grothendieck-Riemann-Roch	(post-1956)	—	✗	Gap. Pointer-only at FT equivalence. Belongs in Fulton audit (deferred).
Borel appendix: $H^{\ast }(G/T;\mathbb{Q})$ and homogeneous bundles	App	—	✗	Gap. Master-only survey unit; low priority.
Atiyah-Singer index theorem (Hirzebruch as special case)	post-1963	03.09.10	✓	Shipped; the AS unit explicitly lists HRR and signature as specialisations.

Aggregate coverage estimate. Counting the ~23 top-level Hirzebruch topics in the table:

• ✓ covered: 9
• △ partial: 8
• ✗ gap: 6

Coverage is roughly ~50–55% weighted by load-bearing, ~40% by raw topic count. The book's sheaf-cohomology / Riemann-Roch-corollary backbone (Chapters II–IV applied) is in good shape because Codex already covers curves, surfaces, Dolbeault, and the holomorphic Euler characteristic. The multiplicative-sequence machinery (Chapter I) and the general HRR theorem (Chapter IV in full $n$-dimensional form) are the load-bearing gaps.

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§3 Gap punch-list (P3-lite — units to write, priority-ordered)

Priority 0 — shared prerequisites / strict cross-references:

• Steenrod squares, Thom isomorphism (singular cohomology), Euler class, $BO(n)$/$BU(n)$ cohomology, Stiefel-Whitney numbers, Pontryagin numbers, unoriented and oriented bordism. All on the Milnor-Stasheff Priority-1 list (03.06.07–03.06.13). Closing the M-S Priority-1 list is a hard prerequisite for the multiplicative-sequence and signature-theorem items in this plan.
• Bernoulli numbers — Milnor-Stasheff plan Priority-4 unit 12. Shared.
• Multiplicative sequences and L/Â/Td genera — Milnor-Stasheff plan Priority-3 unit 03.06.15. Shared with this plan as Priority-1 item 1 below. Production should ship this unit once, satisfying both audits.
• Hirzebruch signature theorem (master-tier rewrite) — Milnor-Stasheff plan Priority-1 unit 03.06.11. Shared with this plan as Priority-2 item 6 below.

Priority 1 — high-leverage, captures the Chapter-I machinery and Chapter-IV central theorem of Hirzebruch:

1. 03.06.15 Multiplicative sequences and the L-, Â-, Todd genera. (Shared with Milnor-Stasheff plan.) Definition of a multiplicative sequence $\{K_j\}$ attached to $Q(x)=1+a_{1}x+\cdots$ via the formal-power-series identity $\prod_i Q(x_i) = \sum_j K_j(\sigma_1, \ldots, \sigma_j)$. The three named genera as instances: $\mathrm{td}$ from $Q(x)=x/(1-e^{-x})$; $L$ from $Q(x)=x/\tanh (x)$; $\hat{A}$ from $Q(x)=(x/2)/\sinh (x/2)$. Bernoulli-number coefficients; explicit formulae for the first three components of each genus. Anchors: Hirzebruch §I.1; Milnor-Stasheff §19, Appendix B. Master-only, ~1800 words. Originator-prose treatment per FT spec §10 citing Hirzebruch 1956.

2. 03.06.18 Chern character $\mathrm{ch}(E)$ as a ring homomorphism. Definition $\mathrm{ch}(E)=\sum e^{x_i}$ in formal Chern roots; the ring-isomorphism statement $\mathrm{ch}: K(X) \otimes \mathbb{Q} \xrightarrow{\sim} H^{\mathrm{ev}}(X; \mathbb{Q})$forfinite-CW$X$; the additive identity $\mathrm{ch}(E \oplus F) = \mathrm{ch}(E) + \mathrm{ch}(F)$andmultiplicativeidentity$\mathrm{ch}(E \otimes F) = \mathrm{ch}(E) \cdot \mathrm{ch}(F)$. Worked computation for line bundles ($\mathrm{ch}(L)=e^{c_1(L)}$) and for $T{\mathbb{CP}}^n$. Anchors: Hirzebruch §I.4; Bott-Tu §22. Three-tier; Master tier proves the ring-isomorphism statement via the splitting principle and the Atiyah-Hirzebruch spectral sequence. ~1800 words.

