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The two-set Mayer-Vietoris sequence is the first gluing ledger many students meet. A cover with several sets produces a larger double complex: one direction takes chains inside intersections, while the other alternates across the cover indices. Filtering the total complex reveals a spectral sequence whose early pages show local intersection homology and whose later pages account for interactions that the nerve alone may miss.
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Chapter contents
Choose the route closest to the decision, or read from Chapter 1. The chapters remain open; these grouped routes are a navigable index.
Route 1 · Define the decision
Route 2 · Work the ledgers
Route 3 · Transfer the method
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE THE DECISION
1. Index the cover and every non-empty intersection
Let U={Ui} cover X under hypotheses that make the chosen chain or sheaf construction valid. For an ordered tuple i0<… The p=0 column contains individual cover members, p=1 contains pairwise intersections, p=2 contains triple intersections and so on. Empty intersections contribute zero rather than a generator.
CHAPTER 2 OF 12 · DEFINE THE DECISION
2. Build the double complex before naming a page
A homological version uses groups Cq(Ui0…ip) assembled over all p-fold cover intersections. The vertical differential is the chain boundary. The horizontal differential is an alternating sum induced by omitting one cover index and including the smaller intersection into the larger one.
A sign convention is needed so the two directions combine into a total differential that squares to zero. State the convention instead of relying on a diagram whose arrows hide signs.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. Work a three-set page ledger
Take three connected contractible sets with connected contractible pairwise intersections and empty triple intersection. On the E1 page, only q=0 survives. The p=0 and p=1 groups reproduce the chain complex of a triangle boundary.
| Cover degree p | Intersections represented | Surviving E1 term | First differential role |
|---|---|---|---|
| 0 | U1,U2,U3 | Three H0 copies | Send vertices to adjacent overlaps |
| 1 | U12,U13,U23 | Three H0 copies | Alternating incidence map |
| 2 | U123 empty | 0 | No face fills the cycle |
| Total degree 1 | Kernel modulo image | One class | Produces H1 of the union |
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. Read d1 before jumping to higher differentials
After vertical homology, E1 records Hq of intersections. The d1 differential moves one step in cover degree and is induced by the alternating inclusion maps. E2 is the homology of that d1 complex.
Higher differentials connect terms separated in both bidegrees. Their direction depends on homological or cohomological indexing. Write the source and target formula for the convention in use before declaring that a class survives or dies.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Change one overlap and expose a second row
Now replace U12 by an annular overlap with H1=R while leaving the ordinary nerve unchanged. The E1 page gains a term in bidegree (1,1). That local cycle can support or receive later differential information unavailable in the q=0 nerve row.
| Changed datum | Nerve sees | E1 sees | Required response |
|---|---|---|---|
| Contractible U12 | One edge | H0 only | Nerve row may suffice |
| Annular U12 | Same edge | Additional H1=R | Track q=1 row |
| Disconnected U12 | Same edge | Extra reduced H0 | Track components |
| Unknown U12 | Same edge | Unverified groups | Withhold collapse claim |
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Recover ordinary Mayer-Vietoris as the two-set case
For a cover by U and V, only cover degrees zero and one occur. The associated short exact sequence of chain complexes produces the familiar long exact Mayer-Vietoris sequence involving U∩V, U⊔V and X.
The multi-set spectral sequence is therefore not a decorative generalisation. It organises the same inclusion-and-boundary bookkeeping when several overlaps and higher intersections must be handled at once.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Connect the established cover routes
The homological nerve guide explains when local acyclicity collapses the ledger to nerve homology. The good-cover guide gives the stronger contractible-intersection route.
Use numerable covers and partitions of unity for controlled gluing maps. Those tools do not by themselves make the spectral sequence collapse.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Distinguish filtration data from the final group
Convergence commonly gives an associated graded object for a filtration of the target homology, not an automatic canonical direct-sum decomposition. Extension problems can remain when reconstructing the final group from E-infinity pieces.
A class surviving to E-infinity represents a graded piece at a stated filtration level. It should not be promoted to a preferred generator of Hn(X) without resolving extensions and identifying the edge maps.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Know the limits and common misconceptions
A page diagram is not a proof of convergence. Finiteness, local finiteness, boundedness, open-cover or subcomplex hypotheses and the chosen homology theory must match the theorem being used.
Zero higher differentials do not follow merely because they are hard to draw. Degree reasons, vanishing lines or an explicit collapse theorem are needed. Page indices and arrow directions also change between homological and cohomological conventions.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Use a proof and source routine
Record the cover class, ordering, coefficient ring, intersection groups, chain or sheaf model, horizontal signs, total differential, filtration, E0/E1 convention, source and target of dr, convergence theorem, edge maps, collapse evidence and unresolved extensions.
Allen Hatcher’s Spectral Sequences chapter develops filtrations, pages and the Mayer-Vietoris spectral sequence. Cornell’s Toric Varieties appendix on spectral sequences states the covering spectral sequence with intersection cohomology and alternating inclusion maps.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Is the E1 page just the nerve?
Only when the relevant intersection homology is concentrated in degree zero in the required way. Otherwise additional rows carry local topology.
Does collapse mean the target is a direct sum?
Not automatically. Collapse identifies associated graded pieces; extension questions may remain.
Why track signs if I only need ranks?
Signs determine kernels, images and compatibility. Rank-only reasoning can miss torsion, cancellation and the actual maps.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Continue with a four-set transfer check
Build the E1 page for four cover members whose proper intersections are connected and acyclic and whose fourfold intersection is empty. Identify the q=0 nerve row. Then give one triple intersection an H1 class and mark the new bidegree and every differential that could interact with it under your convention.
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