Chromatic splitting: the pieces can fit without splitting apart

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Family 318 · Topology
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-22

1. The question

Chromatic homotopy theory studies spectra one periodic height at a time. Reconstructing a spectrum from those layers requires understanding their overlap. A long-standing prediction proposed that the overlap for the sphere splits into a particularly elegant collection of lower-height spheres.

Family 318 contains five manuscripts that give a more complicated picture. They claim that the predicted pieces can form a finite filtration in a large-prime range, while the corresponding direct-sum decomposition fails already at height three. Elsewhere, even a retraction of the canonical map fails, and one small-prime example cannot be assembled from lower local spheres by any finite sequence of the permitted operations.

These statements address different questions. Knowing the pieces of a filtration does not determine how they attach. Knowing that an object is finitely assembled does not make one specified map split. The family’s main theme is that the attaching maps carry information that a list of pieces misses.

2. Background and essential definitions

A spectrum is a stable object in which suspension is invertible. Write $\Sigma^dX$ for a degree shift. A cofiber sequence

$$A\longrightarrow B\longrightarrow C\longrightarrow\Sigma A$$

describes $B$ as an extension of $A$ by $C$. The last map records the attachment. If the extension splits, $B$ is a wedge $A\vee C$, the stable analogue of a direct sum. An unsplit extension can have the same pieces without being their direct sum, just as an abelian-group extension can.

Fix a prime $p$, and let $S=S_p^{\wedge}$ be the derived $p$-completed sphere. Morava localization $L_{K(n)}$ isolates height $n$. Johnson–Wilson localization $L_r=L_{E(r)}$ retains heights zero through $r$, with $L_0$ rationalization. The overlap is

$$X_{n,p}=L_{n-1}L_{K(n)}S.$$

The canonical unit gives a distinguished map

$$i_{n,S}:L_{n-1}S\longrightarrow X_{n,p}.$$

Four notions will be kept separate. Strong splitting prescribes a particular wedge decomposition, usually with specified maps. Weak splitting asks only for a homotopy retraction of $i_{n,S}$. A filtration constructs the overlap by successive cofibers with specified layers, allowing nonzero attachments. Ordinary finite assembly allows finite sums, all shifts, cofibers, and retracts from lower local spheres, without prescribing a layer list.

In M4, finite assembly takes place in the category of $E(2)$-local modules over the completed sphere. It is ordinary thick closure: tensoring with arbitrary additional objects is not an extra permitted operation.

3. Development history and previous partial results

Hopkins’s chromatic splitting proposal, recorded by Hovey, related sphere overlaps to exterior classes in stabilizer cohomology. At height one the expected behavior is classical. At height two, Shimomura–Yabe’s calculations supported the large odd-prime answer, later reorganized and corrected by Behrens. Goerss–Henn–Mahowald proved the height-two splitting at the prime three.

The prime-two height-two case already showed why formulations matter. Beaudry disproved the original strong wedge formula. Beaudry–Goerss–Henn subsequently supplied a refined decomposition with Moore-spectrum terms while retaining the canonical unit retraction. A failure of the original list therefore did not imply failure of weak splitting or of finite assembly.

Barthel–Schlank–Stapleton–Weinstein determined the rational homotopy of the local sphere at every prime and height. It is an exterior algebra over $\mathbb Q_p$ with generator degrees $1-2i$. At height three, its degrees are

$$0,-1,-3,-4,-5,-6,-8,-9.$$

These are exactly the shifts expected in the classical eight-piece pattern. But agreement of rational dimensions does not identify the maps between heights, integral representatives, or extension classes.

The manuscripts also build on Gross–Torii marking towers, recent geometric coheight-one calculations, and descent theory. They distinguish completed geometric coefficient calculations from ordinary spectra: returning from a completion requires control of finite marking stages, pole bounds, and actual comparison maps. The five introductions provide the precise primary sources and the roles of these earlier results.

4. What the new manuscripts claim

M1: the predicted layers form a filtration at generic primes

M1, dated 25 September 2026, states that for every $n\geq1$ and prime $p>n+1$, the overlap has an ordered filtration with $2^n$ layers. For a subset $I\subseteq\{1,\ldots,n\}$, put

$$d(I)=\sum_{i\in I}(2i-1).$$

The empty-set layer is $L_{n-1}S$. For nonempty $I$, the layer is

$$\Sigma^{-d(I)}L_{n-\max I}S.$$

The first-stage map to the overlap is the canonical unit. The theorem retains the attachments; it does not assert a wedge or a retraction. One family of sphere classes supplies compatible product bases at all lower positive heights. The final rational block is chosen from rational dimensions and is not identified with all the remaining products of that family.

