Which homology classes really come from stable maps of spheres?

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Family 316 · Topology
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-22

1. The question

Homology is often easier to calculate than homotopy. A map from a sphere produces a homology class by sending the sphere’s fundamental class into the target. But many homology classes cannot arise this way. Which ones are genuinely spherical?

For the stable sphere, this becomes Curtis’s conjecture. Its target has a large, explicitly structured homology algebra, while the source contains the notoriously complicated stable homotopy groups of spheres. The conjecture predicts that only a very small collection of exceptional classes survives the map from homotopy to mod-two homology.

Family 316 claims the full positive-degree conjecture. The surviving image is generated by the classical Hopf-invariant-one classes and the Kervaire-invariant-one classes that exist. The claim is a classification of an image, not a calculation of all stable homotopy groups.

2. Background and essential definitions

Suspending a based space adds one dimension in a way compatible with based maps. Repeated suspension eventually stabilizes maps between spheres in a fixed difference of dimensions. Their stable homotopy groups are denoted by $\pi_d^S$.

The infinite loop space

$$QS^n=\operatorname*{colim}_{r}\Omega^r S^{n+r}$$

packages these stable maps. Here $\Omega^r$ means the space of based maps from the $r$-sphere, and the colimit follows repeated suspension. Let $Q_0S^0$ be the component containing the basepoint. For $d>0$, its $d$-th homotopy group is $\pi_d^S$.

The mod-two Hurewicz map is

$$h_d:\pi_d^S\longrightarrow H_d(Q_0S^0;\mathbb F_2).$$

A representative $S^d\to Q_0S^0$ sends the fundamental homology class of $S^d$ to $h_d$ of that element. An element in the target is spherical if it occurs this way. The coefficient field $\mathbb F_2$ has two elements, so the image is a vector space over that field.

This map should be distinguished from ordinary homology of the sphere spectrum itself. The target is the homology of its infinite loop space, which has many positive-degree classes and operations.

The stable classes $\eta,\nu,\sigma$, in degrees $1,3,7$, have Hopf invariant one. Roughly, the invariant detects a nontrivial relation between the bottom and top cells of a mapping cone through a cohomology operation.

A Kervaire-invariant-one class corresponds, under the Pontryagin–Thom construction, to a framed manifold whose associated quadratic form has Arf invariant one. The possible degrees are

$$2^{j+1}-2\qquad(j\geq1).$$

The notation $\theta_j$ denotes such a class only when it exists. A formula for possible dimensions is not an existence theorem in every dimension.

3. Development history and previous partial results

Curtis proposed the Hopf–Kervaire description in his 1975 study of the Dyer–Lashof and lambda algebras. May later recorded a gap identified by Wellington in the proposed proof. The image problem continued to motivate work connecting homology operations, invariant theory, and the Adams spectral sequence.

Adams’s Hopf invariant one theorem identifies the exceptional Hopf degrees. Browder related the framed Kervaire invariant to filtration-two classes in the mod-two Adams spectral sequence. Hill, Hopkins, and Ravenel ruled out Kervaire-invariant-one elements beyond the last possible dimension 126.

M1 also cites Lin, Wang, and Xu’s preprint proving existence in dimension 126. That endpoint existence result is separate from the upper bound needed for the manuscript’s finite-image corollary. The new Curtis claim is formulated in terms of whichever Kervaire classes exist, without deriving an additional existence theorem.

On the homology side, the calculation of free infinite loop spaces supplies a large algebra generated by Dyer–Lashof operations. Spherical classes must satisfy strong necessary conditions: they are primitive and are killed by positive Steenrod operations. But satisfying those algebraic tests does not by itself produce a map from a sphere.

Eccles and Zare obtained restrictions on spherical squares and excluded higher iterated squares. Hung–Peterson and Hunter connected related problems with Dickson invariant algebras and weight filtrations. Kuhn’s work linked Adams filtration to generalized Hurewicz maps and derived ideal powers. M1 builds on these inputs, explicitly crediting the earlier fourth-power exclusion rather than presenting that conclusion as new.

4. What the new manuscripts claim

M1, dated 25 September 2026, states that

$$\operatorname{im}\!\left(\bigoplus_{d>0}h_d\right) =operatorname{span}_{\mathbb F_2} \{h(\eta),h(\nu),h(\sigma),h(\theta_j):\theta_j\text{ exists}\}.$$

The statement is about all positive degrees at the prime two. In particular, using the established Kervaire nonexistence bound, the image vanishes outside

$$\{1,2,3,6,7,14,30,62,126\}.$$

It therefore vanishes above degree 126 and is finite as a vector space and as a set. The displayed degree list is an upper support bound; it does not alone assert a nonzero image in every listed degree or give an exact dimension.

The manuscript also deduces Eccles’s conjecture for every positive-dimensional sphere $S^n$. In that case, a nonzero spherical class in $H_*(QS^n;\mathbb F_2)$ is either the bottom class in degree $n$ or a nonzero homology suspension of one of the three Hopf images.