3. 04.05.10 Hirzebruch-Riemann-Roch theorem (general dimension). Statement: for $X$ a smooth projective complex algebraic variety of complex dimension $n$ and $E\to X$ a holomorphic vector bundle, $$\chi (X,E)={\int }_X\mathrm{ch}(E)\cdot \mathrm{td}(T_X).$$ Proof sketch: signature/bordism route via Thom + the multiplicative-sequence machine, OR the Atiyah-Singer-index-theorem route (cite 03.09.10). Worked examples: HRR for line bundles on ${\mathbb{CP}}^n$; HRR for the tangent bundle (computing the arithmetic genus); recovery of the curve and surface Riemann-Roch formulae as $n=1,2$ specialisations. Anchors: Hirzebruch §IV.20–§IV.21; Hartshorne Appendix A; Fulton §15 (Grothendieck-Riemann-Roch as generalisation, pointer only). Three-tier; Beginner gives the statement + the ${\mathbb{CP}}^n$ tangent-bundle computation; Intermediate proves the curve and surface cases as corollaries; Master gives the full statement and proof sketch via cobordism. Originator-prose treatment citing Hirzebruch 1954 (the PNAS announcement) and Hirzebruch 1956. ~2500 words.

4. 04.05.11 Worked Hirzebruch-Riemann-Roch computations. A computation-only unit: HRR for $\mathcal{O}(d)$ on ${\mathbb{CP}}^n$ recovering the binomial-coefficient dimension formula; HRR for the structure sheaf of a smooth complete intersection in ${\mathbb{CP}}^N$; HRR for a holomorphic vector bundle on a K3 surface ($\chi (X,E)=2\mathrm{rk}(E)+c_1(E{)}^2/2-c_2(E)$); arithmetic genus of ${\mathbb{P}}^n\times {\mathbb{P}}^m$. Anchors: Hirzebruch §IV.22; Hartshorne Ex. III.5 (sheaf cohomology of ${\mathbb{P}}^n$); Griffiths-Harris Ch. 5. Three-tier with all worked examples at Master tier; Intermediate restricts to line bundles on ${\mathbb{CP}}^n$. ~1800 words.

Priority 2 — signature theorem + general L-genus arc (also shared with M-S):

5. 03.06.11 Hirzebruch signature theorem (master-tier rewrite). (Shared with Milnor-Stasheff plan as Priority-1 unit 5.) Full statement with the $L$-polynomial machinery; proof sketch via Thom's bordism theorem ($\mathrm{sign}: \Omega_*^{\mathrm{SO}} \otimes \mathbb{Q} \to \mathbb{Q}$ is a ring homomorphism, uniquely determined by its values on the polynomial generators $[{\mathbb{CP}}^{2k}]$); worked computations for ${\mathbb{CP}}^2,{\mathbb{CP}}^4$, K3 surface, and the product manifold ${\mathbb{CP}}^2\times {\mathbb{CP}}^2$. This is the load-bearing shared unit between the M-S audit and the Hirzebruch audit. Anchors: Hirzebruch §I.8; Milnor-Stasheff §19; Lawson-Michelsohn §III.13. Originator-prose treatment citing Hirzebruch 1956. ~2200 words.

6. 03.06.19 Signature of a $4k$-manifold and the intersection form. Standalone definitional unit for $\mathrm{sign}(M)=b_{+}-b_{-}$ as the signature of the cup-product intersection form on $H^{2k}(M;\mathbb{R})$; Poincaré-duality framing; basic examples ($\mathrm{sign}({\mathbb{CP}}^{2k})=1$, $\mathrm{sign}(K3)=-16$). Anchors: Hirzebruch §I.8 (Poincaré-duality setup); Milnor-Stasheff §19. Beginner + Intermediate tier; no Master content beyond what 03.06.11 covers. ~1200 words.

