+++ title = “The Cτ Bridge” section = “mathematics” +++

The $C\tau$ Bridge

The $C\tau$ bridge connects the motivic Adams spectral sequence to the algebraic Novikov spectral sequence via the Gheorghe–Wang–Xu theorem. It provides a powerful tool for computing $\tau$-torsion orders and constraining motivic differentials.

The cofiber of $\tau$

In motivic homotopy theory, $\tau \in \pi_{0,-1}(S^{0,0}{\mathrm{mot}})$ is the element detecting the algebraic $K$-theory class. Its cofiber $C\tau$ fits into a short exact sequence of $A{\mathrm{mot}}$-modules:

$$0 \to \Sigma^{0,1}\mathbb{F}_2 \xrightarrow{;i;} C\tau \xrightarrow{;\pi;} \mathbb{F}_2 \to 0$$

Here $i$ is the top cell inclusion (the generator at weight 1 maps to $\tau \cdot \iota$, where $\iota$ is the bottom cell) and $\pi$ is the bottom cell projection.

As an $A_{\mathrm{mot}}$-module, $C\tau \cong \mathbb{F}_2[\tau]/(\tau^2)$: it has two generators $\iota$ at degree $(0, 0)$ and $\tau\iota$ at degree $(0, 1)$, with $\tau \cdot \iota = \tau\iota$ and all Steenrod squares acting as zero.

The Gheorghe–Wang–Xu theorem

Theorem (Gheorghe–Wang–Xu, 2018). There is an isomorphism of trigraded $\mathbb{F}_2$-algebras:

$$\operatorname{Ext}{A{\mathrm{mot}}}^{s,t,w}(C\tau,; \mathbb{F}2) ;\cong; \operatorname{Ext}{BP_BP}^{s, 2(t-s)-w}(BP_,; BP_*)$$

Moreover, the $C\tau$ Adams spectral sequence collapses at $E_2$: all differentials are zero.

This is a remarkable result: it identifies the Ext of a simple motivic module ($C\tau$, a two-cell complex) with the Adams–Novikov $E_2$ page, which is one of the deepest objects in stable homotopy theory.

Consequence for htpy. Since the $C\tau$ spectral sequence collapses, every generator of $\operatorname{Ext}(C\tau)$ is a permanent cycle. This is used directly by Pass B of the deduction engine.

The long exact sequence in Ext

Applying $\operatorname{Ext}{A{\mathrm{mot}}}(-, \mathbb{F}_2)$ to the short exact sequence above gives a long exact sequence:

$$\cdots \to \operatorname{Ext}^s(\mathbb{F}_2) \xrightarrow{\pi^} \operatorname{Ext}^s(C\tau) \xrightarrow{i^} \operatorname{Ext}^s(\Sigma^{0,1}\mathbb{F}_2) \xrightarrow{;\delta;} \operatorname{Ext}^{s+1}(\mathbb{F}_2) \to \cdots$$

where all Ext groups are taken over $A_{\mathrm{mot}}$.

Detecting $\tau$-torsion

The connecting homomorphism $\delta$ detects $\tau$-torsion:

$$x \in \operatorname{im}(\delta_{s-1}) \iff x \text{ is } \tau\text{-torsion in } \operatorname{Ext}^s(\mathbb{F}_2)$$

This follows from the exact sequence: $x \in \ker(\pi^)$ if and only if $\tau \cdot x = 0$ (since $\pi$ is the quotient by $\tau$), and $\ker(\pi^) = \operatorname{im}(\delta)$ by exactness.

Implementation. htpy resolves $\mathbb{F}2$, $C\tau$, and $\Sigma^{0,1}\mathbb{F}2$ over $A{\mathrm{mot}}$, constructs chain maps lifting $i$ and $\pi$, and computes the connecting homomorphism $\delta$ via the chain-level snake lemma. The rank of $\delta{s-1}$ at each tridegree $(s, t, w)$ gives the number of $\tau$-torsion generators.

Computing higher torsion orders

For $\tau$-torsion order (not just detection), htpy uses the iterated LES. Define $C_k = \mathbb{F}_2[\tau]/(\tau^k)$, so $C_2 = C\tau$. For each $k \geq 2$, there is a short exact sequence:

$$0 \to \Sigma^{0,1}C_{k-1} \xrightarrow{;i_k;} C_k \xrightarrow{;\pi_k;} \mathbb{F}_2 \to 0$$

The connecting homomorphism $\delta_k$ of this sequence detects $\tau^{k-1}$-torsion:

$$T_{k-1} := \operatorname{im}(\delta_k) = {x \in \operatorname{Ext}(\mathbb{F}_2) : \tau^{k-1} \cdot x = 0}$$

The exact torsion order of $x$ is

$$\operatorname{ord}(x) = \min{m \geq 1 : x \in T_m}$$

with $\operatorname{ord}(x) = 0$ if $x$ is not $\tau$-torsion.

Early termination. The iterated LES terminates when an iteration produces no new torsion globally. In practice, for the sphere at $p = 2$:

• Order 1 torsion first appears at stem 1 ($h_1$)
• Order 2 torsion first appears at stem 40
• Orders 3 and 4 appear at still higher stems

Chain map computation

htpy constructs the chain maps $\tilde{i}$ and $\tilde{\pi}$ lifting the module maps $i$ and $\pi$ to the resolution level. At homological degree $s = 0$, the chain maps are given directly by the module maps. At $s > 0$, the chain maps are extended inductively using the lift_generators_via_qi routine, which lifts generators through the quasi-inverse of the resolution differential.

The connecting homomorphism $\delta$ is then computed by the standard construction: given $x \in \operatorname{Ext}^s(\Sigma^{0,1}\mathbb{F}_2)$, lift to a chain in the $C\tau$ resolution via $\tilde{i}^{-1}$ (the chain-level homotopy), apply $d$, and project via $\tilde{\pi}$ to get $\delta(x) \in \operatorname{Ext}^{s+1}(\mathbb{F}_2)$.

Validation

htpy cross-validates the $C\tau$ computation in several ways:

• $d^2 = 0$: verified through $s \leq 6$, $t \leq 15$ for the motivic resolution of $C\tau$.
• GWX isomorphism: $C\tau$ Ext dimensions match the algebraic Novikov $E_2$ page from Isaksen’s CSV through $s \leq 8$, stem $\leq 20$.
• $\tau$-torsion counts: $\operatorname{rank}(\delta_{s-1})$ matches Isaksen’s CSV $\tau$-torsion counts at all 86 tridegrees tested ($s \leq 8$, stem $\leq 30$).
• $\pi^ \circ \tilde{i} = 0$*: the composition of chain maps from the short exact sequence is verified to be zero.

The archive modules

• S_2_mot: The motivic Adams $E_2$ page with $\tau$-torsion orders, computed via the iterated LES and supplemented by Isaksen’s CSV at higher stems.
• S_2_ctau: The $\operatorname{Ext}(C\tau)$ module, with generator names matched from the algebraic Novikov CSV via the GWX isomorphism.
• S_2_nov: The algebraic Novikov $E_2$ page, imported from Isaksen’s CSV data for comparison.

References

• B. Gheorghe, G. Wang, and Z. Xu, The special fiber of the motivic deformation of the stable homotopy category is algebraic, Acta Math. 226 (2021), no. 2, 319–407.
• D. Chua, The $E_3$ page of the Adams spectral sequence, 2021.
• D.C. Isaksen, Stable stems, Mem. Amer. Math. Soc. 262 (2019), no. 1269.