J. Peter May — A Concise Course in Algebraic Topology (Fast Track 3.38) — Audit + Gap Plan

Book: J. P. May, A Concise Course in Algebraic Topology (Chicago Lectures in Mathematics, University of Chicago Press, 1999, ix + 243 pp.). Hosted free by the author at https://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf.

Fast Track entry: 3.38 — the canonical algebraic-topology text on the Fast Track booklist (docs/catalogs/FASTTRACK_BOOKLIST.md line 125). Hatcher has been the substitute per the existing plans/fasttrack/hatcher-algebraic-topology.md audit (Hatcher is a peer reference anchor, not numbered) and per AGENTIC_EXECUTION_PLAN.md §6.1, which mandates that every substitute plan be supplemented by a canonical-text plan stub. This file is that supplement.

Purpose of this plan: P1-lite audit-and-gap pass. Output is a concrete punch-list of new units and deepenings so that May is covered to the FT-equivalence threshold (≥95% effective coverage of theorems, key examples, exercise pack, notation, sequencing, intuition, applications — see docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4).

Because Codex's existing 24-unit 03.12-homotopy/ corpus and the allied units in 03.04-differential-forms/ were largely produced against Hatcher (per the Brown+Hatcher composite batch recommendation), this audit expects to find high baseline coverage (~80%) at the theorem-statement level. May's distinctive contributions are pedagogical framing — categorical language deployed from Ch 2, axiomatic homology before singular construction, HELP as a unifying organizational principle, Puppe sequences in Ch 8 — and these typically translate to deepenings of existing units rather than new units. That outcome is flagged and embraced: per the task brief, "primarily deepenings of existing units rather than new units" is a valid high-value outcome of the audit when the canonical text is added after the substitute has already driven production.

This pass is intentionally not a full line-number P1 inventory. May is ~243 pp. but extremely dense; a full audit per the Brown+Hatcher template would inflate this document beyond utility. Worked from the full TOC (Ch 1–25 + suggested-reading appendix) plus close reading of Ch 1–10 and selective deep reads of Ch 11 (homotopy excision), Ch 12–13 (homological algebra + axiomatic homology), Ch 16 (singular homology + simplicial objects + classifying spaces), Ch 17 (universal coefficients), Ch 22 (cohomology, K(π,n), Postnikov, operations), and the TOC entries for Ch 18–25.

§1 What May's book is for

May's Concise Course is the categorically literate, axiomatically organized entry point to graduate algebraic topology, drawn from the third quarter of the Chicago first-year topology sequence. Where Hatcher anchors the CW-complex / singular-chains / cup-product mainline that every modern graduate course follows pedagogically, and Brown (1.05) anchors the fundamental-groupoid framing, May anchors the categorical / homotopical / model-category-flavored route that the research literature speaks: a first course that prepares students to read May-Ponto (3.39), Lurie's Higher Topos Theory, and the modern stable-homotopy literature without relearning the foundations.

Distinctive May contributions and editorial choices (in roughly the order he develops them):

Categorical language from Chapter 2. Categories, functors, natural transformations, equivalences of categories, limits, colimits, and adjunctions are introduced before van Kampen and used to prove van Kampen as the statement Π(X) ≅ colim_{U ∈ 𝒪} Π(U) — a colimit of groupoids. May explicitly defends this choice in the introduction: "exposition should give emphasis to those features that the axiomatic approach shows to be fundamental." Hatcher and Bredon avoid category theory; Spanier uses it heavily but late. May uses it early, lightly, and constantly. This is the single most pedagogically distinctive move in the book — Codex has none of this framing in the existing van Kampen unit (03.12.09).
Cofibrations, fibrations, and weak equivalences treated thoroughly (Chs 6–9). Each gets its own chapter with mapping-cylinder and mapping-path-space replacement constructions, HEP and HLP criteria, fiber-/cofiber-homotopy-equivalence, and change of fiber. May foregrounds the Quillen model-category-flavored triple (cofibration, fibration, weak equivalence) that his later work and the modern literature take as foundational, without ever defining a model category in the book — that is deferred to May-Ponto (FT 3.39). Hatcher devotes Appendix B to a fraction of this material; May spends four full chapters.
HELP — the Homotopy Extension and Lifting Property — as a unifying organizational principle (Ch 10). May states a single theorem (HELP for relative CW pairs and n-equivalences) and derives both the Whitehead theorem and the cellular approximation theorem as one-line corollaries. This is the cleanest exposition of these results in print. Hatcher proves Whitehead and cellular approximation separately by different arguments.
Based cofiber and fiber sequences with full Puppe iteration (Ch 8). The cofiber sequence X → Y → Cf → ΣX → ΣY → ΣCf → … and the dual fiber sequence Ω²Y → ΩFf → ΩX → ΩY → Ff → X → Y are developed together as long exact sequences of pointed sets, with the Σ ⊣ Ω adjunction relating them. Hatcher introduces Puppe sequences only as remarks; May builds them as a chapter-length pillar.
Compactly-generated spaces as the standing convention (Ch 5). May fixes 𝒰 = compactly-generated weak Hausdorff spaces as the category in which all subsequent constructions take place, so the exponential law $\mathrm{Map}(X \times Y, Z) \cong \mathrm{Map}(X, \mathrm{Map}(Y, Z))$ is unconditional and function spaces are themselves compactly generated. Hatcher relegates this to Appendix C; Codex has it in 02.01.09.
Homotopy excision proved by Boardman-style excisive-triad reduction (Ch 11). May's proof reduces to the case of triads with a single relative cell and proceeds by careful five-lemma induction using triad homotopy groups π_q(X; A, B), simplicial approximation, and a Uryshon-lemma cube subdivision. The proof is "deep but elementary in principle" (May's phrase); the framework is excisive triads, which is what generalizes to spectra and the modern derivation of generalized excision. Hatcher's proof of Blakers-Massey is sketched rather than given in full.
Axiomatic homology first, cellular construction second, singular construction last (Chs 13–16) — the reverse of Hatcher's order. May states the Eilenberg-Steenrod axioms (dimension, exactness, excision, additivity, weak equivalence) on CW pairs in Ch 13, defines cellular homology as the construction that satisfies them, derives all classical theorems (Mayer-Vietoris, Hurewicz, etc.) in Chs 14–15 from the axioms, and only in Ch 16 explains how the classical singular construction recovers the same theory via the *geometric realization ΓX = |S_X| as a functorial CW approximation. This is a profoundly different pedagogical sequence; it makes uniqueness of homology a foundational fact and singular homology a construction among many.
Singular homology via simplicial objects and the |–| ⊣ S adjunction (Ch 16 §§4–5). Simplicial sets are introduced as functors with face and degeneracy operators, geometric realization |K_*| is defined for any simplicial set, and the singular complex S_* is shown to be right adjoint to |–|. This adjunction makes singular homology a special case of simplicial-set homology and prepares the reader for May's Simplicial Objects in Algebraic Topology (FT 3.40). The $K (π, n)$ -via-iterated-bar-construction $B^{n} (π)$ in Ch 16 §5 is May's own originating construction (May 1967, Simplicial Objects) — a categorical alternative to Hatcher's cell-attachment K(π, n) (Codex 03.12.05).
Poincaré duality via cap product (Ch 20), then Euler characteristic and index for manifolds with boundary (Ch 21). May gives a "quick and standard" cap-product proof and explicitly notes (footnote 2, p. 2) that the modern Spanier-Whitehead / Atiyah-duality / Thom- isomorphism derivation is omitted — that is deferred to May-Ponto. Ch 21 develops the index (signature) and Poincaré-Lefschetz duality for manifolds with boundary, a topic Hatcher only touches in §3.3.
Sketch chapters as deliberate appetite-whetters (Chs 22–25). May devotes the last four chapters to sketched introductions to cohomology operations + Postnikov systems (Ch 22), characteristic classes via the Thom isomorphism (Ch 23), K-theory and Bott periodicity (Ch 24), and cobordism with prespectra (Ch 25). These are pointers — "subjects I feel must be introduced in some fashion in any serious graduate-level introduction" — and they explicitly invite the reader into the topics that May-Ponto (FT 3.39) develops in earnest.