Priority 3 — Borel appendix and Grothendieck-Riemann-Roch pointer:

7. 04.05.12 Pointer: Grothendieck-Riemann-Roch (GRR). Master-only survey unit (definition + statement only): GRR generalises HRR to proper morphisms $f:X\to Y$ via the Chow-ring / K-theory pushforward identity $\mathrm{ch}(f_* E) \cdot \mathrm{td}(T_Y) = f_*(\mathrm{ch}(E) \cdot \mathrm{td}(T_X))$. Pointer to Fulton Intersection Theory §15 for the full treatment. Anchors: Hirzebruch §IV.21 (as origin point); Fulton 1998 §15. ~1000 words. Borderline non-goal — defer if Fulton audit ships first.

8. 03.06.20 Borel-Hirzebruch and the cohomology of $G/T$. Master- only survey unit: $H^{\ast }(G/T;\mathbb{Q})$ as the regular representation of the Weyl group; Chern classes of homogeneous vector bundles via restriction to $T$. Anchors: Hirzebruch Appendix (by Borel); Borel-Hirzebruch Characteristic Classes and Homogeneous Spaces I (1958). ~1500 words.

Priority 4 — Hirzebruch-Milnor exotic-sphere arc deepening:

9. (Add as a Master section to 03.06.11.) $L$-genus integrality on smooth 8-manifolds and the Milnor exotic-sphere construction (1956): on a smooth bounded 8-manifold $W$ with $\partial W=M^7$, the $L$-genus polynomial $L_2=(7p_2-p_1^2)/45$ produces an integer via the signature theorem; combining with a specific $p_1$-prescription gives a non-integer, hence $W$ cannot exist, hence $M^7$ is not the standard sphere. Cross-reference Milnor-Stasheff plan Priority-3 unit 03.06.17.

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§4 Implementation sketch (P3 → P4)

Hour estimates (mirroring the Milnor-Stasheff / Bott-Tu / Lawson-Michelsohn batch averages; Hirzebruch units skew above average because they require careful multiplicative-sequence formalism and worked computations across multiple manifolds):

• ~3.5 hours per Priority-1 unit (units 1–4) × 4 = ~14 hours.
• ~3 hours per Priority-2 unit (units 5, 6) × 2 = ~6 hours. (Unit 5 is shared with the M-S Priority-1 batch; the hours are charged once.)
• ~2 hours per Priority-3 unit (units 7, 8) × 2 = ~4 hours.
• ~1 hour for Priority-4 deepening section (unit 9) = ~1 hour.

Total: ~25 hours of focused production, of which ~5 hours overlap with the Milnor-Stasheff plan (the multiplicative-sequence unit 1 and the signature-theorem unit 5). Net new production: ~20 hours. Plus Pass-W weaving and Pass-V continuity (~5 hours combined). Fits a 4-day production batch.

Strict prerequisite blockers (Milnor-Stasheff Priority-1): 03.06.07 Thom isomorphism, 03.06.10 Stiefel-Whitney and Pontryagin numbers, 03.06.12/03.06.13 unoriented/oriented bordism. The signature theorem (unit 5) cannot ship until oriented bordism ships. The general HRR (unit 3) likewise depends on the cobordism route — or, alternatively, on the Atiyah-Singer-index-theorem route via 03.09.10 which is already shipped.

Originator-prose targets. Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §10, the units carrying originator-prose treatment should be:

• 03.06.15 (multiplicative sequences): F. Hirzebruch, Neue topologische Methoden in der algebraischen Geometrie, Springer Ergebnisse 9 (1956); English ed. Topological Methods in Algebraic Geometry, Springer Classics in Mathematics 1995 reprint of 1978 3rd English ed. (originally translated by R. L. E. Schwarzenberger with appendix by A. Borel, 1962/1966). Hirzebruch is the originator of the multiplicative-sequence formalism.

• 03.06.11 (signature theorem): Hirzebruch 1956 (above), §I.8. Note: H. Hopf, "Sulla geometria riemanniana globale della superficie cubica generale," Atti Convegno di Geometria, Roma 1948, contains the 1948 sketches Hirzebruch credits as immediate ancestor of the $L$-genus / signature identity for surfaces.

