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Degree and parity criteria for Serre collapse

Statement

Let $E_{s}$ be either the homological or cohomological Serre spectral sequence, and fix $r \geq 2$ . Write $S_{r}$ for the set of bidegrees where $E_{r}$ is nonzero. If, for every $s \geq r$ , no two points of $S_{r}$ differ by the bidegree of $d_{s}$ , then every $d_{s}$ for $s \geq r$ is zero and the sequence collapses at $E_{r}$ . Equivalently, it is enough that every possible incoming or outgoing endpoint from every point of $S_{r}$ lies outside $S_{r}$ .

In particular:

support in a single row or a single column at $E_{r}$ forces collapse;
if $S_{r}$ is supported in total degrees of only one parity, then collapse occurs at $E_{r}$ ; and
more locally, a specified differential is zero whenever its source and target total-degree parities cannot both occur in the support.

The homological assertion is choice-free. For the cohomological Serre spectral sequence as constructed in this library, assume the Axiom of Choice. Collapse identifies $E_{r} = E_{\infty}$ with the associated graded of the abutment filtration; it does not assert that the filtration splits.

Facts & Assumptions

Given: One of the two Serre sequences in the statement and its support on page $r \geq 2$ .

[A1]

The Axiom of Choice is assumed only in the cohomological Serre branch.

[F1]

Homological Serre spectral sequence supplies the choice-free homological bidegree $(- s, s - 1)$ , persistence of zero terms, and the associated-graded abutment.

[F2]

Cohomological Serre spectral sequence supplies, under [A1], the cohomological bidegree $(s, 1 - s)$ , persistence of zero terms, and the associated-graded abutment.

[F3]

Collapse from one column at page s ≥ 1 or one row at page s ≥ 2 proves the one-row and one-column criterion for every page $r \geq 2$ .

[F4]

Degree reasons force stabilization in a bounded region proves that zero terms persist and explains pointwise stabilization from absence of incident endpoints.

[F5]

Collapse does not in general split the abutment supplies an explicit collapsed filtered $Z / 4$ whose two $Z / 2$ graded pieces do not split.

Proof

technique · inspect both endpoints of every differential

1.1

In homological indexing, a nonzero $d_{s}$ would have a source $(p, q) \in S_{r}$ and target $(p - s, q + s - 1) \in S_{r}$ . Indeed, every point outside $S_{r}$ is zero on page $r$ and stays zero on all later pages by [F1] and [F4]. The support-disjointness hypothesis excludes this pair for every $s \geq r$ , so every later differential has a zero endpoint and vanishes. In cohomological indexing the same argument uses the pair $(p, q)$ and $(p + s, q - s + 1)$ and [F2]. Thus every page transition is the homology of a zero differential, proving collapse at $E_{r}$ .

F1F2F4

2.1

A single row or column has no pair differing by $(- s, s - 1)$ or $(s, 1 - s)$ when $s \geq r \geq 2$ ; this is also exactly [F3]. For parity, a homological differential lowers total degree by one, while a cohomological differential raises total degree by one. Its endpoints therefore have opposite total-degree parity. If only one parity occurs in $S_{r}$ , one endpoint is zero. The same endpoint argument proves both the global collapse assertion and the local criterion for a specified differential.

F1F2F3step 1.1

2.2

The convergence statements in [F1] and [F2] identify the stable page only with the successive quotients of the abutment filtration. The filtered $Z / 4$ in [F5] has zero differentials and two $Z / 2$ stable pieces, but the quotient map $Z / 4 \to Z / 2$ has no homomorphic section. Hence none of the degree arguments supplies a splitting.

F1F2F5step 1.1

3.1

If the base or total space is empty, all page terms are zero and the support is empty. The zero coefficient ring, a zero page, or a single nonzero bidegree satisfies the criterion. Row or column number zero and total degree zero require no separate exception. A degenerate representative has zero or ordinary bidegree and is governed by its page class. Both incoming and outgoing endpoints, both parity values, and both indexing conventions were checked in steps 1.1–2.1. AC is used only to invoke [F2], not in the support argument. There is no biconditional claim: support separation is sufficient, not necessary, because a differential between two nonzero terms may still vanish algebraically.

A1F1F2F3F4F5step 1.1step 2.1step 2.2∎

Source notes

Hatcher, Chapter 5, printed pp. 532–538, repeatedly applies these row, column, and degree obstructions in Serre computations. The endpoint argument is written out above; the nonsplitting warning is supplied internally by [F5].

Depends on

Used by

Gysin sequence from a sphere-fiber Serre spectral sequence Theorem

Dependency tree · two levels

42 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Hatcher, Algebraic Topology, Serre spectral sequence examples (standard reference, not scraped)