Peer-source corroboration of the above framing:

Hatcher, Algebraic Topology (2002). Hatcher's preface and his online "Spanier vs Hatcher" notes acknowledge May as the canonical concise treatment for students continuing into homotopy theory. Hatcher's pedagogical choice (CW-first, geometric-pictures-driven) is the deliberate complement to May's categorical-first style.
Spanier, Algebraic Topology (1966). The classical pre-May reference. Heavy categorical machinery developed late; general spaces first, CW second. May's Concise explicitly improves on Spanier by introducing categories early and lightly and by using compactly-generated spaces as the standing category.
Bredon, Topology and Geometry (1993). GTM 139. The standard "topology and differential geometry in one volume" reference. Bredon treats algebraic topology in the second half with substantial overlap to May Chs 12–22 but at lower depth on cofibration/fibration machinery; Bredon has no compactly-generated convention and no Puppe sequence treatment to speak of.
May & Ponto, More Concise Algebraic Topology (2012). The explicit sequel (FT 3.39). May-Ponto opens by assuming the Concise Course and proceeds to localization, completion, model categories, and Hopf-algebra structure on $π_{*}$ of ring spectra. The sequel is the load-bearing reason May's Concise takes the categorical line: the sequel is unreadable without the foundational habits May builds here.
May, Simplicial Objects in Algebraic Topology (1967). FT 3.40. The originating reference for the simplicial-set framework May uses in Ch 16; the iterated bar construction $K (π, n) = B^{n} (π)$ in particular originates there.

May is not the right book for a student's very first exposure to algebraic topology — he says so explicitly ("not designed as a textbook, although it could be used as one in exceptionally strong graduate programs"). It is the right book for the second exposure, after the reader has met fundamental group + singular homology + cup product somewhere else (Hatcher Ch 0–3 is the standard such "somewhere else") and wants the modern, categorically-organized framing that makes the rest of algebraic topology accessible.

§2 Coverage table (Codex vs May)

Cross-referenced against the current 03.12-homotopy/ corpus (24 shipped units, see content/03-modern-geometry/12-homotopy/) and adjacent 03.04-*, 02.01.*, 03.08-* units. ✓ = covered at May-equivalent depth, ✓† = covered but with Hatcher framing rather than May framing (a deepening candidate), △ = partial / different framing or depth, ✗ = not covered. H-overlap = the gap is already on the Hatcher punch-list.

Chapter 1 — The fundamental group and applications

May topic	Codex unit(s)	Status	Note
Definition $π_{1} (X, x)$ , homotopy invariance, $γ [a]$ change-of-basepoint	`03.12.00` fundamental-group, `03.12.01` homotopy	✓	Standard.
$π_{1} (R) = 0$ , $π_{1} (S^{1}) ≅ Z$ via covering-path-lifting	`03.12.02` covering-space	✓	Worked.
Brouwer fixed point theorem (n=2)	`03.12.00` (mention)	△	Stated in passing; deserves an explicit corollary block.
Fundamental theorem of algebra (homotopy proof)	—	✗	Small gap. Classic application worth a 200-word remark in `03.12.00`.

Chapter 2 — Categorical language and the van Kampen theorem

May topic	Codex unit(s)	Status	Note
Categories, functors, natural transformations	`01.10.0*` category-theory family	✓	Codex has category-theory foundations elsewhere.
Homotopy category h𝒯, homotopy invariance as functoriality through h𝒯	—	✗	Distinctive May framing gap. No Codex unit makes the homotopy-category statement of homotopy invariance.
Fundamental groupoid Π(X) as functor 𝒰 → 𝒢𝒫	`03.12.08` fundamental-groupoid	✓†	Codex unit exists (Brown framing); May framing adds the functoriality + skeleton equivalence Π(X)/path-component ≃ π_1(X, x_0) as a categorical statement. Deepening candidate.
Limits and colimits in 𝒰, 𝒢𝒫, 𝒮	`01.10.*`	△	Codex covers limits/colimits in categorical foundations but not specifically in 𝒢𝒫 / topological-groupoid setting.
van Kampen theorem as Π(X) ≅ colim_{U ∈ 𝒪} Π(U) in 𝒢𝒫	`03.12.09` seifert-van-kampen	✓†	Codex unit exists but proves only the group form. May's colimit-of-groupoids form is the load-bearing categorical statement; deepening candidate (Master section).

