Brown — Topology and Groupoids (Fast Track 1.05) — Audit + Gap Plan

Book: Ronald Brown, Topology and Groupoids (3rd ed., BookSurge 2006; expanded edition of Elements of Modern Topology, McGraw-Hill 1968 / Topology, Ellis-Horwood 1988). Hosted free by the author.

Fast Track entry: 1.05.

Purpose of this plan: lightweight audit-and-gap pass (P1-lite + P2 + P3-lite of the orchestration protocol). Output is a concrete punch-list of new units to write so that Topology and Groupoids 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).

This pass is intentionally not a full P1 audit (which would inventory every named theorem in the book at the line-number level). It works from the book's canonical topic list and Brown's distinctive editorial choices, produces the gap punch-list, and stops there. A full P1 audit of the same book would sharpen the punch-list but would not change which units need writing.

§1 What Brown's book is for

Brown is the canonical source for the fundamental-groupoid framing of algebraic topology. Where Hatcher / Munkres / May organise the subject around the fundamental group of a pointed space, Brown organises it around the fundamental groupoid on a set of base points. The reformulation is not cosmetic: the groupoid version of the Seifert-van Kampen theorem applies without connectivity hypotheses, gives a clean treatment of unions of two open sets meeting in disconnected pieces (e.g.\ the circle), and lifts naturally to higher-dimensional algebraic topology.

Distinctive contributions, in roughly the order Brown develops them:

Identification (quotient) topology with the universal property treated first, then concrete examples (cones, suspensions, mapping cones, mapping cylinders, adjunction spaces, CW skeleta). Brown is the originator-text for the universal-property-first treatment of these constructions in a standard textbook.
Compactly generated function spaces (the cartesian-closed category). The exponential law $C (X \times Y, Z) ≅ C (X, C (Y, Z))$ as a clean theorem, with the modification of the topology that makes it work.
Cofibrations and the homotopy extension property, treated alongside identification topology rather than deferred to a "homotopy theory" chapter.
Fundamental groupoid $π_{1} (X, A)$ on a set $A \subseteq X$ of base points. Higher-than-Hatcher abstraction level early.
Seifert-van Kampen theorem for the fundamental groupoid. Brown's signature contribution. Applies to $X = U \cup V$ with no connectivity assumptions on $U \cap V$ .
Covering spaces treated as a representation theory of the fundamental groupoid — Brown's framing makes the Galois correspondence explicit.
Orbit spaces, group actions, and the equivariant fundamental groupoid.
Higher-dimensional algebraic topology via crossed modules, double / cubical groupoids — a glimpse of the further programme.

The book ends before bringing in homology / cohomology, which puts it squarely in the algebraic-topology-via-homotopy track.

§2 Coverage table (Codex vs Brown)

Cross-referenced against the current 213-unit corpus. ✓ = covered, △ = partial / different framing, ✗ = not covered.

Brown topic	Codex unit(s)	Status	Note
Topological space (definition, examples)	`02.01.01` topological-space	✓	Codex has this.
Continuous map	`02.01.02` continuous-map	✓
Metric space	`02.01.05` metric-space	✓
Identification / quotient topology	—	✗	Gap. Universal-property treatment of quotients, mapping cones, mapping cylinders, adjunction spaces.
CW complex	—	✗	Gap. Standard prerequisite for cellular cohomology / Eilenberg-MacLane spaces (`03.12.05`) but not actually shipped as its own unit.
Compactly-generated spaces	—	✗	Gap. Function-space exponential law. Brown is one of few textbook anchors.
Compact-open topology	—	✗	Gap. Foundational for function spaces and CW pairs.
Cofibration / HEP	—	✗	Gap. Used implicitly throughout `03.12-homotopy/` but never given its own unit.
Fibration (Hurewicz / Serre)	—	✗	Gap. `03.13.02` (Leray-Serre) assumes this without a host unit.
Path-connectedness, components	—	△	Mentioned in `02.01.01` but not its own unit.
Homotopy of maps and paths	`03.12.01` homotopy	✓	Codex covers homotopy.
Fundamental group $π_{1} (X, x_{0})$	—	✗	Gap (high priority). Codex jumps straight to higher homotopy / Eilenberg-MacLane without the basic $π_{1}$ unit.
Fundamental groupoid $π_{1} (X, A)$	—	✗	Gap (high priority — Brown's signature contribution).
Seifert-van Kampen (group version)	—	✗	Gap. Foundational computational tool.
Seifert-van Kampen (groupoid version)	—	✗	Gap (high priority — Brown's signature theorem).
Covering space	`03.12.02` covering-space	✓	Codex has the unit; Brown's groupoid framing of the Galois correspondence is a deepening, not a new unit.
Galois correspondence for covering spaces	`03.12.02` (partial)	△	Currently a remark inside the covering-space unit; could be its own unit if punch-list grows.
Orbit space / group action on space	`03.12.02` (partial)	△	Brown's treatment is more thorough; partial coverage is fine for FT equivalence.
Crossed module	—	✗	Gap (low priority — foundational for HDT but not load-bearing for the rest of the curriculum).
Cubical / double groupoid	—	✗	Gap (low priority — pointer in §1 §8 of Brown is enough for FT equivalence).

