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Connectedness repairs the pieces. CAT(0) geometry explains how local links, global loops and controlled flow can turn that repair into a contractible metric space.
Open the chapter contents · Topology root · Gold Standard of Education · How X Works
Chapter contents
Choose the route closest to the question, or read from Chapter 1 for the complete argument. The chapters remain open on the page; these grouped routes are an index.
Route 1 · Define metric geometry
Route 2 · Work links and flow
Route 3 · Test global implications
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE METRIC GEOMETRY
1. Move from a subset to a metric space
The augmented Guirardel core is not only a selected collection of points in T1 × T2. With its induced piecewise Euclidean path metric, it becomes a square complex whose geometry can be tested. The augmentation joins disconnected pieces using prescribed one-dimensional material while preserving convex fibres.
Three claims must remain separate: connectedness says points can be joined; contractibility says the whole space can be continuously shrunk to a point; CAT(0) supplies a strong non-positive-curvature condition that, in this setting, gives unique geodesics and contractibility.
CHAPTER 2 OF 12 · DEFINE METRIC GEOMETRY
2. Read links as local curvature tests
At a vertex of a square complex, the link records incident edge directions as vertices and square corners as edges. A short loop in the link signals too much positive curvature around the original vertex. For an all-right square complex, local CAT(0) requires links to have no embedded circuit of length less than four.
This is a local test. Global CAT(0) also needs the appropriate simply connected condition. Passing every link test does not by itself prove that a loop elsewhere in the complex contracts.
CHAPTER 3 OF 12 · WORK LINKS AND FLOW
3. Work a vertex-link ledger
Imagine four square sectors meeting around a vertex. Their link is a four-cycle: four direction vertices joined by four edges. Its angular length is 2π, so it passes the no-short-loop test. If a connector created a triangular link, the angular length would be 3π/2 and the local CAT(0) condition would fail.
The augmentation therefore cannot be treated as arbitrary stitching. Its connectors and incident cells must be checked in the induced complex. Guirardel’s construction is controlled by convex fibres, not by drawing whichever diagonal looks convenient.
| Local link | Cycle length | Local CAT(0) reading | Interpretation |
|---|---|---|---|
| tree | no cycle | passes | no closed angular circuit |
| four-cycle | 4 | passes threshold | flat total angle |
| triangle | 3 | fails | too short around the vertex |
CHAPTER 4 OF 12 · WORK LINKS AND FLOW
4. Follow a deformation toward the augmented core
A deformation retraction is a continuous family Ft that begins with the identity on the ambient product and ends inside the augmented core while fixing the augmented core throughout. A useful mental model is fibrewise flow: move a point along a convex fibre toward the minimal invariant connected set.
The proof needs compatible choices across fibres and time. Pointwise nearest-looking moves are not enough if they jump when the input crosses a cell boundary. Continuity and equivariance are part of the theorem, not decorative extras.
CHAPTER 5 OF 12 · WORK LINKS AND FLOW
5. Use CAT(0) geodesics to contract the space
In a nonempty CAT(0) space, any two points are joined by a unique geodesic depending continuously on its endpoints. Fix a basepoint x0. Sending each x along its geodesic to x0 gives a contraction.
This implication explains why CAT(0) is stronger than a vague “no holes” picture. It supplies a canonical path family. Contractibility follows, while the metric information remains useful for comparing points and analysing group actions.
| Statement | Local or global? | What it gives |
|---|---|---|
| convex fibres | fibrewise | controlled slices and projections |
| link condition | local | non-positive curvature near vertices |
| simply connected plus local CAT(0) | global | CAT(0) via Cartan–Hadamard |
| CAT(0) | global | unique geodesics and contractibility |
CHAPTER 6 OF 12 · TEST GLOBAL IMPLICATIONS
6. Changed context: remove one square but keep a loop
Take a square annulus built from cells. Every vertex link may still avoid triangles, so the complex is locally CAT(0), yet the central loop does not contract inside the annulus. Local checks have not supplied simple connectedness.
Fill the central region with compatible squares and the obstruction can disappear. This transfer check shows why a link audit and a global loop audit answer different questions.
CHAPTER 7 OF 12 · TEST GLOBAL IMPLICATIONS
7. Connect augmentation to twice-light rectangles
When the raw core is disconnected, twice-light rectangles identify where prescribed diagonals are added in the augmented construction. The previous article explains the combinatorial repair. Here the next question is metric: do those additions preserve the convex-fibre and link properties required by the CAT(0) argument?
Continue through Twice-Light Rectangles, Augmented Cores and Connectedness Repairs and Core Slices, Convex Fibres and One-Dimensional Compatibility before using dimension as a compatibility test.
CHAPTER 8 OF 12 · TEST GLOBAL IMPLICATIONS
8. Distinguish two contractibility theorems
Guirardel’s augmented core is a CAT(0) object associated with a pair of tree actions. A deformation space of G-trees is a different space: its points are trees in the same elliptic-subgroup class, assembled from simplices of metrics and collapse relations. Guirardel and Levitt prove deformation spaces are contractible by their own global argument.
The two results support one another conceptually but are not interchangeable. Contractibility of each augmented core does not, by itself, prove contractibility of the parameter space of all trees.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and common misconceptions
Connected does not imply contractible, and contractible does not automatically imply CAT(0). A CAT(0) conclusion depends on the metric, not only the underlying topology. A different cell metric can change the curvature test while leaving the point set unchanged.
An induced subcomplex of a CAT(0) complex need not be convex or CAT(0). Equivariance matters when the object is used for a group action. Toy square diagrams illustrate logic but do not replace stabiliser and fibre checks in a concrete action.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Build an evidence check
Record T1, T2, the group action, cell stabilisers, light and heavy quadrants, raw core, augmentation rule and path metric. Enumerate vertex links and search for forbidden short circuits. Test connectedness, convexity of horizontal and vertical fibres, and the global hypothesis used to obtain CAT(0).
For a claimed retraction, state the domain, target, time parameter, fixed-point property on the target, continuity and equivariance. For contractibility, identify the basepoint and continuous contraction rather than relying on a drawing.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Questions readers should ask
Does a four-cycle in a link violate CAT(0)?
Not in an all-right square complex. The forbidden embedded circuits have length less than four; a four-cycle represents total angle 2π.
Is every augmented core a tree?
No. It may contain two-dimensional rectangles. One-dimensionality is a special condition connected to common refinements.
Does contractibility mean every group action has a fixed point?
No. Contractibility is a property of the space; fixed-point properties require additional hypotheses about the action and metric.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Continue through verified routes
Begin with the Topology root, Gold Standard of Education and How X Works. Continue through Guirardel Cores, Heavy Quadrants and Intersection Numbers, Compatibility JSJ Trees, Common Refinements and Universal Compatibility and Deformation Spaces of Trees, Collapse Moves and Non-Unique Splittings.
Primary routes include Guirardel’s Core and intersection number for group actions on trees, Guirardel and Levitt’s Deformation spaces of trees, and their outer space of a free product. The vertex-link and annulus checks are original teaching examples.
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