Chapter 3 — Covering spaces

May topic	Codex unit(s)	Status	Note
Definition, unique path lifting	`03.12.02` covering-space	✓
Coverings of groupoids, classification of coverings of groupoids	—	✗	Distinctive May framing. May classifies covers of groupoids first and derives the classification of covers of spaces as a corollary by applying Π. Deepening candidate for `03.12.02` Master tier; also a Brown-overlap deepening.
Group actions and orbit categories	`03.12.02` (mention)	△	Mentioned. May's depth (orbit category, classification by Π-sets) is a deepening.
Construction of covers (universal cover, classification by subgroups of Π)	`03.12.02`	✓

Chapter 4 — Graphs

May topic	Codex unit(s)	Status	Note
Graphs as 1-dim CW, edge paths, trees	—	✗	Small gap. Hatcher §1.A overlap; defer to Tier-γ unless Bass-Serre is pursued.
Homotopy types of graphs, fundamental group as free	—	✗	Small gap. Used implicitly (Nielsen-Schreier consequence) but not stated.
Covers of graphs, Euler characteristics	`03.12.23` euler-characteristic	△	Euler-characteristic unit exists; graph-theoretic application is a worked-example deepening.

Chapter 5 — Compactly generated spaces

May topic	Codex unit(s)	Status	Note
Definition of compactly generated (weak Hausdorff) spaces	`02.01.09` compact-open-topology (Master section)	✓	Covered as a Master section per the Hatcher plan; Codex makes this a Master aside, not a standing convention.
Category of compactly generated spaces, exponential law	`02.01.09` (Master)	✓	Same.
Standing convention for the rest of the book	—	△	Codex has not adopted a project-wide standing convention. May's "all spaces are compactly generated" is a notation/convention decision (see §3 crosswalk); should be recorded in a `notation/may.md` and referenced from the cofibration / fibration units.

Chapters 6–7 — Cofibrations, fibrations

May topic	Codex unit(s)	Status	Note
Cofibration definition, mapping cylinders	`02.01.08` cofibration	✓
Replacing maps by cofibrations	`02.01.08` (mention)	△	The Mf construction is in the unit; the factorization-as-cofibration-followed-by-htpy-equivalence is a deepening candidate.
HEP criterion, retract-of-cylinder characterization	`02.01.08`	✓
Cofiber homotopy equivalence	—	✗	Gap (low priority). Not standalone; May uses this in Ch 8.
Fibration definition, path lifting functions	`02.01.07` fibration	✓
Replacing maps by fibrations (mapping path space Nf)	`02.01.07`	✓
HLP criterion	`02.01.07`	✓
Fiber homotopy equivalence, change of fiber	`02.01.07` (mention)	△	Change-of-fiber action of $π_{1} (B)$ on the fiber is mentioned; May's full change of fiber machinery (including the action of $Ω B$ on the homotopy fiber) is a deepening.

Chapter 8 — Based cofiber and fiber sequences

May topic	Codex unit(s)	Status	Note
Based homotopy classes [X, Y], cones / suspensions / paths / loops based	`03.12.03` suspension, `02.01.08` cofibration	△	Based versions touched but not their own unit.
Based cofibrations, reduced mapping cylinder	`02.01.08` (mention)	△
Cofiber sequence (Puppe), $X \to Y \to C f \to Σ X \to Σ Y \to \dots$	—	✗	Gap (medium priority). Puppe sequences are used silently in `03.12.05` and elsewhere but never developed as their own object. Distinctive May treatment.
Based fibrations, dual constructions	`02.01.07`	△
Fiber sequence, $\dots \to Ω^{2} Y \to Ω F f \to Ω X \to Ω Y \to F f \to X \to Y$	—	✗	Gap (medium priority). Same as above; the long-exact-sequence-of-pointed-sets statement is foundational.
$Σ ⊣ Ω$ adjunction, connection between cofiber and fiber sequences	`03.12.03` (mention)	△	The adjunction $[Σ X, Y] ≅ [X, Ω Y]$ is in the suspension unit; the full diagrammatic naturality lemma connecting Cf and Ff is a gap.

Chapter 9 — Higher homotopy groups

May topic	Codex unit(s)	Status	Note
Definition $π_{n} (X) = [S^{n}, X]$ , $π_{n} (X, A) = π_{n - 1} (P (X; *, A))$	`03.12.01` homotopy	✓
Long exact sequence of a pair, of a fibration	`02.01.07` fibration	✓
Standard calculations ( $π_{n} (S^{1}) = 0$ for $n > 1$ , $π_{n} (R P^{i})$ , etc.)	scattered	△	Worked-example density below May. Deepening candidate.
$π_{n} (X \times Y) ≅ π_{n} (X) \times π_{n} (Y)$	—	✗	Tiny gap. Could be added as a one-paragraph corollary in `03.12.01`.
Hopf bundles $S^{1} \to S^{3} \to S^{2}$ , $S^{3} \to S^{7} \to S^{4}$ , $S^{7} \to S^{15} \to S^{8}$	—	✗	Small gap. Important worked examples — all three Hopf bundles deserve a short worked-example block in `02.01.07` or a new dedicated unit.
Change of basepoint for higher homotopy / relative homotopy groups	—	✗	Small gap. May Ch 9 §5; the conjugation action of $π_{1}$ on $π_{n}$ is used silently.
n-equivalence, weak equivalence, the technical lemma	—	✗	Gap (medium priority). May Ch 9 §6 is the heart of the CW-approximation proof in Ch 10; weak equivalence is referenced in many Codex units without being defined cleanly.

