Formalizing the Proof that the Fourth Homotopy Group of the 3-Sphere is Z/2Z in Cubical Agda

核心概念

The fourth homotopy group of the 3-sphere, π4(S3), is isomorphic to Z/2Z.

要約

The paper presents a formalization in Cubical Agda of the proof that the fourth homotopy group of the 3-sphere, π4(S3), is isomorphic to Z/2Z. This result was first proved synthetically in Homotopy Type Theory by Brunerie in his 2016 PhD thesis.

The formalization closely follows Brunerie's proof, with some simplifications. The key steps are:

1. Introducing basic HoTT concepts and their formalization in Cubical Agda, including higher inductive types, loop spaces, and homotopy groups.

2. Formalizing the first part of Brunerie's proof, which culminates in the definition of the "Brunerie number" β such that π4(S3) ∼= Z/βZ. This involves the James construction and Whitehead products.

3. Formalizing the second part of Brunerie's proof, which shows that |β| ≡ 2, thereby establishing that π4(S3) ∼= Z/2Z. This part relies on advanced machinery like cohomology, the Hopf invariant, and Gysin sequences.

4. Presenting a new, simpler proof that β = ±2, which avoids the technical difficulties of Brunerie's original proof. This new proof manually computes the image of the generator η of π3(S2) under a sequence of maps, yielding a new "Brunerie number" that normalizes to -2 in Cubical Agda.

The formalization provides the first computer-assisted proof of the classical result that π4(S3) ∼= Z/2Z, and demonstrates the power of Homotopy Type Theory and Cubical Agda for formalizing advanced results in synthetic homotopy theory.

統計

None.

引用

None.

抽出されたキーインサイト

Formalising and Computing the Fourth Homotopy Group of the $3$-Sphere in Cubical Agda

by Axel... 場所 arxiv.org 05-01-2024

https://arxiv.org/pdf/2302.00151.pdf

深掘り質問

What other advanced results in homotopy theory could be formalized using Homotopy Type Theory and Cubical Agda

Homotopy Type Theory and Cubical Agda provide a powerful framework for formalizing advanced results in homotopy theory. Some other results that could be formalized using these tools include the Adams spectral sequence, the Serre spectral sequence, the EHP sequence, and the stable homotopy groups of spheres beyond the first few. These results are fundamental in algebraic topology and having computer-verified proofs of these theorems would greatly enhance the rigor and reliability of the field.

How could the techniques developed in this formalization be applied to simplify or automate the computation of higher homotopy groups of spheres

The techniques developed in the formalization of the Brunerie number could be applied to simplify and automate the computation of higher homotopy groups of spheres. By leveraging the power of Homotopy Type Theory and Cubical Agda, one could potentially streamline the process of computing these groups, reducing the risk of human error and providing a more efficient way to handle complex calculations. Additionally, the formalization could lead to the development of algorithms or tools that automate the computation of homotopy groups, making it more accessible to researchers and mathematicians.

What are the broader implications of having computer-verified proofs of results in synthetic homotopy theory, and how might this impact the field

Having computer-verified proofs of results in synthetic homotopy theory has several broader implications. Firstly, it enhances the reliability and trustworthiness of the results, as they are rigorously verified by a machine rather than relying solely on human proof-checking. This can lead to a higher level of confidence in the validity of the theorems and their applications in various mathematical contexts. Furthermore, computer-verified proofs can potentially uncover new insights or patterns in the data that may not be immediately apparent to human researchers, leading to new discoveries and advancements in the field. Overall, the impact of computer-verified proofs in synthetic homotopy theory could revolutionize the way we approach and understand complex mathematical problems.