Exotic Hopf maps, weight shifting and applications to vector bundles

1. Problem Statement

The paper addresses the problem of finding explicit polynomial representatives for specific elements in the homotopy groups of spheres within the context of motivic homotopy theory and algebraic geometry.

• Context: While P. Baum (1967) proved that stable homotopy groups of spheres can be represented by real quadratic maps, Wood (1968) showed this fails in the unstable range ($n \ge 2m$). Consequently, researchers turned to complex polynomial representatives, leveraging the fact that the complex algebraic sphere $S^n_{\mathbb{C}}$ is homotopy equivalent to the real sphere $S^n$.
• Specific Gap: Previous work ([AFH25]) established the existence of complex polynomial representatives for the suspension of the Hopf map ($\Sigma \eta: S^4 \to S^3$) and the square of the Hopf map using motivic homotopy theory and a "weight-shifting" procedure. However, these constructions were conceptual and non-explicit; they lacked concrete polynomial formulas defined over $\mathbb{Z}$.
• Goal: The authors aim to produce explicit polynomial maps (defined over $\mathbb{Z}$) between algebraic quadrics $Q_4 \to Q_3$ that realize the suspension of the Hopf map upon complex realization. Furthermore, they aim to apply these maps to construct an explicit rank 2 vector bundle on the Jouanolou device of $\mathbb{P}^3$.

2. Methodology

The authors utilize Motivic Homotopy Theory (Morel–Voevodsky) and Hermitian K-theory (Symplectic K-theory). The core strategy involves:

1. Quadric Models: Replacing topological spheres $S^n$ with smooth affine quadrics $Q_n$ defined over a field $k$.

   • $Q_{2n-1} = \{ \sum x_i y_i = 1 \} \subset \mathbb{A}^{2n}$
   • $Q_{2n} = \{ \sum x_i y_i = z(1-z) \} \subset \mathbb{A}^{2n+1}$
   • These quadrics model motivic spheres in the unstable motivic homotopy category $\mathcal{H}(k)$.

2. Two Distinct Constructions: The paper provides two independent methods to construct the map $Q_4 \to Q_3$:

   • Method A: Symplectic K-theory and Coniveau Spectral Sequences.
     • The authors compute the motivic homotopy group $\pi^{A_1}_{2,2}(Q_3)$ using an exact sequence involving Grothendieck-Witt groups ($GW$).
     • They identify the target group with $KSp_1(Q_4)$ (Symplectic K-theory).
     • Using the coniveau spectral sequence, they construct an explicit $8 \times 8$ symplectic matrix representing the generator of the $\mathbb{Z}/2\mathbb{Z}$ summand.
     • They perform a symplectic reduction (using elementary symplectic operations) to reduce this $8 \times 8$ matrix to a $2 \times 2$ matrix (a unimodular row) defining the map $Q_4 \to Q_3$.
   • Method B: Weight Shifting and Adjunction.
     • This method utilizes the "weight-shifting" procedure from [AFH25].
     • They start with the universal rank 2 bundle on $Q_5$ (classified by $Q_5 \to BSL_2$).
     • Using adjunction, they derive a map $S^{1,3} \to SL_2$ (where $S^{1,3}$ is a motivic sphere).
     • Since $S^{1,3}$ lacks a convenient smooth scheme model, they represent it as a quotient of $Q_3 \times \mathbb{G}_m$.
     • They construct a morphism $T: Q_3 \times \mathbb{G}_m \to SL_2$ which is trivial on the wedge sum.
     • By exploiting the fact that $T$ is $2\epsilon$-torsion, they construct a homotopy between $T$ and the identity, allowing them to "shift weights" to produce a map $Q_4 \to Q_3$. This involves modifying a homotopy $G(t)$ to ensure it extends to the whole quadric $Q_4$.

3. $\mathbb{P}^1$-Suspension: The authors develop a general procedure (Section 2.5) to explicitly $\mathbb{P}^1$-suspend maps between quadrics, allowing the construction of maps representing arbitrary elements in $GW(k)$ for higher dimensions.

3. Key Contributions and Results

A. Explicit Exotic Hopf Maps

The paper provides two explicit polynomial maps $Q_4 \to Q_3$ (defined over $\mathbb{Z}$) whose complex realization is the suspension of the Hopf map $\Sigma \eta: S^4 \to S^3$.

• Theorem 1 (Symplectic Construction): Defines the map via a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ with entries $a, b, c, d$ being explicit polynomials in the coordinates of $Q_4$. The construction relies on reducing a symplectic matrix derived from the generator of $GW^0_1(k)$.
• Theorem 2 (Weight Shifting Construction): Defines the map via a unimodular row $(a', b')$ with highly complex polynomial entries (degree up to 10). This construction arises from the weight-shifting procedure applied to the universal bundle on $Q_5$.
• Properties:
  • Both maps are defined over $\mathbb{Z}$.
  • They reduce to the identity map modulo 2 (trivial over $\mathbb{F}_2$).
  • The authors note that while they do not know if the two maps are homotopic over $\mathbb{Z}$, they coincide after base change to an algebraically closed field.

B. Explicit Rank 2 Vector Bundle on the Jouanolou Device

The authors apply the exotic Hopf map to construct a specific vector bundle.

• Construction: They form a composite map:
  $$J_3 \xrightarrow{\pi_3} Q_6 \xrightarrow{\Sigma_{\mathbb{P}^1} \eta'} Q_5 \to BSL_2$$
  where $J_3$ is the Jouanolou device of $\mathbb{P}^3$ (an affine scheme homotopy equivalent to $\mathbb{P}^3$), $\pi_3$ is a clutching map, and $\Sigma_{\mathbb{P}^1} \eta'$ is the suspended exotic Hopf map.
• Result: This yields an explicit rank 2 vector bundle on $J_3$ (represented by an explicit idempotent $3 \times 3$ matrix).
• Invariants:
  • Chern Classes: The bundle has trivial Chern classes ($c_2 = 0$).
  • Atiyah-Rees Invariant: The bundle has a non-trivial Atiyah-Rees invariant (taking values in $\mathbb{Z}/2\mathbb{Z}$).
  • This confirms the motivic analogue of the topological result by Atiyah and Rees (1976), which states that there exist indecomposable rank 2 bundles on $\mathbb{P}^3$ with trivial Chern classes.

4. Significance

• Explicitness in Motivic Homotopy: The paper bridges the gap between abstract motivic existence proofs and concrete algebraic geometry. It moves from "there exists a map" to "here is the exact polynomial formula."
• Resolution of the Unstable Range: It provides the first known explicit polynomial representatives for the suspension of the Hopf map in the unstable range, a problem previously thought to be difficult due to the failure of real polynomial maps.
• Vector Bundles: It constructs a concrete algebraic model for a topological phenomenon (the Atiyah-Rees bundle) over arbitrary fields (specifically $\mathbb{Z}$), demonstrating that the classification of vector bundles by Chern classes and the $\alpha$-invariant holds in the motivic setting.
• Methodological Advancement: The development of explicit $\mathbb{P}^1$-suspension techniques and the detailed symplectic reduction procedures provide new tools for computing motivic homotopy groups and constructing algebraic bundles.

In summary, Fasel and Hornslien successfully translate high-level motivic homotopy theory into explicit algebraic geometry, producing concrete polynomial maps and vector bundles that realize deep topological phenomena over the integers.