The goal of this project was to write a computer program that would aid in the search for relations among the Eilenberg-Zilber maps, which relate to simplicial objects in algebraic topology. This presentation outlines the process of writing this program, the challenges faced along the way, and the final results of the project

Abma, Owen T

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Searching for New Relations among the Eilenberg-Zilber mapsOwen Abma | USRI Summer 2021MotivationThe primary goal of this research project was to write a computer program that would aid in the discovery of new mathematical relations among the Eilenberg-Zilber maps.The Eilenberg-Zilber MapsThe Eilenberg-Zilber Maps relate the chain complex of the product of two simplicial sets with the the tensor product of the chain complexes.The maps interact with one another as shown on this diagram, with some known relations below.From Szczarba's twisting cochain and the Eilenberg-Zilber maps, by M. FranzThe Eilenberg-Zilber MapsAs more factors are added, these diagrams get more complex.(this diagram doesn’t even include the swap maps 𝛵 and 𝜏 - the full sized diagram would have 6 of these quadrilateral structures)The ProgramThe programming language being used for this project is Julia 1.6.1, a young yet powerful language that works very well in mathematical applications.Julia has similar syntax to Python, which made it relatively easy to pick up and start using in a short amount of time.The ProgramThe first version of the program was used to solve for new relations in two factors.However, since solving for relations in two factors requires very little computation, it was unlikely that I would discover anything new.The main point of this first version was to create a framework for how a larger program (for more than two factors) would workThe Program - Main IdeaThe process of this program is as follows:1. Generate all legal, nontrivial permutations of maps2. Sort these permutations by domain, target, and dimension3. For each category, put the resulting simplices into a matrix, and using basic linear algebra, solve for all linear dependenciesThe Program - Step by StepStep 1: Generate all legal, nontrivial permutations of mapsThe “path” of a simplex (maps that have already been applied and their order) and the space it is in both affect which maps are allowed to be applied.The Program - Step by StepStep 1: Generate all legal, nontrivial permutations of mapsThe program was written to avoid discovering a) Trivial relations (e.g. swapping twice is the identity) b) Illegal relations (e.g. AW cannot be applied twice in a row)c) Known properties (e.g. applying H twice yields zero)The Program - Step by StepStep 2: Sort these permutations by domain, target, and dimensionBy sorting these permutations of maps by:a) Domain (initial space)b) Target (final space)c) Dimension (number of times H is applied)The calculations in step 3 were much easier for Julia to handle.The Program - Step by StepStep 3: For each category, put the resulting simplices into a matrix, and using basic linear algebra, solve for all linear dependenciesThis was initially performed with a linear algebra package that I had coded myself.If the program found a linear dependency, it was able to return the relation that was implied by this dependency.The Program - Moving UpThough no new relations were discovered in 2 factors, the program worked as it was supposed to, so it was scaled up to do the same process with three factors.The Program - Moving UpWorking in 3 factors is much more complex and took a significantly larger amount of workAs you can see, this process required many drafts until it was finally right.ChallengesThe largest challenges that came with this project were:1. Avoiding trivial relations2. Locating and fixing inefficient code3. Lack of computing powerChallengesAvoiding trivial relationsDue to the complex definition of the Eilenberg-Zilber maps, it was very difficult to completely avoid trivial consequences, and often required a lot of tweaking to fix.ChallengesLocating and fixing inefficient codeAlthough I was able to quickly pick up Julia’s basic syntax, there were still many instances where I was writing inefficient code, resulting in a very slow program.Over the course of the summer, I was able to improve greatly and quickly recognize any poor programming errors made.ChallengesLack of computing powerUnfortunately since I am using a laptop for this project, I am limited by the amount of power and memory it contains.This became an issue towards the end of the summer, when the length of the permutations that I could generate developed a cap around 7.ResultsOver the course of the summer, I have discovered two new relations, shown below.(H𝑌×𝑍,𝑋 ∘ 𝜏𝑋,𝑍 ∘ 𝜏𝑋,𝑌 ∘ H𝑋×𝑌,𝑍 ∘ H𝑋,𝑌×𝑍) + (H𝑌×𝑍,𝑋 ∘ H𝑌,𝑍×𝑋 ∘ 𝜏𝑋,𝑍 ∘ 𝜏𝑋,𝑌 ∘ H𝑋×𝑌,𝑍)=(H𝑌,𝑍×𝑋 ∘ 𝜏𝑋,𝑍 ∘ 𝜏𝑋,𝑌 ∘ 𝐻𝑋×𝑌,𝑍 ∘ H𝑋,𝑌×𝑍) + (H𝑌×𝑍,𝑋 ∘ H𝑌,𝑍×𝑋 ∘ 𝜏𝑋,𝑍 ∘ 𝜏𝑋,𝑌 ∘ H𝑋,𝑌×𝑍)AW𝑌×𝑍,𝑋 ∘ 𝜏𝑋,𝑍 ∘ 𝜏𝑋,𝑌 ∘ H𝑋×𝑌,𝑍 ∘ H𝑋,𝑌×𝑍=(AW𝑌×𝑍,𝑋 ∘ 𝜏𝑋,𝑍 ∘ H𝑌×𝑋,𝑍 ∘ H𝑌,𝑋×𝑍 ∘ 𝜏𝑋,𝑌) + (AW𝑌×𝑍,𝑋 ∘ H𝑌,𝑍×𝑋 ∘ 𝜏𝑋,𝑍 ∘ 𝜏𝑋,𝑌 ∘ H𝑋,𝑌×𝑍)From Szczarba's twisting cochain and the Eilenberg-Zilber maps, by M. FranzAs you can see, these are much more complex than some of the known relations.ConclusionThis project was a fantastic opportunity for me to learn about programming and high-level mathematics and I am satisfied with my results, though I do believe that there are more relations that can be found with enough time and computing power.Thank you for reading/listening!

Searching for New Relations among the Eilenberg-Zilber maps

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Searching for New Relations among the Eilenberg-Zilber maps

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