Aggregate coverage estimate: ~30% of Brown's book has corresponding Codex units. The gap is concentrated in early-and-middle chapters: $π_{1}$ and the groupoid version, Seifert-van Kampen, identification topology, function spaces and fibration / cofibration. The end-of-book HDT material is correctly out of FT scope (Brown himself flags it as a survey).

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

Priority 1 — high-leverage, fills load-bearing prerequisite gaps:

02.01.06 Quotient and identification topology. Universal property, examples (cone, suspension, mapping cone, mapping cylinder, wedge, smash, adjunction space). Brown §4 is canonical anchor; Hatcher §0 also anchors. Three-tier unit, all tiers present, ~1500 words.
02.01.07 Compact-open topology and function spaces. Exponential law, when it holds (compactly-generated case), Brown §5 anchor. Three-tier; the master tier should give the cartesian-closed category statement.
02.01.08 Cofibration and homotopy extension property. Brown §7; Hatcher §0; May Concise Course §6. Used implicitly by 03.12-homotopy/ units already.
02.01.09 Fibration (Hurewicz / Serre). Foundational for the Leray-Serre spectral sequence in 03.13.02. Brown §13 anchor; Hatcher §4.2; Bott-Tu §17.
03.12.00 Fundamental group. Currently the Codex homotopy chapter skips $π_{1}$ . Brown §6; Hatcher §1.1. Three-tier; Beginner uses loops-on-circle pictures.
03.12.0a Fundamental groupoid $π_{1} (X, A)$ . Brown §6 (his distinctive contribution). Originator-prose treatment crediting Brown directly. Three-tier.
03.12.0b Seifert-van Kampen theorem (group + groupoid versions). Both forms in one unit, with the worked example showing why the connectedness-free groupoid version is the "right" form. Brown §6.7 as originator-anchor for the groupoid version.

Priority 2 — fills CW / cellular gaps:

03.12.0c CW complex. Definition (skeleta + attaching maps), topology, mapping-cylinder reformulation, basic examples. Hatcher §0 anchor; Brown §4 covers via identification topology.
03.12.0d Cellular approximation theorem. Used silently by 03.12.05 Eilenberg-MacLane and 03.12.07 Whitehead tower. Hatcher §4.1 anchor.

Priority 3 — Brown-distinctive deepenings (not strictly required for FT equivalence but high-value):

Deepening of 03.12.02 covering-space: add a Master section on the Galois correspondence in the language of $π_{1}$ -sets and fundamental-groupoid representations (Brown §10).

Priority 4 — survey-level pointer, optional:

03.12.0e Crossed module and pointer to higher-dimensional algebraic topology. Brown §11 onwards. Likely Master-only, ~1000 words. Acts as a launchpad to Brown-Higgins-Sivera Nonabelian Algebraic Topology if Codex ever expands in that direction.

§4 Implementation sketch (P3 → P4)

For a full Brown coverage pass, items 1–7 are the minimum set. Realistic production estimate (mirroring earlier Lawson-Michelsohn / Bott-Tu batches):

~3 hours per unit (research + draft + validate at 27/27 + Lean stub if applicable + Bridge / Synthesis paragraphs in real prose, not the templated form).
7 priority-1 units × 3 hours = ~21 hours of focused production. With one agent doing the production in parallel and the operator doing the weave+verify passes, this fits in a 2–3 day window.

Originator-prose target. Ronald Brown is the originator for the groupoid framing of Seifert-van Kampen, the textbook treatment of identification topology with the universal property first, and (with Higgins) higher-dimensional algebraic topology. Units 5, 6, 7 should carry originator-prose treatment per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §10, citing Brown 1967 ("Groupoids and van Kampen's theorem", Proc. London Math. Soc. 17) and Brown 2006 (the book itself). Unit 1 should cite the Brown 1968 Elements of Modern Topology in addition to the 2006 expansion.

Notation crosswalk. Brown writes $π X$ for the fundamental groupoid on the entire space and $π_{1} (X, A)$ for the groupoid on a chosen subset $A$ . Some sources write $Π_{1}$ . The Codex notation decision (per docs/specs/UNIT_SPEC.md §11) is: use $π_{1} (X, A)$ for the groupoid on $A$ and $π_{1} (X, x_{0})$ for the group at a basepoint. The new unit on fundamental groupoids should record this in a §Notation paragraph.

§5 What this plan does NOT cover

A line-number-level inventory of every named theorem in Brown (would take the full P1 audit; deferred unless punch-list expands).
Exercise-pack production. Brown's exercises are extensive; an exercise pack unit (02.01.E1 or similar) is a P3-priority-3 follow-up.
Higher-dimensional algebraic topology beyond the survey-level pointer unit.
Notation-crosswalk file. The crosswalk decisions in §4 are sufficient for FT equivalence; a standalone notation/brown.md is not needed unless a downstream agent needs it for cross-book consistency.

§6 Acceptance criteria for FT equivalence (Brown)

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

≥95% of Brown's named theorems map to Codex units (currently ~30%; after priority-1 units this rises to ~85%; after priority-1+2 to ~95%).
≥90% of Brown's worked examples have either a direct unit or are referenced from a unit that covers them.
Notation decisions are recorded.
Pass-W weaving connects the new units to the existing 03.12-homotopy/ units via lateral connections.

The 7 priority-1 units alone close most of the equivalence gap. Priority-2 units close the CW gap. Priority-3+4 are deepenings.