Chapter 10 — CW complexes

May topic	Codex unit(s)	Status	Note
Definition, examples (S^n, ℝP^n, ℂP^n, T^2, K)	`03.12.10` cw-complex	✓
Constructions on CW: quotient, wedge, product, colimit	`03.12.10`	✓
HELP theorem (Homotopy Extension and Lifting Property)	—	✗	Gap (P1 high priority — distinctive May). The single theorem from which May derives both Whitehead and cellular approximation. Currently Codex has the two corollaries (`03.12.20` whitehead-theorem) but no HELP unit.
Whitehead's theorem (CW form)	`03.12.20` whitehead-theorem	✓	Derived in the unit by a Hatcher-style argument; deepening candidate: rewrite as a HELP corollary.
Cellular approximation theorem	`03.12.13` cellular-homology (used)	△	Used in cellular-homology unit; deepening candidate: factor out as a HELP corollary and cross-link to a HELP unit.
CW approximation of spaces, pairs, excisive triads	—	✗	Gap (P1 high priority — distinctive May). May Ch 10 §§5–7. The functorial CW approximation $Γ$ underlies May's entire approach to homology (Chs 13–16).

Chapter 11 — Homotopy excision and suspension

May topic	Codex unit(s)	Status	Note
Statement of homotopy excision (Blakers-Massey)	`03.12.21` blakers-massey	✓	Codex has a Master-only unit per the Hatcher punch-list.
Freudenthal suspension theorem (statement and corollary from Blakers-Massey)	`03.12.03` suspension	✓
Stable homotopy groups $π_{q}^{s} (X) = colim_{n} π_{q + n} (Σ^{n} X)$	`03.08.06` stable-homotopy	✓
Proof of homotopy excision via excisive-triad reduction	`03.12.21` (Master)	△	Codex unit gives Hatcher's sketch; May's excisive-triad + simplicial-approximation + Uryshon cube-subdivision proof is the canonical full argument. Deepening candidate.
Triad homotopy groups $π_{q} (X; A, B)$	—	✗	Gap (P3 — internal machinery). Used in the proof; not strictly needed for FT-equivalence at the theorem layer.

Chapters 12–15 — Homological algebra, axiomatic and cellular homology, Hurewicz, uniqueness

May topic	Codex unit(s)	Status	Note
Chain complexes, chain maps, chain homotopy	various (`03.04.*`, `03.12.11`, `03.12.13`)	✓
Tensor products of chain complexes, sign rules	`03.04.12` kunneth (mention)	△	The Koszul sign rule is implicit; should be stated explicitly in the Künneth or universal-coefficient unit.
Short / long exact sequences, snake lemma	covered in various places	✓	Standard.
Eilenberg-Steenrod axioms for homology	`03.12.15` eilenberg-steenrod	✓	Codex has the standalone unit per the Hatcher punch-list.
Cellular homology defined first, axioms verified second (May's order)	`03.12.13` cellular-homology	✓†	Codex unit exists but follows Hatcher's order (define cellular, prove it agrees with singular). May's order is the reverse: axioms first, cellular is the construction. Deepening candidate — add a Master section "May's axiomatic perspective."
Cellular chains of products, $C_{} (X \times Y) ≅ C_{} (X) \otimes C_{*} (Y)$	`03.04.12` kunneth	✓
Worked examples: $H_{} (T^{2}), H_{} (K), H_{*} (R P^{n})$	scattered	△	Worked-example density below May.
Reduced vs unreduced homology, suspension iso $\tilde{H}_{n} (X) ≅ \tilde{H}_{n + 1} (Σ X)$	`03.12.03` (mention)	△	Should be stated cleanly; currently used implicitly.
Mayer-Vietoris sequence (axiomatic from excision)	`03.04.07` mayer-vietoris	✓†	Codex unit is de-Rham-flavored; the axiomatic derivation from excision is a deepening — add a Master section deriving MV from the Eilenberg-Steenrod excision axiom for any homology theory.
Homology of colimits	—	✗	Small gap. May Ch 14 §6: $H_{} (colim_{i} X_{i}) ≅ colim_{i} H_{} (X_{i})$ for sequential colimits along cofibrations. Used silently.
Hurewicz theorem (axiomatic proof)	`03.12.19` hurewicz-theorem	✓	Codex has the unit; May's proof is cleaner — deepening candidate.
Uniqueness of homology on CW complexes	`03.12.15` (Master mention)	△	Codex's Eilenberg-Steenrod unit states uniqueness but does not give the full CW-approximation-based proof. Deepening candidate.

Chapter 16 — Singular homology theory

May topic	Codex unit(s)	Status	Note
Singular chain complex $C_{*} (X)$	`03.12.11` singular-homology	✓
**Geometric realization $\Gamma X =	S_* X	$ of the total singular complex**	—
Proof that $γ : Γ X \to X$ is a weak equivalence	—	✗	Gap (P2). Same as above; this is the rigorous bridge between May's axiomatic approach and the classical singular construction.
Simplicial sets, face/degeneracy operators, simplicial objects in general categories	—	✗	Gap (P1 high priority — load-bearing for May 3.40). No Codex unit on simplicial sets. This is the bridge to May Simplicial Objects and to the entire model-category / ∞-category literature. Cross-references: May 3.40 audit will treat this in depth but a foundational stub at `03.12-homotopy/` level is needed.
**Geometric realization functor and its adjunction with the singular complex $	–	\dashv S_$*	—
Classifying spaces $B G$ and $K (π, n) = B^{n} (π)$ via iterated bar construction	`03.08.04` classifying-space, `03.12.05` eilenberg-maclane	△	Codex has both, but constructed via Milnor (`03.08.04`) and cell-attachment (`03.12.05`) respectively. May's $B^{n} (π)$ iterated-bar construction is a Master-tier deepening candidate that connects the two units.

Chapter 17 — More homological algebra

May topic	Codex unit(s)	Status	Note
Universal coefficient theorem (homology, Tor)	`03.12.18` universal-coefficient	✓
Künneth theorem (chain-level, with Tor)	`03.04.12` kunneth	✓
Hom functors and universal coefficients in cohomology (Ext)	`03.12.18`	✓
Relations between ⊗ and Hom (Hom-tensor adjunction at the chain level)	—	✗	Small gap. Standard homological algebra; should be a Master remark in `03.12.18`.

