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+++ title = “Acknowledgements” section = “root” +++

Acknowledgements

htpy builds on a substantial body of prior work in both software and mathematics. This page records the contributions that have made the project possible.

Software

htpy’s computation engine descends from two open-source projects:

ext-rs — Hood Chatham, Dexter Chua, and Joey Beauvais-Feisthauer. The Rust library ext-rs (MIT / Apache-2.0) implements minimal free resolutions over the Steenrod algebra with Yoneda and Massey products, secondary operations, and $\mathbb{F}_p$ linear algebra. htpy’s crates htpy-fp (vector and matrix arithmetic, M4RI row reduction), htpy-algebra (Adem and Milnor basis implementations, module types, free module homomorphisms), and htpy-resolution (the classical resolution engine, chain map lifting, Yoneda products, Massey products, secondary operations) are direct descendants of ext-rs code. The augmented-matrix resolution step, the quasi-inverse machinery, and the $\mathbb{F}_2$ bit-packed vector representation all originate in ext-rs.

SSeqCpp — Weinan Lin. The C++ library SSeqCpp (Apache-2.0) implements spectral sequence computation using Groebner bases over polynomial rings. htpy-groebner is a Rust port of SSeqCpp’s Buchberger algorithm, including the fixed-size monomial representation, the divisibility trie for leading-term reduction, and the critical-pair management with Buchberger criteria.

Reference data

htpy validates its computations against several published datasets:

Daniel C. Isaksen — Motivic and classical Adams spectral sequence charts. CSV data for the $E_2$ through $E_\infty$ pages of the classical Adams, motivic Adams, and algebraic Novikov spectral sequences. Source: Zenodo 6987157 and Zenodo 6987227. These datasets are the primary reference for cross-validating htpy’s motivic resolution and $C\tau$ computation.

Joey Beauvais-Feisthauer, Hood Chatham, and Dexter Chua — The $E_2$ page of the 2-primary Adams spectral sequence in a large range. Source: Zenodo 7339848 (CC BY 4.0). Provides $\operatorname{Ext}_A$ ranks through stem 256, $d_2$ differentials through stem 200, and all Yoneda products with $h_0$ through $h_6$.

Robert Bruner and John Rognes — The cohomology of the mod 2 Steenrod algebra (arXiv:2109.13117). SQLite database with the full Ext computation including named generators, basis elements, and relations. Source: Zenodo 7786290.

Weinan Lin, Guozhen Wang, and Zhouli Xu — Machine proofs for Adams differentials and extension problems among CW spectra (arXiv:2412.10876). Machine-proved Adams differentials and complete spectral sequence data for 220 CW spectra. Source: Zenodo 14475507 (CC BY 4.0).

Mathematics

The algorithms and theoretical foundations draw on work by many people. The following contributions are directly used in htpy’s code:

Minimal resolutions and Ext computation. The resolution algorithm (row reduce, extend to surjection, compute quasi-inverse) is standard homological algebra. The augmented-matrix formulation with chain-map/differential/identity segments was developed in ext-rs.

The Steenrod algebra. The Adem basis implementation follows the classical construction via admissible sequences. The Milnor basis implementation uses the coproduct formulas of J.W. Milnor (1958). Both support arbitrary primes $p$.

M4RI row reduction. The Method of Four Russians for $\mathbb{F}_2$ matrix operations, as described by Albrecht, Bard, and Hart. htpy’s implementation includes SIMD vectorization for x86-64 (AVX2/AVX-512) and aarch64 (NEON).

The motivic Steenrod algebra. The motivic Adem algebra and its bigraded structure follow Voevodsky’s construction, with conventions as in Dugger–Isaksen.

The Gheorghe–Wang–Xu theorem. The identification $\operatorname{Ext}{A{\mathrm{mot}}}(C\tau, \mathbb{F}2) \cong \operatorname{Ext}{BP_BP}(BP_, BP_*)$ and the collapse of the $C\tau$ Adams spectral sequence at $E_2$ are due to Bogdan Gheorghe, Guozhen Wang, and Zhouli Xu (Acta Math. 226, 2021).

$\tau$-torsion and differentials. The connection between $\tau$-torsion order and Adams differentials — specifically, that $\tau^k$-torsion classes lie in $\operatorname{im}(d_{k+1})$ — is Lemma 12.4 of Dexter Chua’s paper “The $E_3$ page of the Adams spectral sequence” (2021, arXiv:2105.07628). This result powers Passes 1.3 and 3 of the deduction engine.

Massey products and Moss’ theorem. The deduction engine’s use of $E_r$ Massey brackets follows the framework of Eva Belmont and Hana Kong (“An algebraic approach to the Adams spectral sequence”), specifically Definitions 2.4 and 3.11 and Theorem 3.13.

Secondary Steenrod operations. The $d_2$ computation via pair algebras follows Hans-Joachim Baues (“The algebra of secondary cohomology operations”) and Christian Nassau (“On the secondary Steenrod algebra”).

Vanishing lines. The propagation of vanishing lines through cofiber sequences uses Lemma 6.3 of Brian Johnson (“Quantitative Bounds for Nilpotence and Vanishing Lines”).

The Cartan–Eilenberg spectral sequence. The CESS and algebraic Novikov spectral sequence setup follows Douglas Ravenel (“Complex Cobordism and Stable Homotopy Groups of Spheres”, 2003), particularly the treatment of the $P_*$-comodule Ext computation.

Classical and motivic computations. The computation of stable stems via the motivic approach is due to Daniel Isaksen, Guozhen Wang, and Zhouli Xu (“Stable homotopy groups of spheres”, Proc. Natl. Acad. Sci. 117, 2020). The motivic Adams charts and naming conventions follow Isaksen (“Stable stems”, Mem. Amer. Math. Soc. 262, 2019).

The Kervaire invariant problem. The CW-spectrum data and machine-proved differentials used for cross-validation are from Weinan Lin, Guozhen Wang, and Zhouli Xu (“On the Last Kervaire Invariant Problem”, 2024, arXiv:2412.10879).

Textbooks

The following textbooks inform the mathematical background throughout:

  • J.F. Adams, Stable Homotopy and Generalised Homology, Chicago Lectures in Mathematics, 1974.
  • J.P. May, A General Algebraic Approach to Steenrod Operations, Lecture Notes in Mathematics 168, Springer, 1970.
  • D.C. Ravenel, Complex Cobordism and Stable Homotopy Groups of Spheres, 2nd edition, AMS Chelsea Publishing, 2003.