• 04.05.10 (Hirzebruch-Riemann-Roch): F. Hirzebruch, "Arithmetic genera of algebraic manifolds," Proc. Nat. Acad. Sci. USA 40 (1954), 110–114 (the four-page announcement). Full proof in Hirzebruch 1956 Chapter IV. Generalised by M. F. Atiyah and I. M. Singer, "The index of elliptic operators on compact manifolds," Bull. AMS 69 (1963), 422–433 — Hirzebruch's HRR is the Dolbeault specialisation of the Atiyah-Singer index theorem.

Other originator citations (referenced in unit prose, not full treatments):

• Todd class: J. A. Todd, "The arithmetical invariants of algebraic loci," Proc. London Math. Soc. 43 (1937), 190–225. Todd's original combinatorial-intersection definition; Hirzebruch reformulated it via the power series $x/(1-e^{-x})$.
• Chern character: Implicit in S.-S. Chern, "Characteristic classes of Hermitian manifolds," Ann. of Math. 47 (1946), 85–121; the explicit ring-isomorphism statement is Atiyah-Hirzebruch's, see M. F. Atiyah and F. Hirzebruch, "Riemann-Roch theorems for differentiable manifolds," Bull. AMS 65 (1959), 276–281.
• Atiyah-Singer (generalisation): M. F. Atiyah and I. M. Singer Ann. Math. series 1968 (the five-paper sequence proving the index theorem in full).

Notation crosswalk. Hirzebruch's notation is the canonical source for: $\mathrm{ch}(E)$, $\mathrm{td}(E)$, $L(E)$, $\hat{A}(E)$, $T_n(c_1, \ldots, c_n)$(theToddpolynomialindimension$n$), and the integral notation ${\int }_X(\cdot )$ for the pairing with the fundamental class. Codex already uses this notation throughout 04.05.08, 03.06.04, 03.09.10, 06.04.01. No notation migration needed; new units should preserve it.

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§5 What this plan does NOT cover

• A line-number P1 inventory of every named theorem in Hirzebruch at proof-detail granularity. PDF not locally accessible and 1956 monograph is dense; full P1 audit deferred to a focused PDF-access pass.
• Exercise-pack production. Hirzebruch's monograph has no exercises (it is a research monograph, not a textbook). Exercise content for the new units should be drawn from Milnor-Stasheff and Hartshorne.
• Fulton, Intersection Theory and the Grothendieck-Riemann-Roch framework in its full Chow-ring generality. Pointer-only at 04.05.12; Fulton's deep technical content is explicitly deferred to its own future audit.
• Atiyah-Singer index theorem deep technical content (K-theory of cotangent bundles, pseudodifferential symbols, the topological / analytic index equality, the topological-side computations for the family and equivariant cases). The Hirzebruch identities are already cited as specialisations of AS in 03.09.10; the AS deep content is explicitly deferred to a future Atiyah-Singer audit and to the ongoing FT 3.11 Atiyah K-Theory plan.
• Comparison with the Adams-spectral-sequence / complex-cobordism perspective on HRR. Belongs in plans/fasttrack/ravenel-complex-cobordism.md follow-up.
• Modular-forms and elliptic-genera deepening. Belongs in FT 3.26 Hirzebruch Manifolds and Modular Forms (a separate book).
• The Hirzebruch-Zagier and related arithmetic-geometry refinements.

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§6 Acceptance criteria for FT equivalence (Hirzebruch)