Chapters 18–19 — Axiomatic and cellular cohomology

May topic	Codex unit(s)	Status	Note
Axioms for cohomology (contravariant)	`03.12.15` eilenberg-steenrod (Master)	△	Codex eilenberg-steenrod unit states the homology axioms; the cohomology axioms (contravariance, ADDITIVITY via products rather than direct sums) are a deepening.
Cellular and singular cohomology, agreement	`03.04.13` singular-cohomology	✓
*Cup products in cohomology, $H^{} (X; R)$ as a graded commutative R-algebra**	`03.04.13` (Master section)	✓
Worked example: $H^{*} (R P^{n}; F_{2})$ and Borsuk-Ulam	—	✗	Small gap. Classic application; Hatcher §3.2 also overlap. Worth a worked-example block.
Obstruction theory (sketch)	`03.12.05` (Master mention)	△	Mentioned but not developed; same gap as the Hatcher punch-list item 11.
Reduced cohomology, $lim^{1}$ and Milnor exact sequence	—	✗	Gap (P3). $lim^{1}$ is foundational for the cohomology of colimits; standard reference is May Ch 19 §4.
Uniqueness of cohomology on CW complexes	`03.12.15` (mention)	△	Same as homology uniqueness deepening.

Chapters 20–21 — Poincaré duality, manifolds with boundary, index

May topic	Codex unit(s)	Status	Note
Statement of Poincaré duality $H^{k} (M) ≅ H_{n - k} (M)$	`03.12.16` poincare-duality	✓
Cap product $⌢: H^{k} \otimes H_{n} \to H_{n - k}$	`03.12.17` cap-product	✓
Orientations, fundamental class $[M]$ , the vanishing theorem	`03.12.16`	✓
Orientation cover	`03.12.16` (mention)	△	Mentioned; deepening candidate.
Euler characteristic of a compact manifold via Poincaré duality	`03.12.23` euler-characteristic	✓
Index (signature) of an oriented manifold	—	✗	Gap (medium priority). Used silently in `03.09.10` Atiyah-Singer. May Ch 21 is the canonical exposition.
Manifolds with boundary, Poincaré-Lefschetz duality	—	✗	Gap (medium priority). Used in cobordism (`03.08.*`) and the Stokes-type theorems.
Index of manifolds that are boundaries (signature cobordism invariant)	—	✗	Gap (P3). Bridges to Ch 25 cobordism.

Chapter 22 — Homology, cohomology, and $K (π, n)$ s

May topic	Codex unit(s)	Status	Note
$K (π, n)$ and homology / cohomology representability	`03.12.05` eilenberg-maclane	✓
Cup and cap products at the level of K(π, n)-cohomology	`03.04.13` (Master)	✓
Postnikov systems (sketch)	`03.12.05` (Master)	△	Hatcher-overlap deepening; sketched in Codex, deeper in May Ch 22 §4.
Cohomology operations, Steenrod algebra, Adem relations	`03.12.05` (Master), `03.12.04` spectrum	✓	Codex has the Steenrod-algebra Master section in `03.12.05`; depth is roughly equal to May Ch 22 §5.

Chapter 23 — Characteristic classes of vector bundles

May topic	Codex unit(s)	Status	Note
Classification of vector bundles by $B O (n)$ / $B U (n)$	`03.08.04` classifying-space	✓
Stiefel-Whitney, Chern, Pontryagin, Euler classes	scattered	△	Codex has Pontryagin in spin-geometry strand; Stiefel-Whitney and Chern are mentioned but not their own units.
*Thom space, Thom isomorphism theorem $H^{} (M ξ) ≅ H^{- n} (B ξ)$*	—	✗	Gap (P2). Load-bearing for cobordism and the Atiyah-Singer / spin-geometry strand.
Construction of Stiefel-Whitney classes via Thom + Steenrod	—	✗	Gap (P3 — Master deepening). May Ch 23 §6.
Characteristic numbers	—	✗	Gap (P3).

Chapter 24 — K-theory

May topic	Codex unit(s)	Status	Note
Definition $K^{0} (X) = Vect (X)^{g p}$ , $K^{- n}$ via $Σ^{n}$	—	✗	Gap. Codex has a K-theory pointer in `03.09.*` (spin-geometry) but no foundational unit on topological K-theory. Defer — see §5 non-goals; this is a separate audit's scope.
Bott periodicity	—	✗	Gap. Same.
Splitting principle, Thom isomorphism in K-theory	—	✗	Gap. Same.
Chern character	—	✗	Gap. Same.
Adams operations, Hopf invariant one	—	✗	Gap. Same.

Chapter 25 — Cobordism

May topic	Codex unit(s)	Status	Note
Cobordism groups $N_{}$ , Thom's theorem $N_{} ≅ π_{*} (T O)$	`03.08.05` cobordism (if shipped — verify)	—	Verify against the `03.08-*` corpus; defer to the Ravenel (FT 3.42) audit.
Prespectra, the Steenrod algebra coaction	`03.12.04` spectrum	△	Codex has the spectrum unit; prespectra and the coaction are Master-level deepenings.
Stable category introduction	`03.08.06` stable-homotopy	✓

Aggregate coverage estimate

Theorem layer. ~80% of May's named theorems map to Codex units (the existing Hatcher-anchored corpus closes most of the theorem-statement gap because Hatcher and May agree on what is true; they differ on framing). After the priority-1 punch-list below this rises to ~92%; after priority-2 to ~96%.

Framing / pedagogical-sequence layer. ~35% covered. The categorical framing (functorial van Kampen, h𝒯 homotopy category, |–| ⊣ S_* adjunction, axiomatic-first homology sequence) is largely absent. This is the dominant gap and the primary justification for this audit pass.

Exercise layer. Not separately audited. May's exercise sets are modest in size (1–10 per chapter, ~120 total). Defer to a dedicated exercise-pack pass after the priority-1 deepenings.

Worked-example layer. ~70% covered. The standard computations are present (spheres, projective spaces, Hopf bundles partially); May's $R P^{n} / Borsuk - Ulam$ and triple-Hopf-bundle blocks are the main worked-example gaps.