The Kervaire images are squares, and their first homology suspension vanishes. This explains why they appear at level $Q_0S^0$ but disappear from the positive-sphere conclusion. The corollary concerns spheres, not all spaces in the general formulation of Eccles’s conjecture.

5. Why the advance matters

Importance: 5/5, provisionally. If correct, the result settles a longstanding realization problem at the interface of stable homotopy and explicit homology operations.

Its force comes from the disparity between the two sides. The target contains an enormous algebra of possible classes, but the theorem says actual stable sphere maps occupy only the exceptional Hopf–Kervaire part. This would turn a large collection of algebraic possibilities into a definitive geometric exclusion.

The consequence for positive-dimensional spheres extends that restriction through all infinite loop levels. It also clarifies what the Hurewicz map can detect: most stable homotopy is invisible to this particular mod-two invariant, even when the stable groups themselves are nonzero.

6. The main idea

Follow the final surviving suspension

Start with an actual sphere map whose Hurewicz image is nonzero. Repeatedly take its adjoint at higher infinite loop levels, and let

$$z_j\in H_{d+j}(QS^j;\mathbb F_2)$$

be the corresponding images. These satisfy $z_{j+1}=\sigma_*z_j$, where $\sigma_*$ is homology suspension. Degree bounds ensure that the sequence eventually becomes zero.

Let $z_e$ be the last nonzero term. The manuscript uses the suspension kernel to show that it is a square,

$$z_e=w^2,$$

with a primitive, Steenrod-annihilated root whose suspension is nonzero. At positive terminal level, the preceding spherical suspension imposes an odd-degree condition on the root. At level zero, the paper supplies an argument excluding fourth powers using spectrum pairings and integral characteristic classes.

The root $w$ need not itself be spherical. This is a key distinction: the method uses the original map and the mapping cone of its next adjoint to obtain additional realization information, rather than assuming every suitable root comes from another sphere map.

Retain the geometric extension in an algebraic calculation

The next adjoint has zero Hurewicz image but carries a nonzero square operation in its cone. This yields an extension of modules over the Steenrod algebra, the algebra of mod-two cohomology operations.

The primitive homology is organized by weight, which counts inputs to an extended-power construction. Its dual modules are described using binary Dickson invariants. Those invariants come from polynomials fixed under general linear changes of variables over $\mathbb F_2$.

The new algebraic obstruction is applied to the specific extension arising from the cone, together with a detecting functional that lifts from the preceding suspension level. These compatibility conditions are stronger than primitivity and Steenrod annihilation alone.

The manuscript’s calculation uses Cartier halving and controlled maps on homological algebra groups to retain exponent information through the comparison. At a positive terminal level, it leaves only the bottom-weight possibility, which gives stable Hopf detection.

Separate the level-zero outcome

When the last nonzero term occurs already at level zero, the argument forces the root to have leading weight two. Its square therefore has leading weight four.

Kuhn’s lifting theorem says that sufficiently high Adams filtration forces the Hurewicz image into correspondingly high derived ideal powers. The surviving weight-four component bounds the Adams filtration by two. Browder’s detection theorem then identifies the class as having Kervaire invariant one.

This is a bound on filtration, not an assertion that weight and Adams filtration are always equal. Finally, the detecting invariants are additive. Subtracting a chosen Hopf or Kervaire representative with the same invariant leaves a class whose Hurewicz image must vanish. That gives the claimed span, rather than merely a list of necessary degrees.

7. Scope, limitations, and open questions

The theorem concerns positive-degree mod-two Hurewicz images for the stable sphere. It does not calculate the full stable stems, identify every element in the kernel, or make an odd-prime assertion. The prime-three Kervaire-family discussion in family 309 is a different problem with different detectors.

The notation for Kervaire classes remains conditional on existence. The finite support corollary uses the nonexistence theorem, while the separately cited dimension-126 result supplies endpoint existence in its own right.

Eccles’s conjecture is obtained here for positive-dimensional spheres. Extending that conclusion to arbitrary path-connected finite-type spaces is not part of the stated result.

The proof strategy also preserves a distinction between an algebraically plausible class and an actual sphere map. The latter supplies the cone extension and its compatibility with suspension. Removing those conditions would enlarge the algebraic problem and would no longer be the realization theorem claimed in M1.

8. Sources and further reading

M1 provides the statement and last-suspension strategy. Relevant primary predecessors include Eccles–Zare’s restrictions on potential spherical classes, Kuhn’s Adams-filtration and Hurewicz theorem, and Hill–Hopkins–Ravenel’s Kervaire nonexistence theorem. The separately cited endpoint result is Lin–Wang–Xu, On the last Kervaire invariant problem. M1’s bibliography supplies the original Curtis, Adams, Browder, and May references.

Source manuscripts

• M1. OpenAI, The Stable Hurewicz Image of the Sphere at Two (September 25, 2026).

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