M2 and M3: eight layers, but no eight-piece wedge

M2, dated 27 September 2026, independently constructs the height-three filtration for every $p\geq5$. In order, its layers are

$$\begin{aligned} &L_2S,\quad\Sigma^{-1}L_2S,\quad \Sigma^{-3}L_1S,\quad\Sigma^{-4}L_1S,\\ &\Sigma^{-5}H\mathbb Q_p,\quad\Sigma^{-6}H\mathbb Q_p,\quad \Sigma^{-8}H\mathbb Q_p,\quad\Sigma^{-9}H\mathbb Q_p. \end{aligned}$$

It identifies the attachments relative to chosen local maps, a compatibility homotopy, and coherent inverse data. Only the first unit is asserted to be canonical independently of those choices.

M3, dated 25 September 2026, claims that the canonical map

$$L_0L_{K(3)}S\longrightarrow L_0L_{K(2)}L_{K(3)}S$$

is nonzero on $\pi_{-3}$ for every $p\geq5$. Here the use of the ordinary sphere or its completion gives the relevant positive-height local object.

For the proposed eight-piece wedge, the corresponding natural map is zero in that degree. M3 therefore rules out even an equivalence of the underlying $E(2)$-local spectra, without requiring it to preserve named summands. Applied to M2’s explicit filtration, this separate result makes its first height-one attachment nonzero. M2’s construction itself does not depend on M3.

M4: finite assembly fails at the prime three

M4, dated 25 September 2026, states that at $p=n=3$,

$$L_2L_{K(3)}S\notin\operatorname{Thick}\{L_0S,L_1S,L_2S\}$$

in the specified module category. The obstruction allows arbitrarily many but finitely many cofibers and retracts, including Moore corrections. It is stronger than failure of a particular finite layer list.

This prime is outside M1’s range at height three. M4 does not by itself claim that the distinguished unit lacks a retraction; ordinary thick membership and splitting a particular map are different tests.

M5: explicit kernels obstruct weak splitting

M5, dated 27 September 2026, claims nonretraction at height $p$ for every prime $p\geq5$, and at height $p+1$ for every prime $p\geq7$, using the completion of the finite sphere as input.

Let $\beta_1$ be the first beta element, of degree $2p(p-1)-2$. The same named sphere product

$$z_p=\beta_1^{(p-1)^2}$$

survives in the appropriate lower localization but is killed by the upper unit. At $p=5$, this gives $\beta_1^{16}$ in degree 608 as a nonzero kernel element for the height-five map. A map with a left inverse cannot kill a nonzero homotopy class, so this obstructs any homotopy retraction, even an unnatural one.

M5 also gives an independent height-$p$ obstruction using incompatible descent exponents, rather than a named permanent sphere class.

5. Why the advance matters

Importance: 5/5, provisionally. If correct, the family substantially revises a central reconstruction picture in chromatic homotopy theory. It supplies a positive replacement by filtrations in one range and explicit obstructions to several stronger predictions.

The distinction between dimensions and maps is especially instructive. The expected rational degree pattern can be entirely correct while the proposed wedge is wrong. The nonzero degree-minus-three map detects information that counting generators cannot see.

The small-prime and weak-splitting counterexamples also show that one universal repair is unlikely to suffice. Allowing nonsplit attachments works in the generic-prime filtration theorem, but it does not rescue ordinary finite assembly at $p=n=3$. Merely asking for a unit retraction also fails in M5’s separate ranges.

6. The main idea

Construct actual compatible bases before gluing

M1 separates two tasks: calculating coefficient cohomology and constructing sphere classes that act on it. Finite marking towers and compact-support geometry on spaces of bundle extensions supply the coefficient calculation. Cartier transfers, pole control, and finite-stage comparisons return that information to ordinary coefficients.

A second construction produces one family of integral stable classes, normalized using regulator and volume calculations. The comparison identifies their products as bases under every lower positive-height test. Successive cofibers then lower the remaining chromatic height, and rational information supplies the last block. This produces an actual filtration with the prescribed first map.