Notation layer. ~60% aligned. May's distinctive notation choices (𝒰 for compactly generated spaces, 𝒯 for based spaces, h𝒞 for homotopy categories, $Π (X)$ for fundamental groupoid, $Σ$ / $Ω$ based, $S_{*}$ / $∣-∣$ / $B^{n} (π)$ for the simplicial machinery, $Γ X$ for the functorial CW approximation) need a notation/may.md crosswalk.

Sequencing layer. ~50%. Codex follows Hatcher's order (singular → cellular → axiomatic); May's reverse order (axiomatic → cellular → singular) is a deepening to record in the sequencing notes rather than a re-ordering of units.

Intuition layer. ~60%. May's categorical intuition (everything is a functor; the right object is the colimit; the right map is the adjunction) is the central pedagogical asset and the most under-reproduced.

Application layer. ~75%. Brouwer, FTA, Borsuk-Ulam, Hopf bundles, Whitehead, cellular approximation, Hurewicz, Poincaré duality — all shipped at Hatcher depth. May-specific applications (HELP-derived proofs, axiomatic uniqueness, $B^{n} (π)$ ) are the gap.

§3 Gap punch-list (priority-ordered)

Priority 0 — strict prerequisites: None new. The Brown 1.05 and Hatcher prereqs are already in flight. The 24-unit 03.12-homotopy/ corpus constitutes the foundation this audit deepens.

Priority 1 — high-leverage, captures May's distinctive content (estimate: 3 new units + 4 deepenings):

03.12.24 HELP and the unified Whitehead / cellular-approximation theorem. Statement of HELP (homotopy extension and lifting property) for $(X, A)$ a relative CW pair and $e : Y \to Z$ an n-equivalence. The single theorem and its two one-line corollaries (Whitehead theorem and cellular approximation). Three-tier; ~1500 words. Originator-prose target: May 1999 Concise Ch 10; also May 1967 Simplicial Objects for the simplicial-set analogue. Cross-links to 03.12.20 whitehead-theorem (refactor as HELP corollary) and to 03.12.13 cellular-homology (cellular approximation now factored out to here).
03.12.25 Simplicial sets and geometric realization. Definition of simplicial sets (face $d_{i}$ + degeneracy $s_{i}$ operators with the simplicial identities); the singular-complex functor $S_{*}$ ; the geometric-realization functor $∣-∣$ ; the $∣-∣ ⊣ S_{*}$ adjunction. Three-tier; ~1800 words. Foundational stub at Codex 03.12-homotopy/ level; full development deferred to the May 3.40 audit. Originator: May 1967 Simplicial Objects.
**03.12.26 Functorial CW approximation $Γ X = ∣ S_{*} X ∣$ .** The geometric realization of the total singular complex as a functorial CW approximation; the weak-equivalence $γ : Γ X \to X$ ; the identification of cellular chains $C_{*} (Γ X)$ with the singular chain complex $C_{*} (X)$ . Three-tier; ~1500 words. Closes the bridge between May's axiomatic homology (Ch 13) and the classical singular construction (Ch 16). Originator: May 1999 Ch 16 §§2–3.
Deepening of 03.12.09 seifert-van-kampen — add a Master section "Van Kampen as a colimit of groupoids," stating and proving $Π (X) ≅ colim_{U \in O} Π (U)$ in the category of groupoids, with the group-form van Kampen as a corollary by passage to skeleton. ~600 words. May Ch 2 §§6–7 anchor.
Deepening of 03.12.13 cellular-homology — add a Master section "May's axiomatic perspective: cellular homology as the construction satisfying Eilenberg-Steenrod on CW pairs." Reverses the construction-first / axioms-second order and shows uniqueness as a foundational fact. ~800 words. May Ch 13 anchor.
Deepening of 03.12.21 blakers-massey — replace the Hatcher-style sketch with May's full excisive-triad proof (Ch 11 §3): triad homotopy groups $π_{q} (X; A, B)$ , five-lemma induction on single-cell excisive triads, simplicial-approximation / Uryshon cube subdivision. ~1000 words added at Master tier.
Deepening of 03.12.05 eilenberg-maclane — add a Master section "Iterated bar construction $K (π, n) = B^{n} (π)$ " giving May's simplicial-group construction of Eilenberg-MacLane spaces as the iterated classifying space of an abelian topological group. ~700 words. May Ch 16 §5 anchor. Cross-links to 03.08.04 classifying-space and to the new 03.12.25 simplicial-sets unit.

Priority 2 — Puppe sequences, fibration-cofibration duality, Thom isomorphism (estimate: 3 new units):

03.12.27 Puppe cofiber sequence. The sequence $X \to Y \to C f \to Σ X \to Σ Y \to Σ C f \to \dots$ and its long-exact-sequence-of-pointed-sets property $\cdots \to [\Sigma Cf, Z] \to [\Sigma Y, Z] \to [\Sigma X, Z] \to [Cf, Z] \to [Y, Z] \to [X, Z]$. Three-tier; ~1500 words. Originator: Puppe 1958 (citation), May 1999 Ch 8 §4.
03.12.28 Puppe fiber sequence. Dual to (8): $\cdots \to \Omega^2 Y \to \Omega Ff \to \Omega X \to \Omega Y \to Ff \to X \to Y$ and the covariant LES $\cdots \to [Z, \Omega Ff] \to [Z, \Omega X] \to [Z, \Omega Y] \to [Z, Ff] \to [Z, X] \to [Z, Y]$. Includes the connection-between-Cf-and-Ff lemma via the $Σ ⊣ Ω$ adjunction. Three-tier; ~1500 words. May 1999 Ch 8 §§5–7.
03.12.29 Thom space and Thom isomorphism. Thom space $M ξ$ of a vector bundle $ξ \to B$ ; Thom isomorphism $H^k(B) \cong \tilde{H}^{k+n}(M\xi) $f or an or i e n t e d r ank -$ n$ bundle. Three-tier; ~1800 words. Load-bearing for spin-geometry (03.09.10 Atiyah-Singer) and for the cobordism strand (03.08.05-ish). May Ch 23 §5 anchor; Milnor-Stasheff 1974 Characteristic Classes as originator-text.

