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Written by Charlotte Aten on 2024-09-20 at 18:08

I posted a new paper on the arXiv!

https://arxiv.org/abs/2409.12923

In "Higher-dimensional book-spaces" I show that for each (n) there exists an (n)-dimensional compact simplicial complex which is a topological modular lattice but cannot be endowed with the structure of topological distributive lattice. This extends a result of Walter Taylor, who did the (2)-dimensional case.

I think this kind of result is interesting because we can see that whether spaces continuously model certain equations is a true topological invariant. All of the spaces that I discuss here are contractible, but only some can have a distributive lattice structure.

A similar phenomenon happens with H-spaces. The (7)-sphere is an H-space, and it is even a topological Moufang loop, but it cannot be made into a topological group, even though our homotopical tools tell us that it "looks like a topological group".

This is (a cleaned up version of) something I did during my second year of graduate school. It only took me about six years to post it.

[#]math #topology #algebra #AbstractAlgebra #UniversalAlgebra #combinatorics #LatticeTheory

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