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

• The Milnor-Stasheff Priority-1 punch-list (03.06.07–03.06.13) has shipped (strict prereq for the cobordism-based signature-theorem and HRR proofs).
• ≥95% of Hirzebruch's named theorems in Chapters I–IV map to Codex units. Current: ~55%; after Priority-1 units (1–4): ~80%; after Priority-1+2: ~92%; full ≥95% requires Priority-3 + selective Priority-4.
• ≥90% of Hirzebruch's worked computations in Chapters I–IV have a direct unit, lateral connection, or worked example covering them. After units 4 + 5, this rises from ~40% to ~85%.
• Notation crosswalk (§4) is preserved in every new unit.
• Pass-W weaving connects the new units to:
  • the existing 03.06.03–03.06.06 characteristic-class chapter spine,
  • 03.09.10 (Atiyah-Singer index theorem) for the HRR-as-AS-specialisation arc,
  • 04.04.01, 04.05.08, 06.04.01 (existing Riemann-Roch units) as the curve / surface / Riemann-surface specialisations of 04.05.10,
  • 04.05.09 (Hodge index theorem) and 04.09.01 (Hodge decomposition) for the holomorphic Euler-characteristic / signature interplay,
  • the Milnor-Stasheff Priority-1 units once shipped (signature-theorem and bordism arc).
• Pass-V continuity holds on the Hirzebruch-restricted scope.
• Originator-prose treatment present in units 03.06.15, 03.06.11, 04.05.10 per §4 above.

The Priority-1 punch-list (units 1–4, ~14 hours of which ~3 are shared with M-S) closes the master-tier gap on the book's central content. Priority-2 closes the signature-theorem deepening (jointly with M-S audit). Priority-3+4 are deepenings and the Borel-appendix pointer.

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§7 Sourcing

• Local copy. Not present in reference/textbooks-extra/ as of 2026-05-17; not present in reference/fasttrack-texts/.
• License. Springer Classics in Mathematics edition (1995, ISBN 3-540-58663-6) is in print and copyrighted. Cite as Hirzebruch, F., Topological Methods in Algebraic Geometry, Classics in Mathematics, Springer-Verlag 1995 (reprint of the 1978 3rd English edition); first published as Neue topologische Methoden in der algebraischen Geometrie, Springer Ergebnisse 9 (1956).
• Free academic copy. No canonical free legal mirror identified (Ranicki archive does not host it as of 2026-05-17; Springer Link page is paywalled / SSO-gated). Anna's Archive / library-genesis copies exist; per FT sourcing convention these are not the canonical citation. Recommendation: acquire the Classics paperback (~€60 retail) for the production batch; standard graduate-library holding.
• Add to local mirror. When acquired, place in reference/fasttrack-texts/04-algebraic-geometry/ as Hirzebruch-TopologicalMethodsInAlgebraicGeometry.pdf to mirror the FT-text pattern.
• Secondary references consulted for this audit pass:
  • Milnor, J. W. & Stasheff, J. D., Characteristic Classes, Annals of Mathematics Studies 76, Princeton 1974, §19 + Appendix B (multiplicative sequences, signature theorem, Bernoulli numbers).
  • Fulton, W., Intersection Theory, Springer Ergebnisse 3.Folge 2, 2nd ed., 1998, §15, §18 (Grothendieck-Riemann-Roch; pointer-only).
  • Atiyah, M. F. & Singer, I. M., "The index of elliptic operators on compact manifolds," Bull. AMS 69 (1963), 422–433 (Hirzebruch as Dolbeault / signature specialisation).
  • Bott, R. & Tu, L. W., Differential Forms in Algebraic Topology, Springer GTM 82, 1982, §20–§23 (de Rham / Chern-Weil derivation of Chern classes, splitting principle, Pontryagin classes).
  • Hartshorne, R., Algebraic Geometry, Springer GTM 52, 1977, Appendix A (cohomology-theoretic statement of HRR).
  • Griffiths, P. & Harris, J., Principles of Algebraic Geometry, Wiley 1978, Ch. 5 (analytic-geometry derivation of HRR for line bundles).
• Codex internal references. plans/fasttrack/milnor-stasheff-characteristic-classes.md (sibling audit; shared Priority-1 unit 03.06.15 and shared Priority-1/Priority-2 unit 03.06.11 — coordinate during production), plans/fasttrack/bott-tu-differential-forms.md, plans/fasttrack/lawson-michelsohn-spin-geometry.md, plans/fasttrack/hartshorne-algebraic-geometry.md, plans/fasttrack/brown-higgins-sivera-nonabelian-algebraic-topology.md (audit-format mirror), and the existing units 03.06.03–03.06.06, 03.09.10, 04.04.01, 04.05.08, 04.05.09, 04.09.01, 06.04.01.