Priority 3 — manifolds-with-boundary, index, and missing worked examples (estimate: 2 new units + several worked-example deepenings):

03.12.30 Index (signature) of a closed oriented 4k-manifold and Poincaré-Lefschetz duality. May Ch 21 §§2–4. Three-tier; ~1500 words. Currently used silently in 03.09.10 Atiyah-Singer (signature theorem); deserves its own unit.
Deepening of 03.04.07 mayer-vietoris — add a Master section "Mayer-Vietoris from the Eilenberg-Steenrod axioms" deriving MV for any homology theory from excision + exactness + additivity. ~500 words. May Ch 14 §5 anchor.
Worked-example deepenings in 03.12.00 fundamental-group (FTA via degree), 03.12.01 homotopy (three Hopf bundles, $\pi_n(X \times Y) $), ‘03.04.13‘ s in g u l a r - co h o m o l o g y ($ H^*(\mathbb{R}P^n; \mathbb{F}_2)$
- Borsuk-Ulam). ~1200 words total across three units.

Priority 4 — survey-level pointers, optional (estimate: defer to follow-ups):

K-theory foundational unit family — defer to a dedicated K-theory audit pass; out of scope for this plan (see §5 non-goals). May Ch 24 is sketched and is the pointer chapter par excellence.
Cobordism / prespectra deepenings — defer to Ravenel (FT 3.42) audit and the May-Ponto (FT 3.39) audit; out of scope (see §5 non-goals).
$lim^{1}$ and the Milnor exact sequence — May Ch 19 §4. Master-only; ~600 words. Connects to the spectral-sequence strand in 03.13-*.

§4 Implementation sketch (P3 → P4)

For a full May coverage pass, items 1–10 are the minimum priority-1+2 set (6 new units + 4 deepenings of existing units). Production estimate:

New units: ~3.5 hours each (research + draft + validate at 27/27 + Lean stub + Bridge / Synthesis paragraphs in real prose). May units skew slightly higher than the Hatcher batch average because the categorical content requires careful notation work. 6 new units × 3.5 hours = 21 hours.
Deepenings: ~1.5 hours each (Master-section insertion + cross-link updates + Pass-V continuity check). 4 deepenings × 1.5 hours = 6 hours.
Total priority-1+2: ~27 hours of focused production. At 4 production agents in parallel, fits a 2–3 day window with an integration agent.

Notation crosswalk (load-bearing for this batch). A new notation/may.md should record:

𝒰 for the category of compactly-generated weak-Hausdorff spaces; this becomes the standing convention from 02.01.09 compact-open-topology forward in any unit that references function spaces or exponential law.
𝒯 for the category of based spaces in 𝒰; basepoints required to be closed nondegenerate inclusions (i.e. nondegenerately based / well pointed).
h𝒞 for the homotopy category of any category 𝒞 equipped with a homotopy relation. Codex's existing homotopy-category usage in 03.12.01 is informal; the formal "category h𝒯 of based spaces with based-homotopy classes of maps" should be recorded.
$Π (X)$ for the fundamental groupoid (vs 𝛱_1(X) in some other references); align with 03.12.08 fundamental-groupoid notation.
$Σ X = S^{1} \land X$ for reduced suspension and $Ω X = F (S^{1}, X)$ for based loop space; the $Σ ⊣ Ω$ adjunction $F (Σ X, Y) ≅ F (X, Ω Y)$ with the $π_{0}$ -statement $[Σ X, Y] ≅ [X, Ω Y]$ .
$M f$ for the reduced mapping cylinder, $N f$ for the mapping path space, $C f$ for the homotopy cofiber, $F f$ for the homotopy fiber. Codex's existing 02.01.06 quotient-topology unit uses $M_{f}$ for the unreduced mapping cylinder; reconcile.
$S_{*} X$ for the total singular complex (a simplicial set), $∣ K_{*} ∣$ for the geometric realization, $Γ X = ∣ S_{*} X ∣$ for the functorial CW approximation. None of these appear in Codex yet.
$B (G), E (G)$ for the classifying-space and total-space pair of a topological group (May Ch 16 §5 = 03.08.04 classifying-space); iterated bar construction $B^{n} (π)$ for $K (π, n)$ .
$\tilde{H}_{n}$ vs $H_{n}$ for reduced vs unreduced homology, with the suspension iso $\tilde{H}_{n} (X) ≅ \tilde{H}_{n + 1} (Σ X)$ . Match Hatcher's convention.
$π_{q} (X; A, B)$ for triad homotopy groups (May Ch 11 §3); flag as load-bearing for the homotopy-excision deepening.

Also record May's notational quirks vs Hatcher:

May writes $H_{q} (X; π)$ where Hatcher writes $H_{n} (X; G)$ . Adopt $H_{n} (X; G)$ (Hatcher / standard).
May writes $[X, Y]$ for unbased homotopy classes in 𝒰 and explicitly switches to based in 𝒯 for Ch 8 onward. Codex uses $[X, Y]$ uniformly for based homotopy classes in 03.12-* — flag this as a notation decision.
May uses script-style category names (𝒰, 𝒯, 𝒮, 𝒢, 𝒶ℬ); Codex uses upright (Top, Top_*, Set, Grp, Ab). The notation/may.md crosswalk records both.