M2 makes the height-three gluing explicit. The unit and determinant class first supply the two height-two pieces. A height-one orientation class supplies the next two. A nonzero rational comparison scalar is matched with the height-one map so that the fracture square glues them using a chosen compatibility homotopy. The remaining cofiber is rational and supplies the final four pieces.

Detect an attachment through a natural map

M3 transports a degree-three stabilizer cohomology class through two descriptions of a modification space on the Fargues–Fontaine curve. One description uses division-algebra symmetries and the other linear-group symmetries. The classical trace expression

$$(U,V,W)\longmapsto\operatorname{tr}_{\mathrm{red}}(U[V,W])$$

provides the relevant class; $[V,W]=VW-WV$ and the reduced trace generalizes matrix trace.

The rank-three class restricts nontrivially toward rank two. Integral Witt-coefficient comparisons establish that the transported classes remain nonzero, and descent carries that nonvanishing into ordinary degree-minus-three homotopy. Locality of the detector forces its map to factor through the canonical localization map. Naturality then rules out any proposed wedge equivalence, not just one poorly chosen set of summand maps.

Preserve a character that finite assembly cannot create

M4 uses an exact test involving height-two Morava $E$-theory, reduction modulo three, and inversion of its deformation parameter $x$. Every finite construction from the proposed generators has representation constituents that are powers of the periodicity line.

The overlap instead contains a line twisted by a nontrivial norm character from an unramified quadratic extension of $\mathbb Q_3$. The argument retains the full semilinear action over $k((x))$: the group acts on coefficients as well as vectors. Looking only at a residue character would lose essential information.

Finite marking comparisons and all deformation jets carry that character into actual homotopy. Its inequivalence to every periodicity power prevents the relevant differentials from removing it and violates the necessary condition for finite assembly.

Kill the same sphere product at a higher height

For M5, an established ordinary homotopy-fixed-point calculation detects the named beta product at height $p-1$. At the upper heights, the manuscript constructs a separated multiplicative filtration with the beta element in filtration at least two and a finite upper cutoff. The product’s filtration exceeds that cutoff, forcing its upper image to vanish.

Completion and naturality place these two facts on the precise weak-splitting map. The separate exponent argument compares lengths of finite descent constructions: a retraction would transfer an upper bound that contradicts a lower bound at height $p-1$. Its lower bound needs survival to a particular finite spectral-sequence page, not a permanent sphere representative.

7. Scope, limitations, and open questions

The prime and height ranges are essential. M1 assumes $p>n+1$; M2 and M3 use height three with $p\geq5$; M4 treats height three at three; M5 uses the two ranges stated above. The manuscripts do not claim a sharp universal prime threshold.

A nonzero attachment does not automatically prove failure of the canonical unit retraction. Conversely, finite assembly does not imply a prescribed wedge decomposition. The family keeps these implications separate, and the examples occur in different ranges.

The filtration theorem retains extension data rather than computing all homotopy groups. M2 describes attachments relative to compatible choices, not a universally canonical formula independent of choices. M4’s ordinary thick closure is also distinct from a tensor ideal allowing arbitrary additional tensor factors.

These results leave many prime-height combinations and finer attachment calculations outside their statements. They report specific positive constructions and negative obstructions, rather than a complete classification of every chromatic overlap.

8. Sources and further reading

M1–M5 are the sources for the five claims and their dependencies. Their introductions cite Hovey’s account of Hopkins’s conjecture; the height-two work of Shimomura–Yabe, Behrens, and Goerss–Henn–Mahowald; the prime-two correction of Beaudry–Goerss–Henn; and the rational calculation of Barthel–Schlank–Stapleton–Weinstein. The bibliographies also identify the geometric marking comparisons and the Hopkins–Miller detector used by M5. Reading the theorem statements together is particularly useful because the same word “splitting” covers several distinct assertions.

Source manuscripts

• M1. OpenAI, Filtered chromatic splitting at generic primes (September 25, 2026).
• M2. OpenAI, The height-three chromatic overlap: an explicit filtration and its attachments (September 27, 2026).
• M3. OpenAI, A rational obstruction to strong chromatic splitting at height three (September 25, 2026).
• M4. OpenAI, Failure of finite assembly for a chromatic overlap at the prime three (September 25, 2026).
• M5. OpenAI, Counterexamples to weak chromatic splitting: sphere kernels and descent exponents (September 27, 2026).

Sources consulted on 22 October 2026 at repository revision 1581d4c. Download the manuscript citations.