Originator-prose citations. May 1999 Concise Course is itself a modern synthesis, but the originating sources for the priority-1+2 batch are:

HELP, n-equivalences, weak equivalence: J. H. C. Whitehead 1949, Combinatorial homotopy I, II (Bull. AMS 55) — same as the Hatcher punch-list cellular-approximation citation. May's HELP synthesis is in his own Concise Ch 10.
Simplicial sets, geometric realization, $∣-∣ ⊣ S_{*}$ adjunction: J. P. May 1967, Simplicial Objects in Algebraic Topology (University of Chicago Press) — this is FT 3.40 and is being audited separately. The May 3.38 units cite the May 3.40 audit as the originator-text reference.
Bar construction, $K (π, n) = B^{n} (π)$ : J. P. May 1972, The Geometry of Iterated Loop Spaces (Lecture Notes in Math 271) — the canonical originator-text for the iterated-bar formulation.
Puppe sequences: D. Puppe 1958, Homotopiemengen und ihre induzierten Abbildungen I (Math. Z. 69, 299–344) — the originator. May synthesises.
Homotopy excision via excisive triads: A. L. Blakers and W. S. Massey 1951–1953, The homotopy groups of a triad I, II, III (Ann. Math.). May Ch 11 §3 follows this directly.
Thom space and Thom isomorphism: R. Thom 1954, Quelques propriétés globales des variétés différentiables (Comment. Math. Helv. 28); for the modern exposition cite Milnor-Stasheff 1974 Characteristic Classes.

Each priority-1 unit's Master section should cite the originator paper in addition to May.

§5 What this plan does NOT cover

A line-number-level inventory of every named theorem and exercise in May's 243 pages. Worked from the TOC + close reading of Chs 1–10 + selective deep reads of Chs 11, 12–13, 16, 17, 22, and the TOC of Chs 18–25. A full P1 audit would sharpen the punch-list (especially exercise-side) but would not change which units need writing.
May Chs 24 (K-theory) and 25 (cobordism + prespectra). These are May's deliberate "appetite-whetting" sketch chapters and they explicitly point to other texts. K-theory is the subject of a separate, deferred audit pass (Atiyah's K-Theory monograph is the natural canonical-text choice). Cobordism is the subject of Ravenel (FT 3.42) and the May-Ponto (FT 3.39) audits.
May Ch 22 §4–§5 — Postnikov + cohomology-operations depth. Codex has these as Master sections per the Hatcher punch-list (item 11); May's depth is roughly equal and not a separate deepening.
The simplicial-objects machinery beyond the foundational stub. The Eilenberg-Zilber theorem, Dold-Kan correspondence, Moore complex, Kan complex / fibration theory, simplicial-set model structure — all the heart of Simplicial Objects in Algebraic Topology — are the scope of the FT 3.40 May Simplicial audit, not this audit. This plan ships only the foundational $∣-∣ ⊣ S_{*}$ stub (item 2) required to make the May framework legible in Codex.
Model categories. Out of scope; deferred to May-Ponto (FT 3.39). May explicitly notes in his introduction that the Concise Course is influenced by Quillen's model-category framework but does not develop it. Codex mirrors that choice — no model-category units land in this audit's punch-list.
The exercise pack. May's exercises are smaller in number than Hatcher's (~120 vs ~600) but several are open-ended. Defer to a dedicated 03.12.E* exercise-pack pass after the priority-1+2 deepening batch closes.
Figures. May uses many small commutative-diagram pictures (the van-Kampen square, the homotopy-invariance square, etc.); these are load-bearing for intuition. Codex's figure infrastructure is the same curriculum-wide deferred item already flagged in the Hatcher plan §5.

§6 Acceptance criteria for FT equivalence (May)

Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4 and §9, May is at equivalence-coverage when:

≥95% of May's named theorems in Chs 1–22 map to Codex units (currently ~80%; after priority-1 units this rises to ~92%; after priority-1+2 to ~96%; priority-3 deepenings + the index unit close the residue). Chs 23–25 are explicitly out of scope (see §5) and do not count against the threshold.
May's categorical framing is captured at three load-bearing points:
- Van Kampen as a colimit of groupoids (deepening item 4).
- Eilenberg-Steenrod axioms as a foundational definition with cellular construction as the verifier (deepening item 5).
- $∣-∣ ⊣ S_{*}$ adjunction explicit (new unit item 2).
The HELP unification is captured (new unit item 1) with Whitehead and cellular approximation refactored as corollaries.
A notation/may.md crosswalk exists and is cross-linked from 02.01.09 compact-open-topology, 03.12.01 homotopy, 03.12.03 suspension, 03.12.08 fundamental-groupoid, and the new units 1–3 above.
≥80% of May's exercises have a Codex equivalent. Currently <5%; closing this requires the dedicated exercise-pack pass per §5.
≥90% of May's worked examples are reproduced in some Codex unit (currently ~70%; the priority-3 worked-example deepenings bring this to ~88%; the residue is May Ch 22's $K (π, n)$ cohomology calculations).
Pass-W weaving connects the new units (items 1, 2, 3, 8, 9, 10) to the existing 03.12-homotopy/ corpus via lateral connections, and connects the simplicial-sets stub (item 2) forward to the May 3.40 audit.

The 6 priority-1+2 new units + 4 priority-1 deepenings close most of the framing gap; priority-3 closes the manifolds-with-boundary and worked-example residue. Priority-4 is deferred to the K-theory and May-Ponto audits.

Composite recommendation. Because the priority-1 deepenings (items 4–7) touch existing units that have shipped against Hatcher, the production strategy is to dispatch them as one composite May-deepening batch with the three new May-distinctive units (items 1, 2, 3) and run Pass-V continuity once at the end of the batch. This avoids re-running Pass-V between each deepening and preserves the units' existing 27/27 validation scores. The priority-2 batch (items 8, 9, 10) is dispatched separately because the Thom-isomorphism unit (item 10) is load-bearing for the spin-geometry / Atiyah-Singer chain and deserves its own quality review.

§7 Sourcing

Free. Author-hosted PDF at https://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf. The edition served from May's UChicago page is the revised version (published 1999 by Chicago Press; the online PDF carries author corrections through 2007, marked "ConciseRevised").
License. Author has placed the PDF freely available for educational use from his UChicago faculty page. For educational use cite as J. P. May, A Concise Course in Algebraic Topology, Chicago Lectures in Mathematics, University of Chicago Press 1999 (revised PDF, ~2007).
Local copy. Already at reference/fasttrack-texts/03-modern-geometry/May-ConciseAlgebraicTopology.pdf (1.6 MB; matches the canonical Codex Fast Track convention for the modern-geometry book bucket).

J. Peter May — *A Concise Course in Algebraic Topology* (Fast Track 3.38) — Audit + Gap Plan